A concatenation using binary strings

 
(Dall-e creation)

To my surprise, S doesn't seem to be in the OEIS. Each term of S is the concatenation of n and its binary expansion.
Example: we form 971100001 by concatenating the decimal number 97 to its binary expansion 1100001.
S was simply computed using the first 666 lines of this table (not counting the 0 0 line).
We could call such numbers “Enbi numbers” [n + bi(nary); a few questions about Enbi numbers after the hereunder S.]

S = 11, 210, 311, 4100, 5101, 6110, 7111, 81000, 91001, 101010, 111011, 121100, 131101, 141110, 151111, 1610000, 1710001, 1810010, 1910011, 2010100, 2110101, 2210110, 2310111, 2411000, 2511001, 2611010, 2711011, 2811100, 2911101, 3011110, 3111111, 32100000, 33100001, 34100010, 35100011, 36100100, 37100101, 38100110, 39100111, 40101000, 41101001, 42101010, 43101011, 44101100, 45101101, 46101110, 47101111, 48110000, 49110001, 50110010, 51110011, 52110100, 53110101, 54110110, 55110111, 56111000, 57111001, 58111010, 59111011, 60111100, 61111101, 62111110, 63111111, 641000000, 651000001, 661000010, 671000011, 681000100, 691000101, 701000110, 711000111, 721001000, 731001001, 741001010, 751001011, 761001100, 771001101, 781001110, 791001111, 801010000, 811010001, 821010010, 831010011, 841010100, 851010101, 861010110, 871010111, 881011000, 891011001, 901011010, 911011011, 921011100, 931011101, 941011110, 951011111, 961100000, 971100001, 981100010, 991100011, 1001100100, 1011100101, 1021100110, 1031100111, 1041101000, 1051101001, 1061101010, 1071101011, 1081101100, 1091101101, 1101101110, 1111101111, 1121110000, 1131110001, 1141110010, 1151110011, 1161110100, 1171110101, 1181110110, 1191110111, 1201111000, 1211111001, 1221111010, 1231111011, 1241111100, 1251111101, 1261111110, 1271111111, 12810000000, 12910000001, 13010000010, 13110000011, 13210000100, 13310000101, 13410000110, 13510000111, 13610001000, 13710001001, 13810001010, 13910001011, 14010001100, 14110001101, 14210001110, 14310001111, 14410010000, 14510010001, 14610010010, 14710010011, 14810010100, 14910010101, 15010010110, 15110010111, 15210011000, 15310011001, 15410011010, 15510011011, 15610011100, 15710011101, 15810011110, 15910011111, 16010100000, 16110100001, 16210100010, 16310100011, 16410100100, 16510100101, 16610100110, 16710100111, 16810101000, 16910101001, 17010101010, 17110101011, 17210101100, 17310101101, 17410101110, 17510101111, 17610110000, 17710110001, 17810110010, 17910110011, 18010110100, 18110110101, 18210110110, 18310110111, 18410111000, 18510111001, 18610111010, 18710111011, 18810111100, 18910111101, 19010111110, 19110111111, 19211000000, 19311000001, 19411000010, 19511000011, 19611000100, 19711000101, 19811000110, 19911000111, 20011001000, 20111001001, 20211001010, 20311001011, 20411001100, 20511001101, 20611001110, 20711001111, 20811010000, 20911010001, 21011010010, 21111010011, 21211010100, 21311010101, 21411010110, 21511010111, 21611011000, 21711011001, 21811011010, 21911011011, 22011011100, 22111011101, 22211011110, 22311011111, 22411100000, 22511100001, 22611100010, 22711100011, 22811100100, 22911100101, 23011100110, 23111100111, 23211101000, 23311101001, 23411101010, 23511101011, 23611101100, 23711101101, 23811101110, 23911101111, 24011110000, 24111110001, 24211110010, 24311110011, 24411110100, 24511110101, 24611110110, 24711110111, 24811111000, 24911111001, 25011111010, 25111111011, 25211111100, 25311111101, 25411111110, 25511111111, 256100000000, 257100000001, 258100000010, 259100000011, 260100000100, 261100000101, 262100000110, 263100000111, 264100001000, 265100001001, 266100001010, 267100001011, 268100001100, 269100001101, 270100001110, 271100001111, 272100010000, 273100010001, 274100010010, 275100010011, 276100010100, 277100010101, 278100010110, 279100010111, 280100011000, 281100011001, 282100011010, 283100011011, 284100011100, 285100011101, 286100011110, 287100011111, 288100100000, 289100100001, 290100100010, 291100100011, 292100100100, 293100100101, 294100100110, 295100100111, 296100101000, 297100101001, 298100101010, 299100101011, 300100101100, 301100101101, 302100101110, 303100101111, 304100110000, 305100110001, 306100110010, 307100110011, 308100110100, 309100110101, 310100110110, 311100110111, 312100111000, 313100111001, 314100111010, 315100111011, 316100111100, 317100111101, 318100111110, 319100111111, 320101000000, 321101000001, 322101000010, 323101000011, 324101000100, 325101000101, 326101000110, 327101000111, 328101001000, 329101001001, 330101001010, 331101001011, 332101001100, 333101001101, 334101001110, 335101001111, 336101010000, 337101010001, 338101010010, 339101010011, 340101010100, 341101010101, 342101010110, 343101010111, 344101011000, 345101011001, 346101011010, 347101011011, 348101011100, 349101011101, 350101011110, 351101011111, 352101100000, 353101100001, 354101100010, 355101100011, 356101100100, 357101100101, 358101100110, 359101100111, 360101101000, 361101101001, 362101101010, 363101101011, 364101101100, 365101101101, 366101101110, 367101101111, 368101110000, 369101110001, 370101110010, 371101110011, 372101110100, 373101110101, 374101110110, 375101110111, 376101111000, 377101111001, 378101111010, 379101111011, 380101111100, 381101111101, 382101111110, 383101111111, 384110000000, 385110000001, 386110000010, 387110000011, 388110000100, 389110000101, 390110000110, 391110000111, 392110001000, 393110001001, 394110001010, 395110001011, 396110001100, 397110001101, 398110001110, 399110001111, 400110010000, 401110010001, 402110010010, 403110010011, 404110010100, 405110010101, 406110010110, 407110010111, 408110011000, 409110011001, 410110011010, 411110011011, 412110011100, 413110011101, 414110011110, 415110011111, 416110100000, 417110100001, 418110100010, 419110100011, 420110100100, 421110100101, 422110100110, 423110100111, 424110101000, 425110101001, 426110101010, 427110101011, 428110101100, 429110101101, 430110101110, 431110101111, 432110110000, 433110110001, 434110110010, 435110110011, 436110110100, 437110110101, 438110110110, 439110110111, 440110111000, 441110111001, 442110111010, 443110111011, 444110111100, 445110111101, 446110111110, 447110111111, 448111000000, 449111000001, 450111000010, 451111000011, 452111000100, 453111000101, 454111000110, 455111000111, 456111001000, 457111001001, 458111001010, 459111001011, 460111001100, 461111001101, 462111001110, 463111001111, 464111010000, 465111010001, 466111010010, 467111010011, 468111010100, 469111010101, 470111010110, 471111010111, 472111011000, 473111011001, 474111011010, 475111011011, 476111011100, 477111011101, 478111011110, 479111011111, 480111100000, 481111100001, 482111100010, 483111100011, 484111100100, 485111100101, 486111100110, 487111100111, 488111101000, 489111101001, 490111101010, 491111101011, 492111101100, 493111101101, 494111101110, 495111101111, 496111110000, 497111110001, 498111110010, 499111110011, 500111110100, 501111110101, 502111110110, 503111110111, 504111111000, 505111111001, 506111111010, 507111111011, 508111111100, 509111111101, 510111111110, 511111111111, 5121000000000, 5131000000001, 5141000000010, 5151000000011, 5161000000100, 5171000000101, 5181000000110, 5191000000111, 5201000001000, 5211000001001, 5221000001010, 5231000001011, 5241000001100, 5251000001101, 5261000001110, 5271000001111, 5281000010000, 5291000010001, 5301000010010, 5311000010011, 5321000010100, 5331000010101, 5341000010110, 5351000010111, 5361000011000, 5371000011001, 5381000011010, 5391000011011, 5401000011100, 5411000011101, 5421000011110, 5431000011111, 5441000100000, 5451000100001, 5461000100010, 5471000100011, 5481000100100, 5491000100101, 5501000100110, 5511000100111, 5521000101000, 5531000101001, 5541000101010, 5551000101011, 5561000101100, 5571000101101, 5581000101110, 5591000101111, 5601000110000, 5611000110001, 5621000110010, 5631000110011, 5641000110100, 5651000110101, 5661000110110, 5671000110111, 5681000111000, 5691000111001, 5701000111010, 5711000111011, 5721000111100, 5731000111101, 5741000111110, 5751000111111, 5761001000000, 5771001000001, 5781001000010, 5791001000011, 5801001000100, 5811001000101, 5821001000110, 5831001000111, 5841001001000, 5851001001001, 5861001001010, 5871001001011, 5881001001100, 5891001001101, 5901001001110, 5911001001111, 5921001010000, 5931001010001, 5941001010010, 5951001010011, 5961001010100, 5971001010101, 5981001010110, 5991001010111, 6001001011000, 6011001011001, 6021001011010, 6031001011011, 6041001011100, 6051001011101, 6061001011110, 6071001011111, 6081001100000, 6091001100001, 6101001100010, 6111001100011, 6121001100100, 6131001100101, 6141001100110, 6151001100111, 6161001101000, 6171001101001, 6181001101010, 6191001101011, 6201001101100, 6211001101101, 6221001101110, 6231001101111, 6241001110000, 6251001110001, 6261001110010, 6271001110011, 6281001110100, 6291001110101, 6301001110110, 6311001110111, 6321001111000, 6331001111001, 6341001111010, 6351001111011, 6361001111100, 6371001111101, 6381001111110, 6391001111111, 6401010000000, 6411010000001, 6421010000010, 6431010000011, 6441010000100, 6451010000101, 6461010000110, 6471010000111, 6481010001000, 6491010001001, 6501010001010, 6511010001011, 6521010001100, 6531010001101, 6541010001110, 6551010001111, 6561010010000, 6571010010001, 6581010010010, 6591010010011, 6601010010100, 6611010010101, 6621010010110, 6631010010111, 6641010011000, 6651010011001, 6661010011010. 

Questions
—>What Enbi numbers are prime? We have seen that 971100001 is one of them – but they are infinitely many, I guess, and their list T starts like this:
T = 11, 311, 5101, 131101, ...
T is already in the OEIS!
—>What Enbi numbers are divisible by n? 11, of course, and 210, but also 4100, 81000, 101010, ... the list is also infinite, starting like that, I guess:
U = 11, 210, 4100, 81000, 101010, 1610000, 2010100, ...
—>Some Enbi numbers are made of 0s and 1s only. When converted to decimal, what do such strings produce (n in yellow, as usual)?
11 = 3
101010 = 42
111011 = 59
1001100100 = 612
1011100101 = 741
10001111101000 = 9192
10011111101001 = 10217
10101111110010 = 11250
10111111110011 = 12275
110010001001100 = 25676
110110001001101 = 27725
111010001010110 = 29782
111110001010111 = 31831
...
If the above table is correct, the sequence V of such numbers should start like this:
V = 3, 42, 59, 612, 741, 9192, 10217, 11250, 12275, 25676, 27725, 29782, 31831, ...
—>Enbi numbers divisible by 2, by 4 or 5 are easy to spot, but what about Enbi numbers divisible by 3, or 6, or 7, or 8, or 9?

(more to come – dinner time now in Brussels – then tennis time in Miami)
Post-Miami-single-men-tennis-2024-final update (congrats to Sinner)

Giorgos Kalogeropoulos was quick to send this:
> Enbi numbers divisible by n
U = 11, 210, 4100, 81000, 101010, 1610000, 2010100, 2110101, 32100000, 40101000, 42101010, 641000000, 801010000, 841010100, 1001100100, 12810000000, 16010100000, 16810101000, 20011001000, 256100000000, 273100010001, 320101000000, 336101010000, 400110010000, 5121000000000, 5461000100010, 6401010000000, 6721010100000, 8001100100000, 10001111101000, 102410000000000, 109210001000100, 128010100000000, 134410101000000, 160011001000000, 200011111010000, 2048100000000000, 2184100010001000, 2231100010110111, 2510100111001110, 2560101000000000, 2688101010000000, 2730101010101010, 3200110010000000, 3300110011100100, 4000111110100000, 40961000000000000, 43681000100010000, 44621000101101110, 50201001110011100, 51201010000000000, 53761010100000000, 54601010101010100, 64001100100000000, 66001100111001000, 80001111101000000, 819210000000000000, 873610001000100000, 892410001011011100, 1000010011100010000

> Binary Enbi numbers converted to decimal numbers
V = 3, 42, 59, 612, 741, 878, 1007, 9192, 10217, 11250, 12275, 25676, 27725, 29782, 31831, 272144, 288529, 304922, 321307, 337780, 354165, 370558, 386943, 404216, 420601, 436994, 453379, 469852, 486237, 502630, 519015, 4294304, 4425377, 4556458, 4687531, 4818692, 4949765, 5080846, 5211919, 5343880, 5474953, 5606034, 5737107, 5868268, 5999341, 6130422, 6261495, 6401456, 6532529, 6663610, 6794683, 6925844, 7056917, 7187998, 7319071, 7451032, 7582105, 7713186, 7844259, 7975420, 8106493, 8237574, 8368647, 68108864, 69157441, 70206026, 71254603, 72303268, 73351845, 74400430, 75449007, 76498472, 77547049, 78595634, 79644211, 80692876, 81741453, 82790038, 83838615, 84896080, 85944657, 86993242, 88041819, 89090484, 90139061, 91187646, 92236223, 93285688, 94334265, 95382850, 96431427, 97480092, 98528669, 99577254, 100625831, 202426592, 204523745, 206620906, 208718059, 210815300

> Here are also the Enbi numbers mod k with k = 2, 3, 4, ... 16.
In the following compact image you can see how the Enbi numbers behave when we divide them by k = 2, 3, 4, ... 16.
They are plotted in a 100 x 100 array – a white pixel meaning that the corresponding Enbi number is divided by k:
Waooow, painting by numbers, my favorite part of sequence-computing! Bravo Giorgos, great stuff!

GK's last wonderful discovery:

> Here is a bonus prime number p that I found related to Enbi:

1) p = 10215364897 is prime
2) p is pandigital
3) Enbi of p 102153648971001100000111000100001100100100001 is prime
4) Reverse Enbi of p 100001001001100001000111000001100179846351201 is prime

—> p is the least integer with the above 4 properties!

Giorgos, this is for you:
____________________










Commentaires

  1. MFH5 avril 2024 à 19:17

    In some sense the sequence S is already in OEIS, since the terms are listed in https://oeis.org/A007088/b007088.txt (with the first digit(s) separated by a space, but we can ignore that).

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  2. MFH5 avril 2024 à 19:59

    10135847629 is the least pandigital prime whose reversal and "Enbi" are also prime.
    10264758937 is the least pandigital prime whose reversal, Enbi and reversed Enbi are prime. (Using digits 10123...9 the smallest permutation with this property is 10578413269, using the digit 2 twice it is 10542689237.)

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