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!

https://www.youtube.com/watch?v=McIfIx29tSg

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:

____________________