Sums in the right place
In the hereunder sequence S any pair of adjacent terms sums up to a number k in position k. This is the lexicographically earliest sequence of distinct positive integers with this property.
S=1,2,3,4,5,6,7,8,9,11,113,15,17,120,12,14,26,10,16,32,18,36,20,40,22,24,46,48,50,23,54,56,19,60,621,13,27,170,37,375,77,79,21,25,29,31,39,34,69,41,59,8100,103,38,35,110,42,28,45,1120,51,240,128,61,320,71,370,141,81,451,30,115,201,560,33,44,66,58,52,68,64,73,47,53,57,43,55,65,91,890,49,75,197,62,83,106,207,63,74,67,70,82,107,90,99,98,109,80,72,84,105,...
Examples (asterisks follow multiple solutions; in bold are sums out of range here) :
1+2=3
2+3=5
3+4=7
4+5=9
5+6=11
6+7=13
7+8=15
8+9=17
9+11=20
11+113=124*
113+15=128
15+17=32
17+120=137*
120+12=132*
12+14=26*
14+26=40*
26+10=36
10+16=26*
16+32=48
32+18=50
18+36=54*
36+20=56
20+40=60*
40+22=62
22+24=46*
46+48=94
48+50=98*
50+23=73*
23+54=77*
54+56=110*
56+19=75
19+60=79
60+621=681
621+13=634
13+27=40*
27+170=197*
170+37=207*
37+375=412
375+77=452
77+79=156*
79+21=100*
21+25=46*
25+29=54*
29+31=60*
31+39=70*
39+34=73*
34+69=103
69+41=110*
41+59=100*
59+8100=8159
8100+103=8203
103+38=141*
38+35=73*
35+110=145*
110+42=152*
42+28=70*
28+45=73*
45+1120=1165
1120+51=1171
51+240=291
240+128=368
128+61=189*
61+320=381
320+71=391
71+370=441
370+141=511
141+81=222
81+451=532
451+30=481
30+115=145*
115+201=316
201+560=761
560+33=593
33+44=77*
44+66=110*
66+58=124*
58+52=110*
52+68=120*
68+64=132*
64+73=137*
73+47=120*
47+53=100*
53+57=110*
57+43=100*
43+55=98*
55+65=120*
65+91=156*
91+890=981
890+49=939
49+75=124*
75+197=272
197+62=259
62+83=145*
83+106=189*
106+207=313
207+63=270
63+74=137*
74+67=141*
67+70=137*
70+82=152*
82+107=189*
107+90=197*
90+99=189*
99+98=197*
98+109=207*
109+80=189*
80+72=152*
72+84=156*
84+105=189*
….
Smallest absents yet: 76, 78, 85,…
I'm quite sure to have blundered somewhere: hope the first 30 terms (on 111) are correct!
Best,
É.
If we want to avoid depending on the base-10 representation, we might consider indices of terms instead of positions within the string of concatenated digits. But then, if a(n)+a(n+1)=m means that we need m at index m would just give the trivial sequence a(n) = n. But for any (injective?) function f we can consider the restriction: a(m) = f(m) --- or (alternatively -- or maybe in addition?): a(f(m)) = m. For example, f(m)=prime(m) (the m-th prime) gives two nontrivial "lexicographically earliest permutation of the positive integers such that a(m) = prime(m) (resp.: a(prime(m)) = m) for all m = a(n)+a(n+1).
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