A166507
Least n-comma number: smallest nonnegative integer that occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for at least n different splittings a(n)=concat(S[0],S[1]).
Original entry on oeis.org
0, 10, 1023, 2676, 16867, 111688, 1522828, 11386882, 112273999, 1212143716, 11232152998, 121321194596
Offset: 0
There are 0 ways to split a(0)=0 in two substrings, so this is the smallest 0-comma number.
The number a(1)=10 is the smallest 1-comma number, cf. A166511.
The number a(2)=1023 is the smallest 2-comma number: it occurs in S(10,23) and in S(102,3), cf. A166512.
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A166507(k) = { my(a,b,c); for( n=10^k\9*10,1e9, c=k; n%100 | next; for(d=1,#Str(n)-1, d+c>#Str(n) & break /* not possible: next n */; a=n\10^d, b=n%10^d; b<10^(d-1) & d>1 & next /* not legal: next d */; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b>n & next; c-- | return(n)))}
A166508
Hypercomma numbers: n occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for each "legal" splitting n=concat(S[0],S[1]).
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 109, 806, 1023, 1044, 2005, 2676, 3066, 3602, 4051, 6053, 6246, 8011, 8349, 9427, 10022, 10074, 10587, 13090, 15031, 16867, 20088, 20699, 21698, 23108, 29986, 30091, 30306, 32226, 40022
Offset: 1
There is no legal way to split the single-digit numbers 1,...,9, therefore they are included.
More generally, a k-comma number which has exactly k nonzero digits when the last digit is ignored, will be in this sequence: e.g., 2005 can only be cut as (200,5); 10022 can only be cut as (1002,2) and (100,22), and it is a 2-comma number (A166512).
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{for(n=1,1e5,/*is_A166508(n)=*/ n%100 & for(d=1,#Str(n)-1, my( a=n\10^d, b=n%10^d ); b<10^(d-1) & d>1 & next /* not legal */; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b>n & next(2) /* bad */); print1(n", "))}
A166511
Comma numbers: n=concat(a,b) occurs in the sequence S[0]=a, S[1]=b, S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]).
Original entry on oeis.org
10, 20, 30, 40, 50, 60, 70, 80, 90, 109, 123, 148, 160, 163, 185, 198, 199, 241, 340, 362, 398, 422, 423, 444, 522, 540, 541, 550, 564, 597, 621, 622, 667, 683, 693, 724, 769, 770, 780, 806, 811, 835, 842, 888, 972, 1023, 1044, 1053, 1054, 1116, 1146, 1177
Offset: 1
a(1)=10 is in the sequence, because splitting 10 into (1,0) gives the sequence S(1,0) = 1, 0, 0+10=10, 10+01=11, 11+01=12, 12+11=23,... which contains 10.
a(10)=109 is in the sequence, because splitting 109 into (10,9) gives the sequence S(10,9) = 10, 9, 9+09=18, 18+91=109, 109+81=190,... which contains 109.
108 is not in the sequence since S(10,8) does not contain 108 and it is not allowed to split 108 -> 1,08 -> S(1,08).
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{for(n=1,1e4,/*iscomma(n)=*/for(d=1,#Str(n)-1, my( a=n\10^d, b=n%10^d ); b<10^(d-1) & (d>1 || a%10==0) & next; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b>n & next; /*return(1)*/ print1(n", "); break))}
A166512
2-comma numbers: n occurs in the sequence S[k+1] = S[k] + 10*last_digit(S[k-1]) + first_digit(S[k]) for two different splittings n=concat(S[0],S[1]).
Original entry on oeis.org
1023, 1044, 1521, 1657, 1789, 1984, 2191, 2263, 2451, 2466, 2523, 2676, 2783, 2824, 3066, 3268, 3589, 3602, 3631, 4051, 4113, 4149, 4159, 4213, 4315, 4611, 4685, 4781, 4969, 5133, 5526, 6053, 6165, 6246, 6445, 6650, 6712, 6893, 7350, 7668, 8011, 8144
Offset: 1
None of the 3-digit terms in A166511 can be split up in 2 ways such that S(a,bc) and S(ab,c) both contain n=abc (concatenation, not product).
Therefore the smallest term in this sequence is a(1)=1023, which occurs in the sequences S(102,3) and S(10,23).
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{for(n=1,1e4,/*is_A166512(n)=*/ my(c=2); for(d=1,#Str(n)-1, my( a=n\10^d, b=n%10^d ); b<10^(d-1) & (d>1 || a%10==0) & next; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b==n & c--==0 & /*return(1)*/ !print1(n", ") & break))}
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