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", "))}
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))}
A166513
3-comma numbers: n occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for three different splittings n=concat(S[0],S[1]).
Original entry on oeis.org
2676, 6246, 8349, 9427, 10587, 11558, 11756, 11811, 12427, 12788, 13090, 13110, 14328, 15031, 15187, 15493, 15637, 16867, 18322, 18768, 19918, 20699, 21138, 21422, 21698, 22824, 23108, 23242, 23868, 24456, 24854, 25342, 25478, 26583
Offset: 1
The 4-digit terms 2676, 6246, 8349, 9427 occurring in A166512, can be split up in any of the 3 possible ways such that S(a,bcd), S(ab,cd), and S(abc,d) all contain abcd (concatenation, not product). Therefore they are in this sequence, and they are even hypercomma (or "phoenix") numbers (A166508).
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{for(n=1e4,1e5,/*is_A166513(n)=*/ my(c=3); for(d=1,#Str(n)-1, d+c>#Str(n) & break; 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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