A309393 Let f(n) be equal to n + S(n) + S(S(n)) ... + S(S(S..(n))), where the last term is less than 10 and S(n) is the sum of digits. This is the sequence of numbers k such that the equation f(x) = k has a record number of solutions.

1, 30, 66, 204, 819, 70032, 3000000000096

Offset

1,2

Comments

Conjecture from Daniel Starodubtsev and Dmitry Petukhov, Nov 19 2019: (Start)

a(8) = 20000000000000046;

a(9) = 8900000000000000000000127. (End)

Links

Table of n, a(n) for n=1..7.

Example

a(4) = 204, because 204 = f(179) = f(185) = f(191) = f(201), which has more solutions than any smaller number.

Mathematica

T = 0*Range[10^5]; f[n_] := Block[{x=n, s=n}, While[x >= 10, x = Plus@@ IntegerDigits[x]; s += x]; s]; Do[v = f[i]; If[v <= 10^5, T[[v]]++], {i, 10^5}]; Flatten[Position[T, #, 1, 1] & /@ Range[6]] (* Giovanni Resta, Jul 30 2019 *)

Prog

(PARI) f(n) = {s=n; m=n; while(sumdigits(s)>9, s=sumdigits(s); m+=s); if(n<10, m=0); m+sumdigits(s); }
g(n) = sum(k=1, n, f(k)==n);
lista(NN) = {x=1; print1(1); for(n=2, NN, if(g(n)>x, x=g(n); print1(", ", n)))} \\ Jinyuan Wang, Jul 31 2019

Crossrefs

Cf. A007953, A006064.

Sequence in context: A290145 A157346 A219543 * A154055 A270758 A241192

Adjacent sequences: A309390 A309391 A309392 * A309394 A309395 A309396

Keyword

nonn,base,more

Author

Daniel Starodubtsev, Jul 28 2019

Extensions

a(7) from Bert Dobbelaere, Aug 15 2019

Status

approved