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

Author

Daniel Starodubtsev, Jul 28 2019

Keywords

• nonn
• base
• more

Comments

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

a(8) = 20000000000000046;

a(9) = 8900000000000000000000127. (End)

Examples

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

Crossrefs

Cf. A007953, A006064.

Programs

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 *)

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

Extensions

a(7) from Bert Dobbelaere, Aug 15 2019