A125088 a(1)=1. a(n) = sum of the earlier terms equal to any exponent in the prime-factorization of n.

1, 1, 2, 2, 2, 2, 2, 0, 10, 2, 2, 16, 2, 2, 2, 0, 2, 24, 2, 26, 2, 2, 2, 2, 32, 2, 0, 36, 2, 2, 2, 0, 2, 2, 2, 46, 2, 2, 2, 2, 2, 2, 2, 62, 62, 2, 2, 2, 66, 68, 2, 70, 2, 2, 2, 2, 2, 2, 2, 84, 2, 2, 88, 0, 2, 2, 2, 94, 2, 2, 2, 98, 2, 2, 104, 104, 2, 2, 2, 2, 0, 2, 2, 116, 2, 2, 2, 2, 2, 126, 2, 128

Offset: 1

Author

Leroy Quet, Nov 19 2006

Keywords

• nonn

Comments

From Robert G. Wilson v, Nov 22 2006: (Start)

Only a(1) and a(2) are odd. a(n)=0 for n>1 in A036966.

Only possible values: ..., 0, 1, 2, 10, 16, 24, 26, 32, 36, 46, 62, 66, 68, 70, 84, 88, 94, 98, 104, ..., .

Position of first occurrence: 8, 1, 3, 9, 12, 18, 20, 25, 28, 36, 44, 49, 50, 52, 60, 63, 68, 72, 75, ..., .

(End)

Examples

			12 has a prime factorization of 2^2 *3^1. So a(12) is the sum of the terms among the first 11 terms of the sequence which equal 1 or 2. There are seven 2's and two 1's among the first 11 terms; so a(12) = 1+1+2+2+2+2+2+2+2 = 16.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A125087.

Programs

Mathematica

f[l_List] := Append[l, Plus @@ Select[l, MemberQ[Last /@ FactorInteger[Length[l] + 1], # ] &]];Nest[f, {1}, 91] (* Ray Chandler, Nov 21 2006 *)
a[1] = 1; a[n_] := a[n] = Plus @@ Flatten[ Cases[ Array[a, n - 1], # ] & /@ Union@ Last@ Transpose@ FactorInteger@n]; Array[a, 92] (* Robert G. Wilson v, Nov 22 2006 *)

PARI

up_to = 105;
A125088list(up_to) = { my(v=vector(up_to)); v[1] = 1; for(n=2,up_to,my(es = vecsort(factor(n)[,2],,8)); v[n] = sum(k=1,n-1,v[k]*!!vecsearch(es,v[k]))); (v); };
v125088 = A125088list(up_to);
A125088(n) = v125088[n]; \\ Antti Karttunen, Apr 01 2021

Extensions

Extended by Ray Chandler, Nov 21 2006