A378658 a(n) = A337345(A091191(n)), where A337345 is the number of divisors d of n for which A003961(d) > 2*d, and A091191 lists the primitive abundant numbers.

3, 3, 3, 4, 4, 5, 2, 4, 3, 4, 2, 4, 3, 3, 2, 2, 6, 2, 2, 2, 6, 2, 6, 6, 2, 2, 7, 2, 6, 2, 2, 2, 6, 2, 6, 2, 6, 2, 5, 5, 2, 2, 2, 2, 6, 5, 2, 2, 4, 2, 2, 6, 2, 2, 5, 6, 2, 2, 2, 12, 2, 8, 2, 6, 2, 2, 2, 2, 6, 2, 2, 8, 2, 6, 2, 8, 6, 2, 2, 2, 8, 2, 6, 2, 2, 6, 8, 2, 2, 13, 2, 2, 2, 6, 2, 2, 8, 2, 6, 2, 2, 2, 4, 6

Offset: 1

Author

Antti Karttunen, Dec 05 2024

Keywords

• nonn
• look

Comments

For all n, a(n) > 1. This follows from a proof given in A337372. See also A378662.

Among the initial 10 million terms, there are 7835064 2's.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A000203, A003961, A080224, A091191, A337345, A337372, A378662.

Programs

Mathematica

s = Select[Range[2^11], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &];
Map[Length@ Select[Divisors[#], 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &] &, s] (* Michael De Vlieger, Dec 06 2024 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337345(n) = sumdiv(n,d,A003961(d)>(2*d));
is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d2*d, return(0))); (1));
k=0; n=0; while(k<100000, n++; if(is_A091191(n), k++; print1(A337345(n), ", "); write("b378658.txt", k, " ", A337345(n))));

Formula

{A337345(k) for k such that A080224(k) = 1}.

a(n) = 1+A378662(A091191(n)).