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

1,1

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

Formula

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

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

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 *)

Prog

(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(d<n && sigma(d)>2*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))));

Crossrefs

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

Sequence in context: A220845 A361469 A073750 * A270059 A120204 A177018

Adjacent sequences: A378655 A378656 A378657 * A378659 A378660 A378661

Keyword

nonn,look,new

Author

Antti Karttunen, Dec 05 2024

Status

approved