A337538 a(n) is the least k such that A003961(k*A071395(n)) is abundant.

6, 6, 15, 15, 15, 15, 15, 15, 3, 6, 6, 6, 9, 2, 15, 15, 15, 3, 15, 2, 15, 3, 15, 15, 6, 3, 2, 3, 3, 3, 9, 3, 9, 3, 3, 3, 2, 15, 6, 15, 6, 3, 2, 15, 15, 15, 3, 15, 15, 15, 3, 15, 15, 3, 15, 15, 2, 15, 15, 15, 15, 2, 3, 15, 2, 15, 15, 15, 2, 15, 15, 15, 15, 2, 15, 15, 15, 15, 15, 15, 2, 2, 15, 15, 15, 15, 2

Offset: 1

Author

Antti Karttunen and Peter Munn, Sep 07 2020

Keywords

• nonn

Comments

A071395(n) is the n-th primitive abundant number. A003961(k) replaces each prime factor of k with the next larger prime.

See also the table in the example section of A337469.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..13037

Crossrefs

Cf. A000203, A003961, A003973, A071395, A337386, A337469, A337479.

Programs

Mathematica

Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; k] &, Select[Range[10^4], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)

PARI

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k,", "); break))));

Formula

a(n) = A337469(n) / A071395(n).