A342662 a(n) = sigma(n) * A064989(n), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma is the sum of the divisors of n.

1, 3, 8, 7, 18, 24, 40, 15, 52, 54, 84, 56, 154, 120, 144, 31, 234, 156, 340, 126, 320, 252, 456, 120, 279, 462, 320, 280, 690, 432, 928, 63, 672, 702, 720, 364, 1178, 1020, 1232, 270, 1554, 960, 1804, 588, 936, 1368, 2064, 248, 1425, 837, 1872, 1078, 2538, 960, 1512, 600, 2720, 2070, 3180, 1008, 3658, 2784, 2080, 127

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization.

Index entries for sequences related to sigma(n).

Formula

Multiplicative with a(p^e) = q^e * (p^(e+1)-1)/(p-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.

a(n) = A000203(n) * A064989(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (16/63) * Product_{p prime > 2} p^4*(p-1)/((p^3-prevprime(p))*(p^2-prevprime(p))) = 0.1935405..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

Mathematica

f[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]*(p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

Prog

(PARI)
A064989(n) = { my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };
A342662(n) = (sigma(n)*A064989(n));

Crossrefs

Cf. A000203, A064989, A151799, A341528 [= a(A003961(n))], A341529, A342661, A342663, A342664.

Sequence in context: A224847 A309153 A260310 * A122237 A307162 A233418

Adjacent sequences: A342659 A342660 A342661 * A342663 A342664 A342665

Keyword

nonn,mult

Author

Antti Karttunen, Mar 23 2021

Status

approved