A342662 - OEIS

%I #14 Dec 24 2022 03:51:06

%S 1,3,8,7,18,24,40,15,52,54,84,56,154,120,144,31,234,156,340,126,320,

%T 252,456,120,279,462,320,280,690,432,928,63,672,702,720,364,1178,1020,

%U 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

%N 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.

%H Amiram Eldar, <a href="/A342662/b342662.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F 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.

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

%F 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

%t 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 *)

%o (PARI)

%o 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) };

%o A342662(n) = (sigma(n)*A064989(n));

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

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Mar 23 2021