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A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

%I A342661 #14 Dec 24 2022 03:51:58
%S A342661 1,2,9,4,20,18,42,8,63,40,88,36,156,84,180,16,238,126,342,80,378,176,
%T A342661 460,72,325,312,405,168,696,360,930,32,792,476,840,252,1184,684,1404,
%U A342661 160,1558,756,1806,352,1260,920,2068,144,1519,650,2142,624,2544,810,1760,336,3078,1392,3186,720,3660,1860,2646,64,3120
%N A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.
%H A342661 Amiram Eldar, <a href="/A342661/b342661.txt">Table of n, a(n) for n = 1..10000</a>
%H A342661 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%H A342661 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A342661 Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
%F A342661 a(n) = n * A326041(n) = n * A000203(A064989(n)).
%F A342661 Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/9) * Product_{p prime > 2} (p^3/((p+1)*(p^2-prevprime(p)))) = 0.1815217..., where prevprime is A151799. - _Amiram Eldar_, Dec 24 2022
%t A342661 f[p_, e_] := If[p == 2, 2^e, Module[{q = NextPrime[p, -1]}, p^e*(q^(e + 1) - 1)/(q - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 24 2022 *)
%o A342661 (PARI)
%o A342661 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 A342661 A326041(n) = sigma(A064989(n));
%o A342661 A342661(n) = (n*A326041(n));
%Y A342661 Cf. A000203, A064989, A151799, A326041, A341528, A341529 [= a(A003961(n))], A342662, A342663, A342664.
%K A342661 nonn,mult
%O A342661 1,2
%A A342661 _Antti Karttunen_, Mar 23 2021