cp's OEIS Frontend

A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

%I A349381 #13 Nov 26 2021 10:33:18
%S A349381 1,2,3,6,4,6,6,18,15,8,6,18,6,12,12,54,6,30,6,24,18,12,10,54,28,12,75,
%T A349381 36,8,24,8,162,18,12,24,90,10,12,18,72,6,36,6,36,60,20,10,162,66,56,
%U A349381 18,36,12,150,24,108,18,16,8,72,8,16,90,486,24,36,10,36,30,48,6,270,8,20,84,36,36,36,10,216,375,12
%N A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.
%C A349381 Multiplicative because both A003961 and A349125 are.
%C A349381 Convolving this with A349127 gives A003972.
%H A349381 Antti Karttunen, <a href="/A349381/b349381.txt">Table of n, a(n) for n = 1..20000</a>
%H A349381 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349381 a(n) = Sum_{d|n} A003961(n/d) * A349125(d).
%F A349381 a(n) = A349383(n) - A349382(n).
%o A349381 (PARI)
%o A349381 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349381 A064989(n) = { my(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 A349381 A349125(n) = (moebius(n)*A064989(n));
%o A349381 A349381(n) = sumdiv(n,d,A003961(n/d)*A349125(d));
%Y A349381 Cf. A003961, A064989, A349125, A349382 (Dirichlet inverse), A349383 (sum with it).
%Y A349381 Cf. also A003972, A349127, and A349355, A349356 and A349384, A349385, and A349387.
%K A349381 nonn,mult
%O A349381 1,2
%A A349381 _Antti Karttunen_, Nov 17 2021