A347236 - OEIS

%I #28 Sep 18 2021 22:02:56

%S 1,1,2,7,2,2,4,13,19,2,2,14,4,4,4,55,2,19,4,14,8,2,6,26,39,4,68,28,2,

%T 4,6,133,4,2,8,133,4,4,8,26,2,8,4,14,38,6,6,110,93,39,4,28,6,68,4,52,

%U 8,2,2,28,6,6,76,463,8,4,4,14,12,8,2,247,6,4,78,28,8,8,4,110,421,2,6,56,4,4,4,26,8,38,16

%N a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.

%C Dirichlet convolution of A003961 and A061019.

%C Dirichlet convolution of A003973 and A158523.

%C Multiplicative because A003961 and A061019 are.

%C All terms are positive because all terms of A347237 are nonnegative and A347237(1) = 1.

%C Union of sequences A001359 and A108605 (= 2*A001359) seems to give the positions of 2's in this sequence.

%H Antti Karttunen, <a href="/A347236/b347236.txt">Table of n, a(n) for n = 1..16383</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

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

%F a(n) = Sum_{d|n} A003961(n/d) * A061019(d).

%F a(n) = Sum_{d|n} A003973(n/d) * A158523(d).

%F a(n) = Sum_{d|n} A347237(d).

%F a(n) = A347239(n) - A347238(n).

%F For all n >= 1, a(A000040(n)) = A001223(n).

%F Multiplicative with a(p^e) = (A151800(p)^(e+1)-(-p)^(e+1))/(A151800(p)+p). - _Sebastian Karlsson_, Sep 02 2021

%t f[p_, e_] := ((np = NextPrime[p])^(e + 1) - (-p)^(e + 1))/(np + p); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 02 2021 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A061019(n) = (((-1)^bigomega(n))*n);

%o A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));

%Y Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239.

%Y Cf. also A347136.

%Y Cf. A151800.

%K nonn,mult

%O 1,3

%A _Antti Karttunen_, Aug 24 2021