A336845 - OEIS

%I #16 Aug 06 2020 23:27:56

%S 1,6,10,27,14,60,22,108,75,84,26,270,34,132,140,405,38,450,46,378,220,

%T 156,58,1080,147,204,500,594,62,840,74,1458,260,228,308,2025,82,276,

%U 340,1512,86,1320,94,702,1050,348,106,4050,363,882,380,918,118,3000,364,2376,460,372,122,3780,134,444,1650,5103,476,1560

%N a(n) = A000005(n) * A003961(n), where A003961 is the prime shift towards larger primes, and A000005 gives the number of divisors of n, and also of A003961(n).

%C Dirichlet convolution of A003961 with itself.

%C Sequence is not injective, as it has duplicate values, for example: a(162) = a(243) = 18750. See also comments in A336475.

%H Antti Karttunen, <a href="/A336845/b336845.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A336845/a336845.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

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

%F Multiplicative with a(prime(i)^e) = (e+1) * prime(1+i)^e.

%F a(n) = A000005(n) * A003961(n).

%F a(n) = A038040(A003961(n)).

%F a(n) = A336841(n) + A003973(n).

%F a(n) is odd if and only if n is a square.

%o (PARI)

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

%o A336845(n) = (numdiv(n)*A003961(n))

%o (PARI) A336845(n) = { my(f = factor(n)); prod(i=1, #f~, (1+f[i,2]) * (nextprime(1+f[i, 1])^f[i,2])); };

%o (PARI) A336845(n) = sumdiv(n,d,A003961(d)*A003961(n/d));

%Y Cf. A000005, A000203, A000290, A003961, A038040, A336475.

%Y Cf. also A336841, A336846 [= gcd(a(n),A003973(n))], A336847, A336848.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Aug 06 2020