A336845 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).

1, 6, 10, 27, 14, 60, 22, 108, 75, 84, 26, 270, 34, 132, 140, 405, 38, 450, 46, 378, 220, 156, 58, 1080, 147, 204, 500, 594, 62, 840, 74, 1458, 260, 228, 308, 2025, 82, 276, 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

Offset: 1

Author

Antti Karttunen, Aug 06 2020

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A003961 with itself.

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

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

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

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336845(n) = (numdiv(n)*A003961(n))

PARI

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

PARI

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

Formula

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

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

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

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

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