A379028 The number of modified exponential divisors of n.

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Dec 14 2024

Keywords

• nonn
• easy
• mult

Comments

If the prime factorization of n is Product_{i} p_i^e_i, then the modified exponential divisors of n are all the divisors of n that are of the form Product_{i} p_i^b_i such that 1 + b_i | 1 + e_i for all i.

The sum of these divisors is A241405(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Row lengths of A379027.

Cf. A241405.

Similar sequences: A000005 (all divisors), A049419 (exponential), A037445 (infinitary), A034444 (unitary), A286324 (bi-unitary).

Programs

Mathematica

f[p_, e_] := DivisorSigma[0, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> numdiv(x+1), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = d(e+1), where d(k) is the number of divisors of k (A000005).