A348961 Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105

Offset: 1

Author

Amiram Eldar, Nov 05 2021

Keywords

• nonn

Comments

First differs from A005117 at n = 24, from A333634 and A348499 at n = 47, and from A336223 at n = 63.

Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 1, and that an e-perfect number (A054979) is a term of this sequence if and only if at least one of the exponents in its prime factorization is not a perfect square.

Since all the e-perfect numbers are products of a primitive e-perfect number (A054980) and a coprime squarefree number, and all the known primitive e-perfect numbers have a nonsquare exponent in their prime factorizations, there is no known e-perfect number that is not in this sequence.

Examples

			3 is a term since esigma(3) = 3, 3 * d_e(3) = 3 * 1, so esigma(3) | 3 * d_e(3).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
• József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

Crossrefs

A005117 and A348962 are subsequences.

Cf. A049419, A051377, A322791, A333634, A336223, A348499, A348964.

Programs

Mathematica

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^# &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]