A348922 Numbers that are both infinitary and noninfinitary harmonic numbers.

45, 60, 54600, 257040, 1801800, 2789640, 4299750, 47297250, 1707259680, 4093362000

Offset: 1

Author

Amiram Eldar, Nov 04 2021

Keywords

• nonn
• more

Comments

a(11) > 10^10.

For each term the two sets of infinitary and noninfinitary divisors both contain more than one element. The only number with a single infinitary divisor is 1 which does not have noninfinitary divisors. Numbers with a single noninfinitary divisor are the squares of primes which are not infinitary harmonic numbers. Therefore, this sequence is a subsequence of A348715.

Nonsquarefree numbers k such that A049417(k) divides k*A037445(k) and A348271(k) divides k*A348341(k). The sequence also includes: 18779856480, 44425017000, 13594055202000, 27188110404000, 299069214444000, 6824215711404000. - Daniel Suteu, Nov 06 2021

Examples

			45 is a term since the infinitary divisors of 45 are 1, 5, 9 and 45, and their harmonic mean is 3, and the noninfinitary divisors of 45 are 3 and 15, and their harmonic mean is 5.
		

Crossrefs

Intersection of A063947 and A348918.

Subsequence of A348715.

Cf. A348923.

Cf. A037445, A049417, A348271, A348341.

Programs

Mathematica

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; id[1] = 1; id[n_] := Times @@ Flatten[2^DigitCount[#, 2, 1]& /@ FactorInteger[n][[;; , 2]] ]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[3*10^5], !pow2Q[DivisorSigma[0, #]] && IntegerQ[# * (d = id[#])/(s = isigma[#])] && IntegerQ[# * (DivisorSigma[0, #] - d)/(DivisorSigma[1, #] - s)] &]