A334902 Infinitary practical numbers (A334901) whose number of divisors is not a power of 2.

72, 360, 480, 504, 600, 672, 792, 864, 936, 1050, 1056, 1152, 1176, 1224, 1248, 1350, 1368, 1400, 1470, 1650, 1656, 1800, 1950, 1960, 2088, 2200, 2232, 2520, 2600, 2646, 2664, 2952, 3096, 3200, 3234, 3240, 3360, 3384, 3402, 3528, 3816, 3822, 3960, 4200, 4248, 4312

Offset

1,1

Comments

Practical numbers (A005153) whose number of divisors is a power of 2 (A036537) are also infinitary practical numbers (A334901), since all of their divisors are infinitary.

Up to 10^6 there are 34768 infinitary practical numbers; of them only 8858 are in this sequence.

Links

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

Mathematica

bin[n_] := 2^(-1 + Position[Reverse @ IntegerDigits[n, 2], _?(# == 1 &)] // Flatten); f[p_, e_] := p^bin[e]; icomp[n_] := Flatten[f @@@ FactorInteger[n]]; fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; infPracQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, r = Sort[icomp[n]]; Do[If[r[[i]] > 1 + isigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[r]}]; ok]]]; pow2Q[n_] := n/2^IntegerExponent[n, 2] == 1; Select[Range[4400], ! pow2Q[DivisorSigma[0, #]] && infPracQ[#] &]

Crossrefs

Cf. A005153, A036537, A287173, A334899, A334901.

Sequence in context: A107314 A223472 A090788 * A192792 A303621 A084479

Adjacent sequences: A334899 A334900 A334901 * A334903 A334904 A334905

Keyword

nonn

Author

Amiram Eldar, May 16 2020

Status

approved