

A334974


Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k)  2*d = 2*k, where isigma is the sum of infinitary divisors function (A049417).


5



24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 96, 102, 104, 114, 120, 138, 150, 174, 186, 222, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 678, 726, 762, 780, 786, 822, 834, 894, 906, 942
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OFFSET

1,1


COMMENTS

Equivalently, numbers that are equal to the sum of their proper infinitary divisors, with one of them taken with a minus sign.
Admirable numbers (A111592) whose number of divisors is a power of 2 (A036537) are also infinitary admirable numbers, since all of their divisors are infinitary. Terms with number of divisors that is not a power of 2 are 96, 150, 294, 360, 420, 486, 540, 630, 660, 726, 780, 960, 990, ...


LINKS

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


EXAMPLE

150 is in the sequence since 150 = 1 + 2 + 3  6 + 25 + 50 + 75 is the sum of its proper infinitary divisors with one of them, 6, taken with a minus sign.


MATHEMATICA

fun[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 @@ fun @@@ FactorInteger[n]; infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e; infAdmQ[n_] := (ab = isigma[n]  2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && infDivQ[n, ab/2]; Select[Range[1000], infAdmQ]


CROSSREFS

The infinitary version of A111592.
Subsequence of A129656.
Cf. A036537, A049417, A077609, A328328, A334972.
Sequence in context: A334972 A109797 A129656 * A048945 A111398 A030626
Adjacent sequences: A334971 A334972 A334973 * A334975 A334976 A334977


KEYWORD

nonn


AUTHOR

Amiram Eldar, May 18 2020


STATUS

approved



