A064597 -id:A064597 - cp's OEIS frontend

A335196 Nonunitary admirable numbers: numbers k such that there is a nonunitary divisor d of k such that nusigma(k) - 2*d = k, where nusigma is the sum of nonunitary divisors function (A048146).

48, 80, 96, 108, 120, 160, 168, 180, 192, 216, 224, 252, 264, 280, 300, 312, 320, 336, 352, 360, 384, 396, 408, 416, 432, 448, 456, 468, 480, 504, 528, 540, 552, 560, 600, 612, 624, 640, 672, 684, 696, 704, 720, 744, 756, 768, 792, 816, 828, 832, 840, 864, 880

Offset: 1

Amiram Eldar, May 26 2020

nonn

Equivalently, numbers that are equal to the sum of their nonunitary divisors, with one of them taken with a minus sign.

			48 is a term since 48 = 2 - 4 + 6 + 8 + 12 + 24 is the sum of its nonunitary divisors with one of them, 4, taken with a minus sign.

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

The nonunitary version of A111592.
Subsequence of A064597.
Similar sequences: A328328, A334972, A334974.
Cf. A048146.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; nuAdmQ[n_] := (ab = nusigma[n] - n) > 0 && EvenQ[ab] && ab/2 < n && !CoprimeQ[ab/2, 2*n/ab]; Select[Range[1000], nuAdmQ]

A327944 Numbers m that are equal to the sum of their first k consecutive nonunitary divisors, but not all of them (i.e k < A048105(m)).

Original entry on oeis.org

480, 2688, 17640, 131712, 2095104, 3576000, 4248288, 16854816, 41055200, 400162032, 637787520, 788259840, 1839272960, 2423592576

Offset: 1

Amiram Eldar, Sep 30 2019

The nonunitary version of Erdős-Nicolas numbers (A194472).

If all the nonunitary divisors are permitted (i.e. k <= A048105(n)), then the nonunitary perfect numbers (A064591) are included.

			480 is in the sequence since its nonunitary divisors are 2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 40, 48, 60, 80, 120 and 240 and 2 + 4 + 6 + 8 + 10 + 12 + 16 + 20 + 24 + 30 + 40 + 48 + 60 + 80 + 120 = 480.~

Cf. A048105, A064591, A064597, A064599.

Cf. A194472, A293618, A309843.

ndivs[n_] := Block[{d = Divisors[n]}, Select[d, GCD[ #, n/# ] > 1 &]]; ndivs2[n_] := Module[{d=ndivs[n]},If[Length[d]<2,{},Drop[d, -1] ]]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n,ndivs2[n]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

A327947 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in more ways than any smaller nonunitary pseudoperfect number.

Original entry on oeis.org

24, 48, 72, 96, 144, 216, 240, 288, 360, 480, 720, 1080, 1440, 2160, 2880, 3600, 4320, 5040, 7200, 7560, 10080, 15120, 20160, 25200, 30240

Offset: 1

Amiram Eldar, Sep 30 2019

The nonunitary version of A065218.

The corresponding numbers of ways are 1, 2, 4, 5, 15, 28, 34, 63, 211, 279, 6025, 17436, 187794, 2035726, 5965563, 36449982, 250420995, 3426156924, 8991176276, 37016127059, 6770551810345, 1095548357870254, 13524344273940115, 604532928571438678, 33370817837127087825, ...

			24 is the least number which is the sum of its nonunitary divisor, thus a(1) = 24.
48 is the least number which is the sum of a subset of its nonunitary divisor in two ways: 24 + 12 + 8 + 4 and 24 + 12 + 8 + 4 + 2, thus a(2) = 48.

Cf. A064591, A064597, A065218, A327945, A327946.

nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; cm = 0; Do[d = nudiv[n]; If[Total[d] < n, Continue[]]; c = SeriesCoefficient[ Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > cm, cm = c; AppendTo[s, n]], {n, 1, 1000}]; s

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A335196 Nonunitary admirable numbers: numbers k such that there is a nonunitary divisor d of k such that nusigma(k) - 2*d = k, where nusigma is the sum of nonunitary divisors function (A048146).

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A327944 Numbers m that are equal to the sum of their first k consecutive nonunitary divisors, but not all of them (i.e k < A048105(m)).

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A327947 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in more ways than any smaller nonunitary pseudoperfect number.

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