A129656 -id:A129656 - cp's OEIS frontend

A380929 Numbers k such that A380845(k) > 2*k.

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A306983 Infinitary pseudoperfect numbers: numbers n equal to the sum of a subset of their proper infinitary divisors.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504

Offset: 1

Amiram Eldar, Mar 18 2019

nonn

Subsequence of A005835.

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

Cf. A005835, A007357, A049417, A126168, A129656, A293188, A292985, A318100, A306984.

idivs[x_] := If[x == 1, 1, Sort@Flatten@Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; s = {}; Do[d = Most[idivs[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[s, n]], {n, 2, 1000}]; s

A323344 Numbers k whose infinitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

2394, 7544, 10184, 1452330, 2154584, 5021912, 5747994, 5771934, 5786298, 5800662, 5834178, 5843754, 5858118, 5886846, 5905998, 5920362, 5929938, 5992182, 6035274, 6059214, 6078366, 6087942, 6102306, 6107094, 6121458, 6174126, 6202854, 6207642, 6245946, 6265098

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The infinitary version of A171641.

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

Cf. A049417, A077609, A129656, A171641, A323341, A323342, A323343.

infdivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ; 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[n_] := If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]; seq={}; Do[s=isigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = infdivs[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 100000}]; seq (* after Michael De Vlieger at A077609 *)

A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504

Offset: 1

Amiram Eldar, May 26 2020

nonn

			6 is a term since its set of infinitary divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets, {1, 2, 3} and {6}, whose sum is equal: 1 + 2 + 3 = 6.

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

The infinitary version of A083207.
Subsequence of A129656.
Cf. A077609, A323344.

infdivs[n_] := If[n == 1, {1}, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infZumQ[n_] := Module[{d = infdivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[500], infZumQ] (* after Michael De Vlieger at A077609 *)

A333928 Recursive abundant numbers: numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 108, 114, 120, 126, 132, 138, 140, 144, 150, 156, 168, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 240, 246, 252, 258, 260, 270, 276, 282, 294, 300, 306, 308, 318, 324, 330, 336, 340, 342, 348

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

			12 is a term since A333926(12) = 28 > 2 * 12.

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

Cf. A333926, A333927.

Analogous sequences: A005101, A034683 (unitary), A064597 (nonunitary), A129575 (exponential), A129656 (infinitary), A292982 (bi-unitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[350], recDivSum[#] > 2*# &]

A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 40, 42, 54, 56, 66, 72, 78, 88, 104, 120, 128, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 440, 456, 462, 480, 504, 510, 520, 546, 552, 570, 594, 600, 616, 640, 672, 680, 690, 696, 702, 714, 728, 744, 750, 760, 792, 798

Offset: 1

Amiram Eldar, May 16 2020

nonn

Includes the powers of 2 of the form 2^(2^k - 1) for k = 0, 1, ... (A058891). The other terms are a subset of infinitary abundant numbers (A129656) and infinitary pseudoperfect numbers (A306983).

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

The infinitary version of A005153.

Cf. A049417, A058891, A129656, A286652, A306983, A334898.

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]]]; Select[Range[1000], infPracQ]

A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 18 2020

nonn

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, ...

			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.

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

The infinitary version of A111592.
Subsequence of A129656.
Cf. A036537, A049417, A077609, A328328, A334972.

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]

A357605 Numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

36, 48, 72, 80, 96, 108, 120, 144, 160, 162, 168, 180, 192, 200, 216, 224, 240, 252, 264, 270, 280, 288, 300, 312, 320, 324, 336, 352, 360, 378, 384, 392, 396, 400, 408, 416, 432, 448, 450, 456, 468, 480, 486, 500, 504, 528, 540, 552, 560, 576, 588, 594, 600, 612

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

The least odd term is a(470) = A357607(1) = 4725.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 5, 92, 1011, 10160, 102125, 1022881, 10231151, 102249758, 1022781199, 10229781638, ... . Apparently, the asymptotic density of this sequence exists and equals 0.102... .
An analog of abundant numbers, in which the divisor sum is restricted to nonsquarefree divisors. - Peter Munn, Oct 26 2022

			36 is a term since A162296(36) = 79 > 2*36.

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

Cf. A162296.
Subsequence of A005101 and A013929.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 1000], q]

A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

120, 840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3480, 3720, 4440, 4920, 5160, 5640, 5880, 6360, 7080, 7320, 7560, 8040, 8520, 8760, 9240, 9480, 9720, 9960, 10680, 10920, 11640, 11880, 12120, 12360

Offset: 1

Amiram Eldar, Mar 10 2018

nonn

Analogous to 3-abundant numbers (A023197) with isigma (A049417) instead of sigma (A000203).

			840 is in the sequence since isigma(840) = 2880 > 3 * 840.

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

Cf. A007357, A023197, A049417, A129656, A129657.

ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, #]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer ? Positive] := Module[{factors = First /@ FactorInteger[n], d = Divisors[n]}, d[[Flatten[ Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][#, Last[#]]] & /@ Transpose[ Last /@ ExponentList[#, factors] & /@ d]], ?(And @@ # &), {1}]]]]]; properinfinitarydivisorsum[k] := Plus @@ InfinitaryDivisors[k] - k; Infinitary3AbundantNumberQ[k_] :=  If[properinfinitarydivisorsum[k] >= 2 k, True, False]; Select[Range[15000], Infinitary3AbundantNumberQ[#] &] (* after Ant King at A129656 *)

A307820 The number of infinitary abundant numbers below 10^n.

Original entry on oeis.org

0, 12, 114, 1270, 12518, 125634, 1257749, 12570993, 125716733, 1256921422, 12570417639

Offset: 1

Amiram Eldar, Apr 30 2019

			Below 10^2 there are 12 infinitary abundant numbers, 24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, and 96, thus a(2) = 12.

Cf. A049417, A129656, A302992, A302993, A302994, A307821, A307823.

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];  c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ isigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq

Conjecture: Lim_{n->oo} a(n)/10^n = 0.125... is the density of infinitary abundant numbers.

a(11) from Amiram Eldar, Sep 09 2022

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A380929 Numbers k such that A380845(k) > 2*k.

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A306983 Infinitary pseudoperfect numbers: numbers n equal to the sum of a subset of their proper infinitary divisors.

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A323344 Numbers k whose infinitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

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A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

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A333928 Recursive abundant numbers: numbers k such that A333926(k) > 2*k.

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A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

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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).

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A357605 Numbers k such that A162296(k) > 2*k.

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A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

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A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).