A127666 -id:A127666 - cp's OEIS frontend

A129656 Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.

24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 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

Offset: 1

Ant King, Apr 29 2007

For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~7.95n. It follows that the density of the infinitary abundant numbers is 1/7.95, which is about 0.126.

			The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A126168, A049417, A127666, A129657, A007357.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;InfinitaryAbundantNumberQ[k_]:=If[properinfinitarydivisorsum[k]>k,True,False];Select[Range[500],InfinitaryAbundantNumberQ[ # ] &]
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]; Select[Range[1000], isigma[#]>2# &] (* Amiram Eldar, May 12 2019 *)

A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

Original entry on oeis.org

21735, 21944, 43064, 58695, 188055, 262184, 414855, 520695, 567944, 611415, 687015, 764504, 792855, 809864, 812889, 833624, 874664, 911624, 945944, 976184, 991304, 1019655, 1026375, 1065015, 1073709, 1157624, 1201095, 1218944, 1248344, 1254015, 1272375, 1272704

Offset: 1

Amiram Eldar, Sep 20 2019

nonn

The least k such that k, k+1 and k+2 are all infinitary abundant numbers is a(75976) = 2666847104.

			21735 is in the sequence since both 21735 and 21736 are infinitary abundant: isigma(21735) = 46080 > 2 * 21735, and isigma(21736) = 50400 > 2 * 21736 (isigma is the sum of infinitary divisors, A049417).

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

Cf. A049417, A127666, A129656.

Cf. A096399, A292704, A318167.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); abQ[n_] := isigma[n] > 2n; s={}; ab1 = 0; Do[ab2 = abQ[n]; If[ab1 && ab2, AppendTo[s, n-1]]; ab1 = ab2, {n, 2, 10^5}]; s

A129657 Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84

Offset: 1

Ant King, Apr 29 2007

For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.

			The sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189, (1990), pp. 395-411.
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A126168, A049417, A127666, A129656, A007357.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k] 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[100], isigma[#] < 2 # &] (* Amiram Eldar, Jun 09 2019 *)

A127667 Odd integers that do not generate monotonically decreasing infinitary aliquot sequences.

Original entry on oeis.org

945, 1743, 2175, 2655, 2823, 2865, 3105, 3375, 3537, 3585, 3729, 4209, 4665, 5775, 6559, 6681, 6969, 7257, 7263, 7785, 8457, 8583, 9657, 10017, 10047, 10113, 10395, 10599, 10743, 12285, 13815, 14055, 14145, 15015, 15597, 16065, 17955, 18529, 18777, 19305, 19635

Offset: 1

Ant King, Jan 26 2007

nonn

Based on empirical evidence, approximately 98.9 % of the infinitary aliquot sequences generated by the odd integers are monotonically decreasing. This sequence represents the 1.1 % of odd integers that are the exceptions to this.

			a(5)=2823 because 2823 is the fifth odd integer whose infinitary aliquot sequence is not monotonically decreasing.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A126168, A127661, A127666.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];u[n_]:=Table[n[[k+1]]

More terms from Amiram Eldar, Sep 16 2019

A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

Original entry on oeis.org

99225, 1091475, 1289925, 1334025, 1576575, 1686825, 1715175, 1863225, 1885275, 2027025, 2061675, 2282175, 2304225, 2395575, 2401245, 2436525, 2480625, 2650725, 2723175, 2789325, 2877525, 2962575, 3031875, 3075975, 3132675, 3185325, 3186225, 3296475, 3353805, 3501225

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The number of terms below 10^k, for k = 5, 6, ..., are 1, 113, 630, 7771, 73685, ... Apparently this sequence has an asymptotic density 0.000007...

			99225 is a term since A348271(99225) = 107207 > 99225.

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

Cf. A348271.
Subsequence of A005231 and A348274.
Similar sequences: A094889, A127666, A129485, A293186, A321147.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1, 2*10^6, 2], s[#] > # &]

A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11025, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 17325, 18585, 19215, 19635, 20475, 21105, 21945, 22365, 22995, 23205, 24255, 24885, 25935, 26145, 26565, 26775

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			1575 is a term since it is odd and A333926(1575) = 3224 > 2 * 1575.

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

Intersection of A005408 and A333928.
Cf. A333926.
Analogous sequences: A005231, A094889 (nonunitary), A129485 (unitary), A127666 (infinitary), A293186 (bi-unitary), A321147 (exponential).

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[2*Range[15000] + 1, recDivSum[#] > 2*# &]

A360526 Odd numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655

Offset: 1

Amiram Eldar, Feb 10 2023

nonn

First differs from A112643, A129485, A249263 at n = 46: a(46) = 165165 is not a term of these sequences.

			15015 is a term since A360522(15015) = 32256 > 2*15015.

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

Cf. A360522.
Subsequence of A005101, A005231 and A360525.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 10^5, 2], q]

isab(n) = { my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) > 2*n;}
is(n) = n%2 && isab(n);

A127663 Infinitary aspiring numbers.

Original entry on oeis.org

30, 42, 54, 66, 72, 78, 100, 140, 148, 152, 192, 194, 196, 208, 220, 238, 244, 252, 268, 274, 292, 296, 298, 300, 336, 348, 350, 360, 364, 372, 374, 380, 382, 386, 400, 416, 420, 424, 476, 482, 492, 516, 520, 532, 540, 542, 544, 550, 572, 576, 578, 586, 592

Offset: 1

Ant King, Jan 26 2007

Numbers whose infinitary aliquot sequences end in an infinitary perfect number, but are not infinitary perfect numbers themselves.

			a(5) = 72 because the fifth non-infinitary perfect number whose infinitary aliquot sequence ends in an infinitary perfect number is 72.

Amiram Eldar, Table of n, a(n) for n = 1..72
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

Cf. A007357, A126168, A127661, A127662, A127664, A127665, A127666, A127667.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryPerfectNumberQ[0]=False;InfinitaryPerfectNumberQ[k_Integer] :=If[properinfinitarydivisorsum[k]==k,True,False];Select[Range[750],InfinitaryPerfectNumberQ[Last[iTrajectory[ # ]]] && !InfinitaryPerfectNumberQ[ # ]&]
f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; s[n_] := Times @@ f @@@ FactorInteger[n] - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, n != v[[-2]] == v[[-1]] > 0]; Select[Range[839], q]  (* Amiram Eldar, Mar 11 2023 *)

A127665 Numbers whose infinitary aliquot sequences end in an infinitary amicable pair.

Original entry on oeis.org

102, 114, 126, 210, 246, 258, 270, 318, 330, 342, 354, 366, 378, 388, 390, 408, 426, 436, 438, 450, 474, 484, 486, 498, 510, 522, 534, 536, 546, 552, 570, 582, 594, 600, 606, 618, 630, 642, 648, 654, 666, 672, 702, 726, 738, 750, 760, 762, 774, 786, 798

Offset: 1

Ant King, Jan 26 2007

Sometimes called the infinitary 2-cycle attractor set.

			a(5)=246 because 246 is the fifth number whose infinitary aliquot sequence ends in an infinitary amicable pair.

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A126168, A007357, A127661, A127662, A127663, A127664, A127666, A127667, A126169, A126170, A126171.

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}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryAmicableNumberQ[k_]:=If[Nest[properinfinitarydivisorsum,k,2]==k && !properinfinitarydivisorsum[k]==k,True,False];Select[Range[820],InfinitaryAmicableNumberQ[Last[iTrajectory[ # ]]] &]

A335055 Odd infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller odd infinitary abundant number.

Original entry on oeis.org

945, 29835, 33345, 43065, 46035, 49875, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 77967015, 132335385, 210124665, 719709375, 724239285, 1756753845, 9665740455, 10394173335

Offset: 1

Amiram Eldar, May 21 2020

nonn

The infinitary abundancy of a number k is isigma(k)/k, where isigma(k) is the sum of infinitary divisors of k (A049417).

			The infinitary abundancies of the first terms are 2.031..., 2.027..., 2.015..., 2.006..., 2.001..., ...

Cf. A049417, A127666, A188263, A335052, A335053, A335054.

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]; seq = {}; r = 3; Do[s = isigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^5, 2}]; seq

cp's OEIS Frontend

A129656 Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A129657 Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A127667 Odd integers that do not generate monotonically decreasing infinitary aliquot sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A360526 Odd numbers k such that A360522(k) > 2*k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A127663 Infinitary aspiring numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs