A007357 -id:A007357 - cp's OEIS frontend

A097464 5-infinitary perfect numbers: numbers k such that 5-infinitary-sigma(k) = 2*k.

6, 28, 496, 47520, 288288, 308474880

Offset: 1

Here 5-infinitary-sigma(k) means sum of 5-infinitary-divisors of k. If k = Product p_i^r_i and d = Product p_i^s_i, each s_i has a digit a <= b in its 5-ary expansion everywhere that the corresponding r_i has a digit b, then d is a 5-infinitary-divisor of k.
Is it certain that 308474880 is the 6th term? M. F. Hasler, Nov 20 2010
Data is verified. a(7) > 10^11, if it exists. - Amiram Eldar, Oct 24 2024

			Factorizations: 2*3, 2^2*7, 2^4*31, 2^5*3^3*5*11, 2^5*3^2*7*11*13, 2^10*3*5*7*19*151.

Cf. A007357, A038182, A074849, A097863.

f[p_, e_] := Module[{d = IntegerDigits[e, 5]}, m = Length[d]; Product[(p^((d[[j]] + 1)*5^(m - j)) - 1)/(p^(5^(m - j)) - 1), {j, 1, m}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300000], s[#] == 2*# &] (* Amiram Eldar, Oct 24 2024 *)

{k: A097863(k) = 2*k}.

Missing a(4) inserted by R. J. Mathar, Nov 20 2010

A074849 4-infinitary perfect numbers: numbers k such that 4-infinitary-sigma(k) = 2*k.

Original entry on oeis.org

6, 28, 36720, 222768, 12646368, 5154170112, 34725010231296

Offset: 1

Yasutoshi Kohmoto, Sep 10 2002

Here 4-infinitary-sigma(k) means sum of 4-infinitary-divisors of k. If k = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a <= b in its 4-ary expansion everywhere that the corresponding r(i) has a digit b, then d is a 4-infinitary-divisor of k.

			Factorizations: 2*3, 2^2*7, 2^4*3^3*5*17, 2^4*3^2*7*13*17, 2^5*3^4*7*17*41, 2^8*3^2*7*13^2*31*61, 2^12*3^5*7*11*41*43*257.

Cf. A007357, A038182, A074847, A097464.

f[p_, e_] := Module[{d = IntegerDigits[e, 4]}, m = Length[d]; Product[(p^((d[[j]] + 1)*4^(m - j)) - 1)/(p^(4^(m - j)) - 1), {j, 1, m}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300000], s[#] == 2*# &] (* Amiram Eldar, Oct 24 2024 *)

{k: A074847(k) = 2*k}. - R. J. Mathar, Mar 13 2024

A327633 Noninfinitary perfect numbers: numbers k whose sum of noninfinitary divisors equals k.

Original entry on oeis.org

112, 1344, 32512, 390144, 483840, 5930176, 2952609792

Offset: 1

Amiram Eldar, Sep 20 2019

Numbers k such that sigma(k) - isigma(k) = A000203(k) - A049417(k) = k.

No more terms below 3 * 10^10.

			112 is in the sequence since its noninfinitary divisors are {2, 4, 8, 14, 28, 56} whose sum is 112.

Cf. A000203, A007357, A049417, A064591, A077609, A282900, A282940.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); nisigma[1] = 0; nisigma[n] := DivisorSigma[1, n] - Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); Select[Range[500000], nisigma[#] == # &]

A185079 a(n) = A064380(n) * A049417(n).

Original entry on oeis.org

3, 8, 15, 24, 36, 48, 60, 80, 90, 120, 140, 168, 192, 216, 255, 288, 330, 360, 390, 448, 504, 528, 600, 624, 672, 720, 760, 840, 936, 960, 1020, 1056, 1134, 1200, 1300, 1368, 1440, 1512, 1620, 1680, 1632, 1848, 1920, 1980, 2088, 2208, 2312, 2400, 2496, 2592, 2730, 2808, 2880, 3024, 3240, 3200, 3330

Offset: 2

Vladimir Shevelev, Feb 18 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A007357, A185078, A185373, A185383.

a(n) = n^2 + o(n^(1+eps)).

Corrected and extended by T. D. Noe, Feb 18 2011

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 *)

A004607 Infinitary sociable numbers (smallest member of cycle).

Original entry on oeis.org

1026, 10098, 10260, 12420, 41800, 45696, 100980, 241824, 448800, 512946, 685440, 830568, 4938136, 6732000, 9424800, 12647808, 13959680, 14958944, 17878998, 25581600, 28158165, 32440716, 36072320, 55204500, 74062944

Offset: 1

N. J. A. Sloane

If n = Product p_i^a_i, d = Product p_i^c_i is an infinitary divisor of n if each c_i has a zero bit in its binary representation everywhere that the corresponding a_i does.
From Amiram Eldar, Mar 25 2023: (Start)
Analogous to A003416 with the sum of the aliquot infinitary divisors function (A126168) instead of A001065.
Only cycles of length greater than 2 are here. Cycles of length 1 correspond to infinitary perfect numbers (A007357), and cycles of length 2 correspond to infinitary amicable pairs (A126169 and A126170).
The corresponding cycles are of lengths 4, 4, 4, 6, 4, 4, 4, 4, 11, 6, 4, 6, 4, 11, 6, 23, 4, 4, 85, 4, 4, 4, 4, 4, 4, ...
It is conjectured that there are no missing terms in the data, but it was not proven. For example, it is not known that the infinitary aliquot sequence that starts at 840 does not reach 840 again (see A361421). (End)

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
J. O. M. Pedersen, Tables of Aliquot Cycles.
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. A001065, A003416, A007357, A126168, A126169, A126170, A361421.

A007358 Infinitary multi-perfect numbers.

Original entry on oeis.org

1, 6, 60, 90, 120, 36720, 73440, 12646368, 22276800, 44553600, 126463680, 133660800, 252927360, 758782080, 4201148160, 8402296320, 28770487200, 287704872000, 575409744000

Offset: 1

N. J. A. Sloane

The sequence contains numbers n such that A049417(n) = k*n for some integer k>=1. A007357 is the subsequence with quotient k=2. Cohen lists n=120, 73440, 44553600, 252927360, 575409744000 as entries with k=3, provides seven entries with k=4 and two entries with k=5.

G. L. Cohen, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Michel Marcus, Unexhaustive list of terms

Cf. A049417, A007357.

Cf. A007691 (analog for sigma).

a049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
isok(n) = frac(a049417(n)/n) == 0; \\ Michel Marcus, Sep 05 2018

a(10)-a(18) from Donovan Johnson, Nov 21 2013

a(1)=1 prepended by Michel Marcus, Sep 04 2018

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[ # ]]] &]

A185078 Numbers k for which A064380(k) = k/2.

Original entry on oeis.org

2, 6, 8, 10, 60, 70, 128, 136, 9822, 18632, 32768, 32896, 36720, 69726, 73662, 73686, 73734, 85962, 86046, 87114, 87198, 87222, 87258, 87294, 87306, 87342, 87366, 87546, 87558, 88014, 88278, 88302, 88338, 88386, 127326, 128046, 128082, 128382, 128406, 128598, 128802

Offset: 1

Vladimir Shevelev, Feb 18 2011

nonn

Note that, if there exist infinitely many infinitary perfect numbers (A007357), then, as k tends to infinity over such numbers, A064380(k)/k = 1/2 + o(k^(-1+eps)). We conjecture that here A064380(k)/k = 1/2 infinitely many times, and thus the sequence contains infinitely many infinitary perfect numbers.

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A064380, A007357, A064380, A185373, A185383.

a(7)-a(13) from Amiram Eldar, Sep 13 2019

a(14)-a(41) from Amiram Eldar, Mar 26 2023

cp's OEIS Frontend

A097464 5-infinitary perfect numbers: numbers k such that 5-infinitary-sigma(k) = 2*k.

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A074849 4-infinitary perfect numbers: numbers k such that 4-infinitary-sigma(k) = 2*k.

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A327633 Noninfinitary perfect numbers: numbers k whose sum of noninfinitary divisors equals k.

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A185079 a(n) = A064380(n) * A049417(n).

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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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A004607 Infinitary sociable numbers (smallest member of cycle).

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A007358 Infinitary multi-perfect numbers.

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A127663 Infinitary aspiring numbers.

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

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