A007357 -id:A007357 - cp's OEIS frontend

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440

Offset: 1

Amiram Eldar, Apr 10 2020

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Cf. A049603, A282446, A333926.

Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

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[10^5], recDivSum[#] == 2*# &]

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Original entry on oeis.org

40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770

Offset: 1

Amiram Eldar, Apr 25 2024

			40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.

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

Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.

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}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];

A127664 Infinitary amicable numbers.

Original entry on oeis.org

114, 126, 594, 846, 1140, 1260, 4320, 5940, 7920, 8460, 8640, 10744, 10856, 11760, 12285, 13500, 14595, 17700, 25728, 35712, 43632, 44772, 45888, 49308, 60858, 62100, 62700, 67095, 67158, 71145, 73962, 74784, 79296, 79650, 79750, 83142, 83904, 86400, 88730

Offset: 1

Ant King, Jan 26 2007

nonn

			a(5)=1140 because 1140 is the fifth infinitary amicable number.

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, A127663, A127665, 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[50000],InfinitaryAmicableNumberQ[ # ] &]
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}]]; infs[n_] := Times @@ (fun @@@ FactorInteger[n]) - n; s = {}; Do[k = infs[n]; If[k != n && infs[k] == n, AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Mar 16 2019 *)

Non-infinitary perfect numbers which satisfy A126168(A126168(n)) = n.

More terms from Amiram Eldar, Mar 16 2019

A185088 a(n) = |n^2 - A185079(n)|.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 4, 1, 10, 1, 4, 1, 4, 9, 1, 1, 6, 1, 10, 7, 20, 1, 24, 1, 4, 9, 24, 1, 36, 1, 4, 33, 22, 25, 4, 1, 4, 9, 20, 1, 132, 1, 16, 45, 28, 1, 8, 1, 4, 9, 26, 1, 36, 1, 104, 49, 34, 1, 0, 1, 4, 49, 16, 25, 36, 1, 34, 57, 140, 1, 84, 1, 4, 9, 76, 73, 36, 1, 26, 1, 80, 1, 16, 11, 128, 9, 4, 1, 180, 105, 64, 55, 92, 25, 36

Offset: 2

Vladimir Shevelev, Feb 18 2011

nonn

Zeros a(z)=0 occur at z=6, 60, 120, 360, 816,... For these z, A049417(z) | z^2, but there may be other numbers like 90, 180, 540,... satisfying this divisibility criterion which are not places of zeros (the criterion is necessary, not sufficient), see A185288.

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

Cf. A185079, A050376, A064380, A049417, A007357, A185078, A185373, A185383.

a(A050376(n)) = 1.

A185288 Numbers n for which the terms of the multiplicative sequence {n^2/A049417(n)} are integers.

Original entry on oeis.org

1, 6, 60, 90, 120, 180, 360, 540, 816, 840, 1080, 1740, 1980, 2280, 2520, 3060, 3960, 5712, 6120, 8280, 9540, 11880, 12240, 16920, 18360, 19260, 24480, 25296, 25560, 32760, 36720, 42840, 48960, 54672, 57240, 63700, 73440, 74256, 84360, 85680, 97920, 103320, 115560

Offset: 1

Vladimir Shevelev, Feb 20 2011

nonn

The sequence contains all infinitary perfect numbers (see A007357).

			Let n=120. Its representation over distinct terms of A050376 is 2*3*4*5. Therefore A049417(n)=(2+1)*(3+1)*(4+1)*(5+1)=360. Since 360 is a divisor of 120^2, 120 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007357, A049417, A064380, A185079, A185088, A185098, A185373, A185383.

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]; aQ[n_] := Divisible[n^2, isigma[n]]; Select[Range[58000], aQ] (* Amiram Eldar, Jul 21 2019 *)

More terms from Nathaniel Johnston, Mar 16 2011

More terms from Amiram Eldar, Jul 21 2019

A331108 Zeckendorf-infinitary perfect numbers: numbers k such that A331107(k) = 2*k.

Original entry on oeis.org

6, 60, 90, 3024, 133056, 1330560, 6879600, 28828800, 302702400, 698544000, 11763214848

Offset: 1

Amiram Eldar, Jan 09 2020

No more terms below 4*10^10.

			6 is a term since A331107(6) = 12 = 2*6.

Cf. A007357, A038182, A074849, A097464, A331107.

fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Fibonacci[1 + Position[Reverse@fr, ?(# == 1 &)]]]; f[p, e_] := p^fb[e]; zsigma[1] = 1; zsigma[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); zPerfectQ[n_] := zsigma[n] == 2 n; Select[Range[10^4], zPerfectQ] (* after Robert G. Wilson v at A014417 *)

A309843 Numbers m that equal the sum of their first k consecutive aliquot infinitary divisors, but not all of them (i.e k < A037445(m) - 1).

Original entry on oeis.org

24, 360, 4320, 14688, 1468800, 9547200, 50585472, 54198720, 189695520, 1680459264

Offset: 1

Amiram Eldar, Sep 14 2019

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

If all the aliquot infinitary divisors are permitted (i.e. k <= A037445(n) - 1), then the infinitary perfect numbers (A007357) are included.

			24 is in the sequence since its aliquot infinitary divisors are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24.

Cf. A007357, A077609, A037445, A049417, A194472, A293618.

idivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[ x ] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[idivs[n], -2]]; s= {}; Do[If[selDivs[n] == 0, AppendTo[s, n]], {n, 2, 10^6}]; s(* after Alonso del Arte at A194472 *)

A331111 Dual-Zeckendorf-infinitary perfect numbers: numbers k such that A331110(k) = 2*k.

Original entry on oeis.org

6, 60, 90, 655200, 28828800, 238140000, 10478160000

Offset: 1

Amiram Eldar, Jan 09 2020

No more terms below 2.8*10^10.

			6 is a term since A331110(6) = 12 = 2*6.

Cf. A007357, A038182, A074849, A097464, A331108, A331110.

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeck[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, {}, v[[i[[1, 1]] ;; -1]]]];
f[p_, e_] := p^Fibonacci[1 + Position[Reverse @ dualZeck[e], _?(# == 1 &)]];
dzsigma[1] = 1; dzsigma[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); seqQ[n_] := dzsigma[n] == 2n; Select[Range[10^6], seqQ]

A335937 Infinitary pseudoperfect numbers (A306983) that equal to the sum of a subset of their aliquot infinitary divisors in a single way.

Original entry on oeis.org

6, 60, 72, 78, 88, 90, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 426, 438, 474, 486, 498, 534, 582, 606, 618, 642, 654, 678, 726, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194

Offset: 1

Amiram Eldar, Jun 30 2020

nonn

			72 is a term since its set of infinitary aliquot divisors is {1, 2, 4, 8, 9, 18, 36}, and {1, 8, 9, 18, 36} is its only subset whose sum is equal to 72.

Amiram Eldar, Table of n, a(n) for n = 1..3000

The infinitary version of A064771.
Subsequence of A306983.
A007357 is a subsequence.
Similar sequences: A295829, A295830, A321145.
Cf. A049417, A077609, A335199.

idivs[x_] := If[x == 1, 1, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infpspQ[n_] := Module[{d = Most @ idivs[n], x}, Plus @@ d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 1]; Select[Range[2, 1200], infpspQ]

A360524 Numbers k such that A360522(k) = 2*k.

Original entry on oeis.org

6, 12, 198, 240, 264, 270, 396, 540, 6720, 7920, 11880, 13770, 27540, 221760, 337440, 605880, 2500344, 6072570, 11135520, 12145140, 267193080, 441692160, 1112629770, 2225259540, 14575841280, 48955709880

Offset: 1

Amiram Eldar, Feb 10 2023

Analogous to perfect numbers (A000396) with A360522 instead of A000203.

a(27) > 10^11, if it exists.

			6 is a term since A360522(6) = 12 = 2 * 6.

Cf. A000203, A360522.

Similar sequences: A000396, A002827, A007357, A054979, A322486, A324707.

f[p_, e_] := p^e + e; q[n_] := Times @@ f @@@ FactorInteger[n] == 2*n; Select[Range[10^6], q]

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

cp's OEIS Frontend

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A127664 Infinitary amicable numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185088 a(n) = |n^2 - A185079(n)|.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A185288 Numbers n for which the terms of the multiplicative sequence {n^2/A049417(n)} are integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A331108 Zeckendorf-infinitary perfect numbers: numbers k such that A331107(k) = 2*k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A309843 Numbers m that equal the sum of their first k consecutive aliquot infinitary divisors, but not all of them (i.e k < A037445(m) - 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A331111 Dual-Zeckendorf-infinitary perfect numbers: numbers k such that A331110(k) = 2*k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A335937 Infinitary pseudoperfect numbers (A306983) that equal to the sum of a subset of their aliquot infinitary divisors in a single way.

Views

Author