A126171 -id:A126171 - cp's OEIS frontend

A126173 Larger element of a reduced infinitary amicable pair.

2295, 75495, 817479, 1902215, 1341495, 1348935, 2226014, 2421704, 3123735, 3010215, 5644415, 4282215, 7509159, 10106504, 12900734, 24519159, 31356314, 41950359, 43321095, 80870615, 42125144, 85141719, 87689415, 87802407, 86477895, 105993657, 168669879, 129081735

Offset: 1

Ant King, Dec 23 2006

nonn

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			a(3)=817479 because 817479 is the largest member of the third reduced infinitary amicable pair, (573560,817479)

Amiram Eldar, Table of n, a(n) for n = 1..278
Jan Munch Pedersen, Tables of Aliquot Cycles.

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126172, A126174, A126175, A126176.

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; ReducedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] - 1] == n + 1 && n > 1, True, False]; ReducedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], ReducedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] - 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data1 = ReducedInfinitaryAmicablePairList[10^7]; Table[Last[data1[[k]]], {k, 1, Length[data1]}]
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] - 1; If[k > n && infs[k] == n + 1, AppendTo[s, k]], {n, 2, 10^5}]; s (* Amiram Eldar, Jan 22 2019 *)

The values of n for which isigma(m)=isigma(n)=m+n+1, where n>m and isigma(n) is given by A049417(n).

a(15)-a(28) from Amiram Eldar, Jan 22 2019

A126174 Smaller member of an augmented infinitary amicable pair.

Original entry on oeis.org

1252216, 1754536, 2166136, 2362360, 6224890, 7626136, 7851256, 9581320, 12480160, 12494856, 13324311, 15218560, 15422536, 19028296, 29180466, 36716680, 37542190, 40682824, 45131416, 45495352, 56523810, 67195305, 71570296, 80524665, 89740456, 93182440, 101304490

Offset: 1

Ant King, Dec 23 2006

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			a(3)=2166136 because 2166136 is the smaller element of the third augmented infinitary amicable pair, (2166136,2580105).

Amiram Eldar, Table of n, a(n) for n = 1..276
Jan Munch Pedersen, Tables of Aliquot Cycles.

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126173, A126175, A126176.

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; AugmentedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] + 1] == n - 1 && ! properinfinitarydivisorsum[n] + 1 == n, True, False]; AugmentedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], AugmentedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[ Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] + 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data = AugmentedInfinitaryAmicablePairList[10^7]; Table[First[data[[k]]], {k, 1, Length[data]}]
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] + 1; If[k > n && infs[k] == n - 1, AppendTo[s, n]], {n, 2, 10^9}]; s (* Amiram Eldar, Jan 20 2019 *)

The values of m for which isigma(m)=isigma(n)=m+n-1, where mA049417(n).

a(9)-a(27) from Amiram Eldar, Jan 20 2019

A126175 Larger member of an augmented infinitary amicable pair.

Original entry on oeis.org

1483785, 2479065, 2580105, 4895241, 7336455, 9100905, 10350345, 16367481, 17307105, 24829945, 15706090, 27866241, 15439545, 23872185, 53763535, 63075321, 41337555, 60923577, 51394665, 56802249, 110691295, 73809496, 89870985, 82771336, 92586585, 150672921, 108212055

Offset: 1

Ant King, Dec 23 2006

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			a(3)=2580105 because 2580105 is the larger member of the third augmented infinitary amicable pair, (2166136,2580105).

Amiram Eldar, Table of n, a(n) for n = 1..276
Jan Munch Pedersen, Tables of Aliquot Cycles.

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126173, A126174, A126176.

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; AugmentedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] + 1] == n - 1 && ! properinfinitarydivisorsum[n] + 1 == n, True, False]; AugmentedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], AugmentedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[ Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] + 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data = AugmentedInfinitaryAmicablePairList[10^7]; Table[Last[data[[k]]], {k, 1, Length[data]}]
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] + 1; If[k > n && infs[k] == n - 1, AppendTo[s, k]], {n, 2, 10^9}]; s (* Amiram Eldar, Jan 20 2019 *)

The values of n for which isigma(m)=isigma(n)=m+n-1, where n>m and isigma(n) is given by A049417(n).

a(9)-a(27) from Amiram Eldar, Jan 20 2019

A126172 Smaller element of a reduced infinitary amicable pair.

Original entry on oeis.org

2024, 62744, 573560, 1000824, 1173704, 1208504, 1921185, 2140215, 2198504, 2312024, 2580864, 4012184, 5416280, 9247095, 12500865, 13496840, 23939685, 26409320, 34093304, 37324584, 40818855, 52026920, 66275384, 76011992, 79842104, 101366342, 101589320, 106004024

Offset: 1

Ant King, Dec 23 2006

nonn

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			a(3)=573560 because 573560 is the smaller element of the third reduced infinitary amicable pair, (573560, 817479)

Amiram Eldar, Table of n, a(n) for n = 1..278
Jan Munch Pedersen, Tables of Aliquot Cycles.

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126173, A126174, A126175, A126176.

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; ReducedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] - 1] == n + 1 && n > 1, True, False]; ReducedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], ReducedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] - 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data1 = ReducedInfinitaryAmicablePairList[ 10^7]; Table[First[data1[[k]]], {k, 1, Length[data1]}]
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] - 1; If[k > n && infs[k] == n + 1, AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Jan 22 2019 *)

The values of m for which isigma(m)=isigma(n)=m+n+1, where mA049417(n).

a(15)-a(28) from Amiram Eldar, Jan 22 2019

A126176 Number of augmented infinitary amicable pairs (i,j) with i

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 26, 48, 104, 227

Offset: 1

Ant King, Dec 24 2006

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			a(9)=48 because there are 48 augmented infinitary amicable pairs (m,n) with m<n and m<=10^9

Pedersen J. M., Known amicable pairs.

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126172, A126173, A126174, A126175.

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; AugmentedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] + 1] == n - 1 && ! properinfinitarydivisorsum[n] + 1 == n, True, False]; AugmentedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], AugmentedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[ Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] + 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data = AugmentedInfinitaryAmicablePairList[10^7]; Table[Length[Select[data, First[ # ] < 10^k &]], {k, 1, 7}]

augmented infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n-1, with m

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[ # ]]] &]

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

A124663 Number of reduced infinitary amicable pairs (i,j) with i

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 14, 25, 51, 112, 213

Offset: 1

Ant King, Dec 23 2006

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			a(7)=14 because there are 14 reduced infinitary amicable pairs (m,n) with m<n and m<=10^7

Pedersen J. M., Known amicable pairs.

Cf. A126169, A049417, A126168, A037445, A126170, A126171, A126172, A126173, A126174, A126175, A126176.

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; ReducedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] - 1] == n + 1 && n > 1, True, False]; ReducedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], ReducedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] - 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data1 = ReducedInfinitaryAmicablePairList[10^7]; Table[Length[Select[data1, First[ # ] < 10^k &]], {k, 1, 7}]

Reduced infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n+1, with m

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A126173 Larger element of a reduced infinitary amicable pair.

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A126174 Smaller member of an augmented infinitary amicable pair.

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A126175 Larger member of an augmented infinitary amicable pair.

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A126172 Smaller element of a reduced infinitary amicable pair.

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A126176 Number of augmented infinitary amicable pairs (i,j) with i

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

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A127664 Infinitary amicable numbers.

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