A126173 -id:A126173 - cp's OEIS frontend

A126174 Smaller member of an augmented infinitary amicable pair.

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

A306871 Larger of reduced bi-unitary amicable pair.

Original entry on oeis.org

2295, 20735, 75495, 1148735, 817479, 774375, 1902215, 1341495, 1348935, 2226014, 2421704, 3123735, 3010215, 5644415, 3894344, 4282215, 4994055, 7509159, 12251679, 10106504, 12900734, 20444319, 24519159, 28206815, 31356314, 33362175, 41950359, 36129375, 43321095

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

A pair m < n are a reduced bi-unitary amicable pair if bsigma(m) = bsigma(n) = m + n + 1, where bsigma(n) is the sum of bi-unitary divisors of n (A188999).

The terms are ordered according to their lesser counterparts (A306870).

			2295 is in the sequence since it is the larger of the amicable pair (2024, 2295): bsigma(2024) = bsigma(2295) = 4320 = 2024 + 2295 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..528 (terms with lesser member below 2*10^11, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers.

Cf. A003503, A126173, A188999, A292981, A306870.

fun[p_, e_]:=If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); f[n_] := bsigma[n] - n - 1; s={}; Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, m]], {n, 1, 10^7}]; s

A306876 Larger of reduced unitary amicable pair.

Original entry on oeis.org

175742294, 6263852385, 16082297385, 18625120185, 32553626105, 38947446285, 37626449685, 41194817265, 103052922665, 87988279533, 103552755405, 126755126589, 131943742994, 192245655405, 226960409585, 181521732405, 502566224565, 399451768365, 403080683565, 461943100905

Offset: 1

Amiram Eldar, Mar 14 2019

nonn

A pair m < n is a reduced unitary amicable pair if usigma(m) = usigma(n) = m + n + 1, where usigma(n) is the sum of unitary divisors of n (A034460). The terms are ordered according to their lesser counterparts (A306875).

			175742294 is in the sequence since it is the larger of the amicable pair (172622505, 175742294): usigma(172622505) = usigma(175742294) = 348364800 = 172622505 + 175742294 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..28 (terms with lesser member below 10^12, from David Moews's site).
David Moews, Perfect, amicable and sociable numbers
J. O. M. Pedersen, Tables of Aliquot Cycles

Cf. A003503, A002953, A126173, A034460, A306875.

us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  s={}; Do[m = us[n] - 1; If[m > n && us[m] == n + 1, AppendTo[s, m]], {n, 1, 10^9}]; s

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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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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A306871 Larger of reduced bi-unitary amicable pair.

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A306876 Larger of reduced unitary amicable pair.

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

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