A077609 -id:A077609 - cp's OEIS frontend

A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).

24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 96, 102, 104, 114, 120, 138, 150, 174, 186, 222, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 678, 726, 762, 780, 786, 822, 834, 894, 906, 942

Offset: 1

Amiram Eldar, May 18 2020

nonn

Equivalently, numbers that are equal to the sum of their proper infinitary divisors, with one of them taken with a minus sign.

Admirable numbers (A111592) whose number of divisors is a power of 2 (A036537) are also infinitary admirable numbers, since all of their divisors are infinitary. Terms with number of divisors that is not a power of 2 are 96, 150, 294, 360, 420, 486, 540, 630, 660, 726, 780, 960, 990, ...

			150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper infinitary divisors with one of them, 6, taken with a minus sign.

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

The infinitary version of A111592.
Subsequence of A129656.
Cf. A036537, A049417, A077609, A328328, A334972.

fun[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 @@ fun @@@ FactorInteger[n]; infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e; infAdmQ[n_] := (ab = isigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && infDivQ[n, ab/2]; Select[Range[1000], infAdmQ]

A335199 Infinitary Zumkeller numbers (A335197) whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

Original entry on oeis.org

6, 56, 60, 70, 72, 88, 90, 104, 3040, 3230, 3770, 4030, 4510, 5170, 5390, 5800, 5830, 6808, 7144, 7192, 7400, 7912, 8056, 8968, 9272, 9656, 9928, 10744, 10792, 11016, 11096, 11288, 11392, 12104, 12416, 12928, 13184, 13192, 13696, 13736, 13952, 14008, 14464, 14552

Offset: 1

Amiram Eldar, May 26 2020

nonn

			6 is a term since there is only one partition of its set of nonunitary divisors, {1, 2, 3, 6}, into two disjoint sets of equal sum: {1, 2, 3} and {6}.

The infinitary version of A083209.

Subsequence of A335197.

infdivs[n_] := If[n == 1, {1}, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infZumQ[n_] := Module[{d = infdivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]]; Select[Range[15000], infZumQ] (* after Michael De Vlieger at A077609 *)

A360721 a(n) is the number of infinitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 18 2023

Cf. A000120, A001694, A037445, A077609, A360723.

Similar sequences: A005361 (number of powerful divisors), A323308 (number of unitary powerful divisors).

f[p_, e_] := 2^DigitCount[e, 2, 1] - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^hammingweight(f[i, 2]) - f[i, 2]%2);}

Multiplicative with a(p^e) = 2^A000120(e) - (e mod 2).
a(n) <= A037445(n) with equality if and only if n is a square.
a(n) <= A005361(n) with equality if and only if n is not in A360723.
Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} ((1-1/p) * Sum_{k>=1} ((2^A000120(k)- k mod 2)/p^k)) = 1.72717... .

A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 128, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 256, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 09 2023

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).

Similar sequences: A099377, A103339.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1)))); }

a(n) = numerator(n*A037445(n)/A049417(n)).
a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.
a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.

A361317 Denominators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 15, 5, 9, 6, 5, 7, 3, 2, 17, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 51, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 17, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 85, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57

Offset: 1

Amiram Eldar, Mar 09 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A037445, A049417, A077609, A063947 (positions of 1's), A361316 (numerators).

Similar sequences: A099378, A103340.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; a[1] = 1; a[n_] := Denominator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); denominator(n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1)))); }

a(n) = denominator(n*A037445(n)/A049417(n)).

A363329 a(n) is the number of divisors of n that are both coreful and infinitary.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 28 2023

For the definition of a coreful divisor see A307958, and for the definition of an infinitary divisor see A037445.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an infinitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 such that the bitwise AND of e and k is equal to k.
The least term that does not equal 1 or 3 is a(128) = 7.
The range of this sequence is A282572.

			a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are infinitary and 3 of them (2, 4 and 8) are also coreful.

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

Cf. A000120, A005361 (number of coreful divisors), A007947, A037445, A077609, A138302, A282572, A359411.

f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = factorback(apply(x -> 2^hammingweight(x) - 1, factor(n)[, 2]));

from math import prod
from sympy import factorint
def A363329(n): return prod((1<Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = 2^A000120(e) - 1.
a(n) = 1 is and only if n is in A138302.
a(n) >= A359411(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (-1/p + (1-1/p)*Product_{k>=0} (1 + 2/p^(2^k))) = 1.29926464312956239535... .

A365296 The smallest coreful infinitary divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 6, 25, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 6, 55, 14, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Aug 31 2023

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.
The number of coreful infinitary divisors of n is A363329(n).
All the terms are in A138302.

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

Cf. A006519, A037445, A077609, A007947, A034444, A138302, A268335, A302792, A339597, A363329.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

from math import prod
from sympy import factorint
def A365296(n): return prod(p**(e&-e) for p, e in factorint(n).items()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = p^A006519(e).
a(n) = n if and only if n is in A138302.
a(n) >= A007947(n) with equality if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.4459084041..., where f(k) = 2*k - A006519(k) = A339597(k-1).
A037445(a(n)) = A034444(n). - Amiram Eldar, Oct 19 2023

A366538 The number of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 8, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 12 2023

Also, the number of infinitary divisors of the terms of A138302, since A138302 is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.

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

Cf. A034444, A037445, A077609, A077610, A138302, A366539.

Similar sequences: A366534, A366536.

f[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(2^#e, ", ")));

a(n) = A034444(A138302(n)).

a(n) = A037445(A138302(n)).

A374786 Numerator of the mean infinitary abundancy index of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 45, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 105, 51, 135, 133, 135, 59, 77, 63, 165, 161, 175, 33, 19, 75, 195, 63, 99, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 665, 23, 675, 91, 295, 119, 231, 123, 315

Offset: 1

Amiram Eldar, Jul 20 2024

The infinitary abundancy index of a number k is A049417(k)/k.
The record values of a(n)/A374787(n) are attained at the terms of A037992.
The least number k such that a(k)/A374787(k) is larger than 2, 3, 4, ..., is A037992(6) = 7560, A037992(33) = 1370819010042780920891599455129161859473627856000, ... .

			For n = 4, 4 has 2 infinitary divisors, 1 and 4. Their infinitary abundancy indices are isigma(1)/1 = 1 and isigma(4)/4 = 5/4, and their mean infinitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

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

Cf. A005117, A037445, A037992, A049417, A050376, A077609, A138302, A327574, A374787 (denominators).

Similar sequences: A374777/A374778, A374783/A374784.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Numerator[Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n])))]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(prod(i = 1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1 + 1/(2*f[i, 1]^(2^(#b-k))), 1))));}

Let f(n) = a(n)/A374787(n). Then:
f(n) = (Sum_{d infinitary divisor of n} isigma(d)/d) / id(n), where isigma(n) is the sum of infinitary divisors of n (A049417), and id(n) is their number (A037445).
f(n) is multiplicative with f(p^e) = Product{k>=1, e_k=1} (1 + 1/(2*p^(2^(k+1)))), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
f(n) = (Sum_{d infinitary divisor of n} d*id(d)) / (n*id(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{P} (1 + 1/(2*P*(P+1))) = 1.21407233718434377029..., where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376). For comparison, the asymptotic mean of the infinitary abundancy index over all the positive integers is 1.461436... = 2 * A327574.
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).
f(n) >= A374783(n)/A374784(n) with equality if and only if n is in A138302.

A374787 Denominator of the mean infinitary abundancy index of the infinitary divisors of n.

Original entry on oeis.org

1, 4, 6, 8, 10, 24, 14, 32, 18, 8, 22, 16, 26, 56, 60, 32, 34, 72, 38, 80, 4, 88, 46, 64, 50, 104, 108, 112, 58, 48, 62, 128, 132, 136, 28, 16, 74, 152, 52, 64, 82, 16, 86, 176, 180, 184, 94, 64, 98, 40, 204, 208, 106, 432, 20, 448, 76, 232, 118, 160, 122, 248

Offset: 1

Amiram Eldar, Jul 20 2024

			For n = 4, 4 has 2 infinitary divisors, 1 and 4. Their infinitary abundancy indices are isigma(1)/1 = 1 and isigma(4)/4 = 5/4, and their mean infinitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = denominator(9/8) = 8.

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

Cf. A037445, A049417 (isigma), A077609, A374786 (numerators).

Similar sequences: A374777/A374778, A374783/A374784.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Denominator[Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n])))]; Array[a, 100]

a(n) = {my(f = factor(n), b); denominator(prod(i = 1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1 + 1/(2*f[i, 1]^(2^(#b-k))), 1))));}

cp's OEIS Frontend

A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2*d = 2*k, where isigma is the sum of infinitary divisors function (A049417).

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A335199 Infinitary Zumkeller numbers (A335197) whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

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A360721 a(n) is the number of infinitary divisors of n that are powerful (A001694).

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A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

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A361317 Denominators of the harmonic means of the infinitary divisors of the positive integers.

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A363329 a(n) is the number of divisors of n that are both coreful and infinitary.

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A365296 The smallest coreful infinitary divisor of n.

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A366538 The number of unitary divisors of the exponentially 2^n-numbers (A138302).

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A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).