A077609 -id:A077609 - cp's OEIS frontend

A331109 The number of dual-Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the dual Zeckendorf expansion (A104326) of each s(i) contains only terms that are in the dual Zeckendorf expansion of r(i).

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

Offset: 1

Amiram Eldar, Jan 09 2020

Dual-Zeckendorf-infinitary divisors are analogous to infinitary divisors (A077609) with dual Zeckendorf expansion instead of binary expansion.

First differs from A286324 at n = 32.

			a(32) = 4 since 32 = 2^5 and the dual Zeckendorf expansion of 5 is 110, i.e., its dual Zeckendorf representation is a set with 2 terms: {2, 3}. There are 4 possible exponents of 2: 0, 2, 3 and 5, corresponding to the subsets {}, {2}, {3} and {2, 3}. Thus 32 has 4 dual-Zeckendorf-infinitary divisors: 2^0 = 1, 2^2 = 4, 2^3 = 8, and 2^5 = 32.

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

Cf. A037445, A038148, A074848, A077609, A104326, A112310, A286324, A318465.

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 == {}, 1, 2^Total[v[[i[[1, 1]] ;; -1]]]]];
f[p_, e_] := dualZeck[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = 2^A112310(e).

A349258 a(n) is the number of prime powers (not including 1) that are infinitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 3, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 4, 3

Offset: 1

Amiram Eldar, Nov 12 2021

The total number of prime powers (not including 1) that divide n is A001222(n).

For each n, all the prime powers that are infinitary divisors of n are "Fermi-Dirac primes" (A050376).

			12 has 4 infinitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

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

Cf. A000120, A000961, A001222, A036537, A037445, A050376, A077609, A077761, A246655.

f[p_,e_] := 2^DigitCount[e, 2, 1] - 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a,100]

A349258(n) = if(1==n,0,vecsum(apply(x->(2^hammingweight(x))-1,factor(n)[,2]))); \\ Antti Karttunen, Nov 12 2021

Additive with a(p^e) = 2^A000120(e) - 1.
a(n) <= A001222(n), with equality if and only if n is in A036537.
a(n) <= A037445(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.28135949730844648114..., where f(x) = -(x+1) + (1-x) * Product_{k>=0} (1 + 2*x^(2^k)). - Amiram Eldar, Sep 29 2023

Wrong comment removed by Amiram Eldar, Sep 22 2023

A358260 a(n) is the number of infinitary square divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 06 2022

First differs from A007424 at n = 36, from A323308 at n = 64, and from A278908 and A307848 at n = 128.

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

Cf. A000120, A048881, A037445, A077609, A358261.
Similar sequences: A046951, A056624, A056626.
Sequences with the same initial terms: A007424, A278908, A307848, A323308.

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

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

Multiplicative with a(p^e) = 2^A000120(e) if e is even, and 2^A000120(e-1) if e is odd.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((1-1/p) * Sum_{k>=1} a(p^k)/p^k) = 1.55454884667440993654... .

A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 3, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 19 2023

First differs from A295878 at n = 32.

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

Cf. A000005, A000120, A005117, A037445, A077609, A268335, A295878, A358260.

f[p_, e_] := 1 + If[OddQ[e], 2^DigitCount[e-1, 2, 1], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = 1 if e is even, and a(p^e) = 2^A000120(e-1) + 1 if e is odd.

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

A302792 a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 4, 1, 5, 1, 4, 1, 7, 5, 1, 1, 9, 1, 5, 7, 11, 1, 12, 1, 13, 9, 7, 1, 15, 1, 16, 11, 17, 7, 9, 1, 19, 13, 20, 1, 21, 1, 11, 9, 23, 1, 16, 1, 25, 17, 13, 1, 27, 11, 28, 19, 29, 1, 20, 1, 31, 9, 16, 13, 33, 1, 17, 23, 35, 1, 36, 1, 37, 25, 19, 11, 39, 1, 16, 1, 41, 1, 28, 17, 43, 29, 44, 1, 45, 13, 23, 31, 47, 19, 48, 1

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

The positive integers that are absent from this sequence are A036554, integers that have 2 as a Fermi-Dirac factor. - Peter Munn, Apr 23 2018

a(n) is the largest aliquot infinitary divisor of n, for n > 1 (cf. A077609). - Amiram Eldar, Nov 19 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001511, A036554, A050376, A052331, A077609, A223490, A302023, A302024, A302776, A300841.
Cf. A084400 (gives the positions of 1's).
Cf. also A032742.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := n / Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 27 2020 *)

up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A001511(n) = 1+valuation(n,2);
A223490(n) = if(1==n,n,A050376(A001511(A052331(n))));
A302792(n) = (n/A223490(n));

a(n) = {if(n==1, 1, my(f = factor(n)); for(i=1, #f~, f[i,1] = f[i,1]^(1<Amiram Eldar, Nov 19 2022

a(n) = n / A223490(n).

a(n) = A300841(A302023(A032742(A302024(n)))).

A323344 Numbers k whose infinitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

2394, 7544, 10184, 1452330, 2154584, 5021912, 5747994, 5771934, 5786298, 5800662, 5834178, 5843754, 5858118, 5886846, 5905998, 5920362, 5929938, 5992182, 6035274, 6059214, 6078366, 6087942, 6102306, 6107094, 6121458, 6174126, 6202854, 6207642, 6245946, 6265098

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The infinitary version of A171641.

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

Cf. A049417, A077609, A129656, A171641, A323341, A323342, A323343.

infdivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ; 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[n_] := If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]; seq={}; Do[s=isigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = infdivs[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 100000}]; seq (* after Michael De Vlieger at A077609 *)

A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504

Offset: 1

Amiram Eldar, May 26 2020

nonn

			6 is a term since its set of infinitary divisors, {1, 2, 3, 6}, can be partitioned into the two disjoint sets, {1, 2, 3} and {6}, whose sum is equal: 1 + 2 + 3 = 6.

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

The infinitary version of A083207.
Subsequence of A129656.
Cf. A077609, A323344.

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]] > 0]]; Select[Range[500], infZumQ] (* after Michael De Vlieger at A077609 *)

A361175 The sum of the exponential infinitary divisors of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67

Offset: 1

Amiram Eldar, Mar 03 2023

First differs from A322857 at n = 256.
The exponential infinitary divisors of n = Product_i p(i)^e(i) are all the numbers of the form Product_i p(i)^d(i) where d(i) is an infinitary divisor of e(i).
The number of exponential infinitary divisors of n is A307848(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT] (2014).
Index entries for sequences related to divisors of numbers.

Cf. A077609, A307848.

Similar sequences: A051377, A322857, A323309, A361174.

idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
f[p_, e_] := Total[p^idivs[e]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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); } \\ Michel Marcus at A077609
ff(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), p^d, 0));
a(n) = {my(f=factor(n)); prod(i=1, #f~, ff(f[i, 1], f[i, 2])); }

Multiplicative with a(p^e) = Sum_{d infinitary divisor of e} p^d.

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[#] == # &]

A331973 a(n) is the number of values of m such that the sum of proper infinitary divisors of m (A126168) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 4, 1, 4, 3, 4, 3, 5, 0, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 6, 2, 5, 2, 5, 3, 7, 1, 6, 2, 4, 2, 7, 1, 5, 4, 5, 3, 8, 0, 5, 2, 6, 1, 8, 2, 5, 4, 6, 4, 9, 0, 6, 1, 5, 3, 10, 2, 8, 2

Offset: 2

Amiram Eldar, Feb 03 2020

nonn

The infinitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A126168(k) = 1.

			a(8) = 2 since 8 is the sum of the proper infinitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Amiram Eldar, Table of n, a(n) for n = 2..30000

Cf. A048138, A049417, A077609, A126168, A324938, A331971.

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]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; Rest@v

cp's OEIS Frontend

A331109 The number of dual-Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the dual Zeckendorf expansion (A104326) of each s(i) contains only terms that are in the dual Zeckendorf expansion of r(i).

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A349258 a(n) is the number of prime powers (not including 1) that are infinitary divisors of n.

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A358260 a(n) is the number of infinitary square divisors of n.

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A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

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A302792 a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).

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A323344 Numbers k whose infinitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

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A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

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A361175 The sum of the exponential infinitary divisors of n.

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