A037445 -id:A037445 - cp's OEIS frontend

A382291 a(n) = A037445(n)/A034444(n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Mar 21 2025

First differs from A368168 at n = 64, and from A359411, A367516 and A368979 at n = 128.

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

Cf. A000120, A034444, A037445, A138302, A243036, A382290, A382292.

Cf. A359411, A367516, A368168, A368979.

f[p_, e_] := 2^(DigitCount[e, 2, 1] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 105]

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

a(n) = 2^A382290(n).
Multiplicative with a(p^e) = 2^(A000120(e)-1) = A048896(e-1) (= A243036(e) for e >= 2).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) = 2 if and only if n is in A382292.

A260084 Infinite sequence starting with a(0)=0 such that A(a(k)) = a(k-1) for all k>=1, where A(n) = n - A037445(n).

Original entry on oeis.org

0, 2, 6, 10, 14, 18, 22, 30, 34, 42, 46, 54, 58, 62, 70, 78, 82, 90, 94, 102, 106, 114, 118, 122, 130, 138, 142, 146, 154, 158, 162, 166, 174, 182, 190, 194, 210, 214, 222, 230, 238, 242, 250, 254, 270, 274, 278, 286, 294, 298, 302, 310, 314, 330, 334, 342, 346, 354, 358, 366, 374, 390, 394, 402, 410, 418, 426, 434, 442

Offset: 0

Vladimir Shevelev, Jul 15 2015

nonn

The first infinitary analog (see also A260124) of A259934 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.
All the first differences are powers of 2 (A260085). The infinitary case is interesting because here we have at least two analogs of sequences A259934, A259935 (respectively A260084, A260124 and A260085, A260123).
It is a corollary of the fact that all terms of A037445, except for n=1, are even (powers of 2). Therefore, in the analogs of A259934 we can begin with not only 0,2 (as in this sequence), but also with 0,1 (as in A260124). Then this sequence contains only the even terms, while A260124 - only the odd ones.
A generalization. For an even m, the multiplication of A260124 by 2^m and 2^(m+1) gives two infinite solutions of the system of equations for integer x_n, n>=1: A037445(x_1 + ... + x_n) = x_n/2^A005187(m), n>=1. In particular, for m=0, we obtain A260124 and A260084.

Cf. A037445, A259934, A259935, A260085, A260124.

a(n) = 2 * A260124(n).

A260123 The second infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1 (see also A260085).

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, Jul 17 2015

nonn

The second after A260085 infinitary analog of A259935 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.

All terms are powers of 2.

Cf. A037445, A259935, A260085.

A327573 Partial sums of the number of infinitary divisors function: a(n) = Sum_{k=1..n} id(k), where id is A037445.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 41, 43, 45, 49, 51, 55, 59, 63, 65, 73, 75, 79, 83, 87, 89, 97, 99, 103, 107, 111, 115, 119, 121, 125, 129, 137, 139, 147, 149, 153, 157, 161, 163, 167, 169, 173, 177, 181, 183, 191, 195, 203, 207, 211, 213, 221

Offset: 1

Amiram Eldar, Sep 17 2019

nonn

Differs from A306069 at n >= 16.

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A037445, A327576.

Cf. A006218 (all divisors), A064608 (unitary), A306069 (bi-unitary), A145353 (exponential).

f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ (f @@@ FactorInteger[n]); Accumulate[Array[id, 100]]

a(n) ~ 2 * c * n * log(n), where c = 0.366625... (A327576). [Corrected by Amiram Eldar, May 07 2021]

A327576 Decimal expansion of the constant that appears in the asymptotic formula for average order of the number of infinitary divisors function (A037445).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 17 2019

			0.366625276945381909556532720687001563033612155971009...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A037445, A050376, A327573.

Cf. A059956 (corresponding constant for unitary divisors), A306071 (bi-unitary).

m = 1000; em = 10; f[x_] := Sum[Log[1 - 1/(1 + 1/x^(2^e))^2], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; $MaxExtraPrecision = 1500; RealDigits[(1/2)*Exp[f[1/2] + f[1/3]]* Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - (1/2)^k - (1/3)^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals Limit_{n->oo} A327573(n)/(2 * n * log(n)). [Corrected by Amiram Eldar, May 07 2021]

Equals (1/2) * Product_{P} (1 - 1/(P+1)^2), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

A260085 Infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1.

Original entry on oeis.org

2, 4, 4, 4, 4, 4, 8, 4, 8, 4, 8, 4, 4, 8, 8, 4, 8, 4, 8, 4, 8, 4, 4, 8, 8, 4, 4, 8, 4, 4, 4, 8, 8, 8, 4, 16, 4, 8, 8, 8, 4, 8, 4, 16, 4, 4, 8, 8, 4, 4, 8, 4, 16, 4, 8, 4, 8, 4, 8, 8, 16, 4, 8, 8, 8, 8, 8, 8, 4, 16, 4, 8, 4, 8, 8, 16, 4, 8, 4, 8, 4, 4, 8, 8, 4, 4, 8, 4, 16, 8, 8, 4, 16, 4, 8, 8, 8, 4, 8, 4, 16, 4, 4, 16

Offset: 1

Vladimir Shevelev, Jul 15 2015

nonn

The first infinitary analog (see also A260123) of A259935 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.

All terms are powers of 2.

Cf. A037445, A259935, A260084, A260123.

a(n) = 2 * A260123(n).

A273011 Numbers n such that d_i(n) >= d_i(k) for k = 1 to n-1, where d_i(n) is the number of infinitary divisors of n (A037445).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 30, 40, 42, 54, 56, 60, 66, 70, 72, 78, 84, 88, 90, 96, 102, 104, 105, 108, 110, 114, 120, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 420, 440, 456, 462, 480, 504, 510, 520, 540, 546, 552, 570, 594, 600, 616

Offset: 1

Vladimir Shevelev, May 13 2016

nonn

An infinitary (or Fermi-Dirac) analog of the Ramanujan sequence A067128.
Between the smallest number b_k which is product of k distinct terms of A050376 and b_(k+1) all terms are products of k distinct terms of A050376.
Thus every subsequence of terms, having in Fermi-Dirac factorization a fixed number of distinct factors from A050376, is finite.
These subsequences have cardinalities: 1, 4, 10, 21, 47, ...

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

Cf. A037445, A064547, A050376, A067128.

a = {}; b = {0}; Do[If[# >= Max@b, AppendTo[a, k] && AppendTo[b, #]] &@ If[k == 1, 1, Times @@ Flatten@ Map[2^First@ DigitCount[#, 2] &, FactorInteger[k][[All, 2]]]], {k, 10^3}]; a (* Michael De Vlieger, May 13 2016, after Jean-François Alcover at A037445 *)

A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, -11, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, -5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Aug 22 2018

Multiplicative because A037445 is.

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

Cf. A037445, A317934 (denominators).

Cf. also A317933, A317940.

up_to = 1+(2^16);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ From A037445
v317941aux = DirSqrt(vector(up_to, n, A037445(n)));
A317941(n) = numerator(v317941aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A037445(n) - Sum_{d|n, d>1, d 1.

A260124 The second infinite sequence starting with a(0)=0 such that A(a(k)) = a(k-1) for all k>=1, where A(n) = n - A037445(n) (cf. A260084).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 15, 17, 21, 23, 27, 29, 31, 35, 39, 41, 45, 47, 51, 53, 57, 59, 61, 65, 69, 71, 73, 77, 79, 81, 83, 87, 91, 95, 97, 105, 107, 111, 115, 119, 121, 125, 127, 135, 137, 139, 143, 147, 149, 151, 155, 157, 165, 167, 171, 173, 177, 179, 183, 187, 195, 197, 201, 205, 209, 213, 217, 221, 223, 231, 233, 237, 239, 243, 247, 255, 257, 261, 263, 267, 269, 271, 275, 279, 281, 283, 287, 289, 297, 301, 305

Offset: 0

Vladimir Shevelev, Jul 17 2015

nonn

The second infinitary analog (after A260084) of A259934 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.
All the first differences are powers of 2 (A260123).
See also comment in A260084.

Cf. A037445, A259934, A260084, A260123.

a(n) = A260084(n)/2.

A282843 Numbers k for which id(k) = id(k + id(k)), where id(k) = A037445(k) is the number of infinitary divisors of k.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 9, 10, 11, 14, 17, 18, 22, 23, 28, 29, 32, 34, 35, 41, 44, 46, 47, 48, 51, 58, 59, 64, 65, 70, 71, 76, 79, 81, 82, 87, 88, 91, 94, 95, 96, 101, 102, 107, 111, 112, 115, 118, 119, 125, 128, 129, 130, 132, 137, 141, 142, 143, 144, 149, 152, 155

Offset: 1

Vladimir Shevelev, Feb 22 2017

nonn

The infinitary analog of A175304. The sequence contains smaller of pairs(p,q) of terms of A050376 if q-p=2 (Fermi-Dirac twin primes). Unlike A175304, this sequence contains some squares (64,81,144,...).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037445, A050376, A175304.

f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], id[#] == id[# + id[#]] &] (* Amiram Eldar, Apr 11 2025 *)

id(n)=2^vecsum(apply(hammingweight, factor(n)[,2]))
is(n)=my(i=id(n)); id(n+i)==i \\ Charles R Greathouse IV, Feb 22 2017

More term from Peter J. C. Moses, Feb 22 2017

cp's OEIS Frontend

A382291 a(n) = A037445(n)/A034444(n).

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A260084 Infinite sequence starting with a(0)=0 such that A(a(k)) = a(k-1) for all k>=1, where A(n) = n - A037445(n).

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A260123 The second infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1 (see also A260085).

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A327573 Partial sums of the number of infinitary divisors function: a(n) = Sum_{k=1..n} id(k), where id is A037445.

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A327576 Decimal expansion of the constant that appears in the asymptotic formula for average order of the number of infinitary divisors function (A037445).

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A260085 Infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1.

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A273011 Numbers n such that d_i(n) >= d_i(k) for k = 1 to n-1, where d_i(n) is the number of infinitary divisors of n (A037445).

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A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

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A260124 The second infinite sequence starting with a(0)=0 such that A(a(k)) = a(k-1) for all k>=1, where A(n) = n - A037445(n) (cf. A260084).

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