A048691 -id:A048691 - cp's OEIS frontend

A061391 a(n) = t(n,3) = Sum_{d|n} tau(d^3), where tau(n) = number of divisors of n, cf. A000005.

1, 5, 5, 12, 5, 25, 5, 22, 12, 25, 5, 60, 5, 25, 25, 35, 5, 60, 5, 60, 25, 25, 5, 110, 12, 25, 22, 60, 5, 125, 5, 51, 25, 25, 25, 144, 5, 25, 25, 110, 5, 125, 5, 60, 60, 25, 5, 175, 12, 60, 25, 60, 5, 110, 25, 110, 25, 25, 5, 300, 5, 25, 60, 70, 25, 125, 5, 60, 25, 125, 5, 264

Offset: 1

Vladeta Jovovic, Apr 29 2001

Inverse Mobius transform of A048785. - R. J. Mathar, Feb 09 2011

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. t(n, 0) = A000005(n), t(n, 1) = A007425(n), t(n, 2) = A035116(n).

Cf. A048691.

f[p_, e_] := (3*e^2 + 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

A061391 = n -> sumdiv(n, d, numdiv(d^3));
for(n=1, 10000, write("b061391.txt", n, " ", A061391(n)));
\\ Antti Karttunen, Jan 17 2017

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

t(n, k) = Sum_{d|n} tau(d^k) is multiplicative: if the canonical factorization of n = Product p^e(p) over primes then t(n, k) = Product t(p^e(p), k), t(p^e(p), k) = (1/2) *(k*e(p)+2)*(e(p)+1).
For k=2 we get an interesting identity: Sum_{d|n} tau(d^2)=(tau(n))^2, cf. A048691, A035116.
a(n) = Sum_{d|n} tau(n*d). - Benoit Cloitre, Nov 30 2002
G.f.: Sum_{n>=1} tau(n^3)*x^n/(1-x^n). - Joerg Arndt, Jan 01 2011
Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 + 2/p^s). - Vaclav Kotesovec, May 15 2021
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers.

Original entry on oeis.org

1, 4, 4, 7, 4, 13, 4, 10, 7, 13, 4, 22, 4, 13, 13, 13, 4, 22, 4, 22, 13, 13, 4, 31, 7, 13, 10, 22, 4, 40, 4, 16, 13, 13, 13, 37, 4, 13, 13, 31, 4, 40, 4, 22, 22, 13, 4, 40, 7, 22, 13, 22, 4, 31, 13, 31, 13, 13, 4, 67, 4, 13, 22, 19, 13, 40, 4, 22, 13, 40, 4, 52

Offset: 1

Ctibor O. Zizka, Nov 01 2008

In general, if n is a prime p then a(p)=4, and k is from {p-1, p+1, 2*p, p^2+p}.
In general, if n is a squared prime p^2 then a(p^2)=7, and k is from {p^2-p, p^2-1, p^2+1, p^2+p, p^3-p^2, p^3+p^2, p^4+p^2}.
The sequence counts solutions with k>0 and any sign of c, or, alternatively, solutions with c>0 and any sign of k. If solutions were constrained to k>0 and c>0, A048691 would result. - R. J. Mathar, Nov 21 2008

			For n=7 we search the number of integer solutions of the equation 7*k/(k-7). This holds for k from {6,8,14,56}. Then a(7)=4. For n=10 we search the number of integer solutions of the equation 10*k/(k-10). This holds for k from {5,6,8,9,11,12,14,15,20,30,35,60,110}. Then a(10)=13.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 2-4.

Cf. A191973.

[# [k:k in {1..n^2+n} diff {n}| IsIntegral(k*n/(k-n))]:n in [1..75]]; // Marius A. Burtea, Oct 18 2019

A146564 := proc(n) local b,d,k,c ; b := numtheory[divisors](n^2) ; kbag := {} ; for d in b do k := d+n ; if k > 0 then kbag := kbag union {k} ; fi ; k := -d+n ; if k > 0 then kbag := kbag union {k} ; fi; end do; RETURN(nops(kbag)) ; end: for n from 1 to 800 do printf("%d,",A146564(n)) ; od: # R. J. Mathar, Nov 21 2008

psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times@@(pp+1) * Times@@(pp^(ee-1))]];
a[n_] := Sum[psi[2^PrimeNu[d]], {d, Divisors[n]}]-1;
a /@ Range[72] (* Jean-François Alcover, Jan 18 2020 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
dedekindpsi(n)=jordantot(n,2)/eulerphi(n);
A146564(n)=sumdiv(n, d, dedekindpsi(2^omega(d)));
for(n=1, 200, print(n" "A146564(n))) \\ Enrique Pérez Herrero, Apr 14 2012

Conjecture: a(n) = A048691(n)+A063647(n). - R. J. Mathar, Nov 21 2008 (See Corollary 4 in Cerruti's paper.)

a(n) = Sum_{d|n} psi(2^omega(d)), where psi is A001615 and omega is A001221. - Enrique Pérez Herrero, Apr 13 2012

Extended beyond a(11) by R. J. Mathar, Nov 21 2008

A252505 Number of biquadratefree (4th power free) divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 8, 2, 6, 6, 9

Offset: 1

Geoffrey Critzer, Mar 21 2015

Equivalently, a(n) is the number of divisors of n that are in A046100.
a(n) is also the number of divisors d such that the greatest common square divisor of d and n/d is 1.
The number of divisors d of n such that gcd(d, n/d) is squarefree. - Amiram Eldar, Aug 25 2023

			a(16) = 4 because there are 4 divisors of 16 that are 4th power free: 1,2,4,8.
a(16) = 4 because there are 4 divisors d of 16 such that the greatest common square divisor of d and 16/d is 1: 1,2,8,16.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A252505, Sequence Machine.
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046100 (biquadratefree numbers).
Cf. A034444 (squarefree divisors), A073184 (cubefree divisors).
Cf. A001620.
Cf. A000005, A058035, A307430.
Also obtained as a Dirichlet convolution of the following pairs: A034444 and A227291, A007427 and A286779, A008966 and A323308, A048691 and A363552, A271102 and A322327, A307445 and A370296, and A018892 and A378214 (conjectured).

Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]

isA046100(n) = (n==1) || vecmax(factor(n)[, 2])<4;
a(n) = {d = divisors(n); sum(i=1, #d, isA046100(d[i]));} \\ Michel Marcus, Mar 22 2015

a(n) = vecprod(apply(x->min(x, 3) + 1, factor(n)[, 2])); \\ Amiram Eldar, Aug 25 2023

Dirichlet g.f.: zeta(s)^2/zeta(4*s).
Sum_{k=1..n} a(k) ~ 90*n/Pi^4 * (log(n) - 1 + 2*gamma - 360*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} mu(gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020
From Antti Karttunen, May 14 2025: (Start)
Following formulas have been generated for this sequence by Sequence Machine:
a(n) = A000005(A058035(n)).
a(n) = Sum_{d|n} A307430(d).
a(n) = Sum_{d|n} A034444(d)*A227291(n/d).
a(n) = Sum_{d|n} A007427(d)*A286779(n/d).
a(n) = Sum_{d|n} A008966(d)*A323308(n/d).
a(n) = Sum_{d|n} A048691(d)*A363552(n/d).
a(n) = Sum_{d|n} A271102(d)*A322327(n/d).
a(n) = Sum_{d|n} A307445(d)*A370296(n/d).
a(n) = Sum_{d|n} A018892(d)*A378214(n/d). [Conjectured]
(End)

A063520 Sum divides product: number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t).

Original entry on oeis.org

1, 3, 6, 5, 8, 8, 8, 14, 13, 9, 14, 17, 8, 18, 23, 18, 14, 17, 13, 33, 23, 10, 19, 36, 15, 22, 32, 22, 19, 26, 17, 39, 24, 18, 50, 45, 8, 22, 39, 38, 22, 27, 13, 50, 45, 16, 27, 52, 24, 39, 38, 27, 20, 50, 45, 72, 24, 12, 31, 58, 15, 28, 69, 45, 49, 39, 12, 52, 40, 33, 33, 66, 12, 33, 64

Offset: 1

Jud McCranie and Vladeta Jovovic, Aug 01 2001

nonn

Number of solutions (r,s) in positive integers to the equation rs = n(r+s) is tau(n^2), cf. A048691. Number of solutions (r,s), r>=s>0, to the equation rs = n(r+s) is (tau(n^2)+1)/2, cf. A018892.

Conjecturally, includes all positive integers except 2, 4, 7 and 11 - David W. Wilson

			There are 8 such solutions to rst = 5(r+s+t): (5, 4, 3), (7, 5, 2), (10, 4, 2), (11, 10, 1), (15, 8, 1), (20, 7, 1), (25, 3, 2), (35, 6, 1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. J. Pelling, Problem 10745, Amer. Math. Monthly, vol. 106 (1999), p. 587.
M. J. Pelling and F. W. Roush, The Sum Divides the Product: Problem 10745, Amer. Math. Monthly, vol. 108, (no. 7, Aug. 2001), pp. 668-669. [Gives upper bound]

Cf. A018892, A004194, A063525.

(* Assuming s <= 2n and t <= n*(n+2) *) redu[n_] := Reap[ Do[ red = Reduce[0 < r <= s <= t && r*s*t == n*(r+s+t), r, Integers]; If[red =!= False, Sow[{r, s, t} /. ToRules[red] ] ], {s, 1, 2*n}, {t, s, n*(n+2)}] ][[2, 1]]; a[n_] := redu[n] // Length; a[1] = 1; Table[ Print[n, " ", an = a[n]]; an, {n, 1, 75}] (* Jean-François Alcover, Feb 22 2013 *)

a(n)=sum(t=1,sqrtint(3*n),sum(s=t,sqrtint(n^2+t)+n,my(N=n*(s+t), D=s*t-n);D&&denominator(N/D)==1&&N/D>=s)) \\ Charles R Greathouse IV, Feb 22 2013

More terms from David W. Wilson, Aug 01 2001

A070919 a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.

Original entry on oeis.org

1, 7, 7, 19, 7, 49, 7, 37, 19, 49, 7, 133, 7, 49, 49, 61, 7, 133, 7, 133, 49, 49, 7, 259, 19, 49, 37, 133, 7, 343, 7, 91, 49, 49, 49, 361, 7, 49, 49, 259, 7, 343, 7, 133, 133, 49, 7, 427, 19, 133, 49, 133, 7, 259, 49, 259, 49, 49, 7, 931, 7, 49, 133, 127, 49, 343, 7, 133

Offset: 1

Benoit Cloitre, May 20 2002

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Antti Karttunen, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.

Cf. A000005, A008683, A048691, A070920, A070921, A086222, A247513.

Join[{1},Table[Product[(k + 1)^3 - k^3, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^3*moebius(n/d)),","))

a(n) = vecprod(apply(x->(x+1)^3-x^3, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

a(n) = Sum_{d|n} A000005(d)^3*A008683(n/d).
Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^3/k^s.
Multiplicative with a(p^e) = 1+3*e+3*e^2 for prime p and e >= 0. - Werner Schulte, Nov 30 2018

A061503 a(n) = Sum_{k=1..n} tau(k^2), where tau is the number of divisors function A000005.

Original entry on oeis.org

1, 4, 7, 12, 15, 24, 27, 34, 39, 48, 51, 66, 69, 78, 87, 96, 99, 114, 117, 132, 141, 150, 153, 174, 179, 188, 195, 210, 213, 240, 243, 254, 263, 272, 281, 306, 309, 318, 327, 348, 351, 378, 381, 396, 411, 420, 423, 450, 455, 470, 479, 494, 497

Offset: 1

N. J. A. Sloane, Jun 14 2001

nonn

a(n) is the number of pairs of positive integers <= n with their LCM <= n. - Andrew Howroyd, Sep 01 2019

Mentioned by Steven Finch in a posting to the Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Jun 13 2001.

Harry J. Smith, Table of n, a(n) for n = 1..1024
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 19.
Vaclav Kotesovec, Graph - the asymptotic ratio
Eric Weisstein's World of Mathematics, Stieltjes Constants

Cf. A000005, A061502. Partial sums of A048691.

List([1..60],n->Sum([1..n],k->Tau(k^2))); # Muniru A Asiru, Mar 09 2019

with(numtheory): a:=n->add(tau(k^2),k=1..n): seq(a(n),n=1..60); # Muniru A Asiru, Mar 09 2019

DivisorSigma[0, Range[60]^2] // Accumulate (* Jean-François Alcover, Nov 25 2013 *)

for (n=1, 1024, write("b061503.txt", n, " ", sum(k=1, n, numdiv(k^2)))) \\ Harry J. Smith, Jul 23 2009

t=0;v=vector(60,n,t+=numdiv(n^2)) \\ Charles R Greathouse IV, Nov 08 2012

from math import prod
from sympy import factorint
def A061503(n): return sum(prod(2*e+1 for e in factorint(k).values()) for k in range(1,n+1)) # Chai Wah Wu, May 10 2022

def A061503(n) :
    tau = sloane.A000005
    return add(tau(k^2) for k in (1..n))
[ A061503(i) for i in (1..19)] # Peter Luschny, Sep 15 2012

a(n) = Sum_{j=1..n^2} floor(n/A019554(j)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002
a(n) = Sum_{i=1..n} 2^omega(i) * floor(n/i). - Enrique Pérez Herrero, Sep 15 2012
a(n) ~ 3/Pi^2 * n log^2 n. - Charles R Greathouse IV, Nov 08 2012
a(n) ~ 3*n/Pi^2 * (log(n)^2 + log(n)*(-2 + 6*g - 24*z/Pi^2) + 2 - 6*g + 6*g^2 - 6*sg1 + 288*z^2/Pi^4 - 24*(-z + 3*g*z + z2)/ Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant (see A082633), z = Zeta'(2) (see A073002), z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Jan 30 2019
a(n) = Sum_{k=1..n} A064608(floor(n/k)). - Daniel Suteu, Mar 09 2019

Name corrected by Peter Luschny, Sep 15 2012

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

A078644 a(n) = tau(2*n^2)/2.

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 3, 4, 5, 6, 3, 9, 3, 6, 9, 5, 3, 10, 3, 9, 9, 6, 3, 12, 5, 6, 7, 9, 3, 18, 3, 6, 9, 6, 9, 15, 3, 6, 9, 12, 3, 18, 3, 9, 15, 6, 3, 15, 5, 10, 9, 9, 3, 14, 9, 12, 9, 6, 3, 27, 3, 6, 15, 7, 9, 18, 3, 9, 9, 18, 3, 20, 3, 6, 15, 9, 9, 18, 3, 15, 9, 6, 3, 27, 9, 6, 9, 12, 3, 30, 9, 9, 9, 6, 9

Offset: 1

Vladeta Jovovic, Dec 13 2002

Inverse Moebius transform of A068068. Number of elements in the set {(x,y): x is odd, x|n, y|n, gcd(x,y)=1}.
The number of Pythagorean points (x,y), 0 < x < y, located on the hyperbola y = 2n(x-n)/(x-2n) and having "excess" x+y-z = 2n. - Seppo Mustonen, Jun 07 2005
a(n) is the number of Pythagorean triangles with radius of the inscribed circle equal to n. For number of primitive Pythagorean triangles having inradius n, see A068068(n). - Ant King, Mar 06 2006
Dirichlet convolution of A048691 and A154269. - R. J. Mathar, Jun 01 2011
Number of distinct L-shapes of thickness n where the L area equals the rectangular area that it "contains". Visually can be thought as those areas of A156688 (surrounded by equal border of thickness n: 2xy = (x+2n)(y+2n), x and y positive integers) where both x and y are even, so they can be split into L-shapes. So L-shapes have formula: 2xy = (x+n)(y+n). - Juhani Heino, Jul 23 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
S. Mustonen, Visualization and characterization of Pythagorean triples
Seppo Mustonen, Visualization and characterization of Pythagorean triples [Local copy]
T. Omland, How many Pythagorean triples with given inradius?, J. Numb. Theory 170 (2017) 1-2.

Cf. A000005, A001105, A048691.

[NumberOfDivisors(2*n^2)/2 : n in [1..100]]; // Vincenzo Librandi, Aug 14 2018

with(numtheory): seq(add(mobius(2*d)^2*tau(n/d), d in divisors(n)), n=1..100); # Ridouane Oudra, Nov 17 2019

Table[DivisorSigma[0, 2 n^2] / 2, {n, 100}] (* Vincenzo Librandi, Aug 14 2018 *)

a(n) = numdiv(2*n^2)/2; \\ Michel Marcus, Oct 04 2013

[sigma(2*n^2,0)/2 for n in range(1,100)] # Joerg Arndt, May 12 2014

Multiplicative with a(2^e) = e+1, a(p^e) = 2*e+1, p > 2. a(n) = tau(n^2) if n is odd, a(n) = tau(n^2) - a(n/2) if n is even.
Dirichlet g.f.: zeta^3(s)/(zeta(2s)*(1+1/2^s)). - R. J. Mathar, Jun 01 2011
Sum_{k=1..n} a(k) ~ 2*n / (9*Pi^2) * (9*log(n)^2 + 6*log(n) * (-3 + 9*g + log(2) - 36*Pi^(-2)*z1) + 18 + 54*g^2 + 18*g * (log(2) - 3) - 6*log(2) - log(2)^2 - 54*sg1 + 2592*z1^2/Pi^4 - 72*Pi^-2*(9*g*z1 + (log(2) - 3)*z1 + 3*z2)), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} mu(2d)^2*tau(n/d), Dirichlet convolution of A323239 and A000005. - Ridouane Oudra, Nov 17 2019
a(n) = A361689(n)/2. - R. J. Mathar, Mar 21 2023

A360908 Multiplicative with a(p^e) = 2*e - 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A048691, A322327, A361430, A367822.

a[n_] := Times @@ ((2*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+(1+1/X)/(1-1/X)^2))[n], ", "))

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]=2*f[k,2]-1; f[k,2]=1); factorback(f); \\ Michel Marcus, Feb 25 2023

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/(p^s*(p^s-1))).
Sum_{k=1..n} a(k) ~ c*n, where c = A367822 = Product_{p prime} (1 + 2/(p*(p-1))) = 3.279577150984783607372919498914633983999130708105267540952619534539808381...
a(n) = A361430(n^2). - Amiram Eldar, Feb 11 2024

A059907 a(n) = |{m : multiplicative order of n mod m = 2}|.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 6, 4, 6, 3, 12, 2, 10, 6, 8, 4, 13, 2, 18, 6, 10, 4, 16, 4, 12, 9, 12, 4, 26, 2, 20, 6, 8, 12, 20, 4, 15, 6, 16, 4, 32, 2, 24, 10, 10, 6, 20, 4, 26, 9, 18, 4, 26, 6, 32, 12, 12, 4, 28, 2, 20, 10, 12, 18, 25, 4, 24, 6, 26, 4, 52, 2, 18, 10, 12, 18, 26, 4, 40, 8, 14, 5, 28

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, GCD(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{3}| = 1, a(3) = |{4,8}| = 2, a(4) = |{5,15}| = 2, a(5) = |{3,6,8,12,24}| = 5, a(6) = |{7,35}| = 2, a(7) = |{4,8,12,16,24,48}| = 6,...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002329, A059908-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A055205, A048691, A048785, A347191.

Row n=2 of A212957. - Alois P. Heinz, Oct 24 2012

with(numtheory):f := n->tau(n^2-1)-tau(n-1):for n from 1 to 100 do printf(`%d,`,f(n)) od:

a[n_] := Subtract @@ DivisorSigma[0, {n^2-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^2-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^2-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

cp's OEIS Frontend

A061391 a(n) = t(n,3) = Sum_{d|n} tau(d^3), where tau(n) = number of divisors of n, cf. A000005.

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A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers.

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A252505 Number of biquadratefree (4th power free) divisors of n.

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A063520 Sum divides product: number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t).

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A070919 a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.

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A061503 a(n) = Sum_{k=1..n} tau(k^2), where tau is the number of divisors function A000005.

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