A064608 -id:A064608 - cp's OEIS frontend

A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813

Offset: 1

Labos Elemer, Sep 24 2001

The corresponding quotients are 1, 3, 3, 5, 7, 7, 7, 7, 11, 13, 13, 13, 13, ...

a(14) > 7.5*10^10, if it exists. - Amiram Eldar, Jun 04 2021

			For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.

Cf. A064608.

Analogous "integer-mean" sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A048290, A063986, A063971, A064911, A062982, A045345.

s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)

{n: A064608(n) == 0 (mod n)}.

a(10)-a(13) from Donovan Johnson, Jul 20 2012

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

A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 41, 45, 47, 51, 53, 57, 61, 65, 67, 75, 77, 81, 85, 89, 91, 99, 101, 107, 111, 115, 119, 123, 125, 129, 133, 141, 143, 151, 153, 157, 161, 165, 167, 175, 177, 181, 185, 189, 191, 199, 203, 211, 215, 219, 221

Offset: 1

Amiram Eldar, Jun 19 2018

nonn

The bi-unitary version of A006218 and A064608.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

Amiram Eldar, Table of n, a(n) for n = 1..10000
D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.

Cf. A001620, A006218, A064608, A286324, A306071, A306072.

fun[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; bdivnum[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Accumulate@ Array[bdivnum, {60}]

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = sum(k=1, n, #biudivs(k)); \\ Michel Marcus, Jun 20 2018

a(n) = A*n*(log(n) + 2*gamma - 1 + B) + O(n^(1/2)*exp(-A * log(n)^(3/5) * log(log(n))^(-1/5))), where gamma = A001620, A = A306071 and B = A306072.

A069212 a(n) = Sum_{k=1..n} 3^omega(k).

Original entry on oeis.org

1, 4, 7, 10, 13, 22, 25, 28, 31, 40, 43, 52, 55, 64, 73, 76, 79, 88, 91, 100, 109, 118, 121, 130, 133, 142, 145, 154, 157, 184, 187, 190, 199, 208, 217, 226, 229, 238, 247, 256, 259, 286, 289, 298, 307, 316, 319, 328, 331, 340, 349, 358, 361, 370, 379, 388, 397

Offset: 1

Benoit Cloitre, Apr 14 2002

More generally, if b is an integer =>3, Sum_{k=1..n} b^omega(k) ~ C(b)*n*log(n)^(b-1) where C(b)=1/(b-1)!*prod((1-1/p)^(b-1)*(1+(b-1)/p)).

G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. (2015). See page 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Partial sums of A074816.

Cf. A001222, A064608, A065473, A350961.

Accumulate @ Table[3^PrimeNu[n], {n, 1, 57}] (* Amiram Eldar, May 24 2020 *)

from sympy.ntheory.factor_ import primenu
def A069212(n): return sum(3**primenu(m) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

Asymptotic formula: a(n) ~ C*n*log(n)^2 with C = (1/2) * Product_{p} ((1-1/p)^2*(1+2/p)) where the product is over all the primes.
The constant C is A065473/2. - Amiram Eldar, May 24 2020
From Ridouane Oudra, Jan 01 2021: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} mu(i*j)^2*floor(n/(i*j));
a(n) = Sum_{i=1..n} mu(i)^2*tau(i)*floor(n/i);
a(n) = Sum_{i=1..n} 2^Omega(i)*mu(i)^2*floor(n/i), where Omega = A001222. (End)
From Vaclav Kotesovec, Feb 16 2022: (Start)
More precise asymptotics:
Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)), then
a(n) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2),
where f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927127898384464343318440970569956414778593366522...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = 0.8023233847630974628467999132875783526536954420333140745016349208975965...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} -6*p*(2*p+1) * log(p)^2 / (p^2 + p - 2)^2 = -0.255987592484328884627082229528266165335336670389046663124468278519...
and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A328331 a(n) is the least k such that the average number of unitary divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 35, 190, 1015, 5304, 27417, 142142, 736782, 3816852, 19774690, 102446730, 530743749, 2749606626, 14244797910

Offset: 1

Amiram Eldar, Oct 22 2019

The unitary version of A085829.

			a(2) = 6 because the average number of unitary divisors of {1..6} is  A064608(6)/6 = 13/6 > 2.

Cf. A013661, A034444, A064608, A085829.

seq={}; s = 0; k = 1; Do[While[s += 2^PrimeNu[k]; s < k*n, k++]; AppendTo[seq, k]; k++, {n, 1, 10}]; seq

Lim_{n->oo} a(n+1)/a(n) = exp(zeta(2)) = exp(Pi^2/6) = 5.180668... (since A064608(n) ~ n*log(n)/zeta(2)).

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]

A368673 Number of squarefree numbers less than n that do not divide n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 4, 4, 3, 6, 4, 7, 6, 7, 9, 10, 8, 11, 9, 10, 11, 14, 12, 14, 13, 15, 13, 16, 11, 18, 18, 17, 18, 19, 19, 22, 21, 22, 22, 25, 20, 27, 25, 25, 26, 29, 27, 29, 27, 28, 28, 31, 29, 30, 30, 31, 32, 35, 29, 36, 35, 35, 37, 36, 33, 40, 38, 39, 36, 43

Offset: 1

Wesley Ivan Hurt, Jan 02 2024

			a(12) = 4 since there are 4 squarefree numbers less than 12 that do not divide 12, namely: 5, 7, 10, and 11.

Cf. A008683 (mu), A064608, A368642, A368674.

Table[Sum[MoebiusMu[k]^2 (Ceiling[n/k] - Floor[n/k]), {k, n}], {n, 100}]

a(n) = sum(k=1, n-1, (n % k) && issquarefree(k)); \\ Michel Marcus, Jan 03 2024

from math import isqrt
from sympy import factorint, mobius
def A368673(n): return sum(mobius(k)*((n-1)//k**2) for k in range(1,isqrt(n-1)+1))-(1<Chai Wah Wu, Jan 03 2024

a(n) = Sum_{k=1..n} mu(k)^2 * (ceiling(n/k) - floor(n/k)).

a(n) = A368642(n) - A064608(n).

A074787 Sum of squares of the number of unitary divisors of k from 1 to n.

Original entry on oeis.org

1, 5, 9, 13, 17, 33, 37, 41, 45, 61, 65, 81, 85, 101, 117, 121, 125, 141, 145, 161, 177, 193, 197, 213, 217, 233, 237, 253, 257, 321, 325, 329, 345, 361, 377, 393, 397, 413, 429, 445, 449, 513, 517, 533, 549, 565, 569, 585, 589, 605, 621, 637, 641, 657, 673

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034444, A064608, A069212, A285052, A319592.

Equals 4*A069811(n) + 1, for n <= 29.

with(numtheory): seq(add(2^(2*nops(ifactors(k)[2])),k=1..n),n=1..100);

Accumulate[Table[Count[Divisors[n],?(GCD[#,n/#]==1&)],{n,60}]^2] (* _Harvey P. Dale, Dec 06 2012 *)
Accumulate[Table[4^PrimeNu[n], {n, 1, 50}]] (* Amiram Eldar, Jul 02 2022 *)

a(n) = Sum_{k=1..n} ud(k)^2 = Sum_{k=1..n} A034444(k)^2 . a(n) = Sum_{k=1..n} 2^(2*omega(k)) = Sum_{k=1..n} 2^(2*A001221(k)).

a(n) ~ c * n * log(n)^3, where c = (1/6) * Product_{p prime} ((1-1/p)^3*(1+3/p)) = A319592 / 6. - Amiram Eldar, Jul 02 2022

A180361 Sum of number of unitary divisors (A034444) from 1 to 10^n.

Original entry on oeis.org

1, 23, 359, 4987, 63869, 778581, 9185685, 105854997, 1198530315, 13385107495, 147849112851, 1618471517571, 17584519050293, 189843229312125, 2038412681323151, 21783930695524161, 231837345778656901

Offset: 0

Andrew Lelechenko, Jan 19 2011

nonn

Henri Lifchitz, Table of n, a(n) for n = 0..23

Cf. A034444, A064608

a(n)=A064608(10^n)

a(3) corrected from 4983 to 4987 by Henri Lifchitz, Nov 07 2017

a(17)-a(23) from Henri Lifchitz, Nov 07 2017

A309192 a(n) = Sum_{k=1..n} mu(k)^2 * k * floor(n/k).

Original entry on oeis.org

1, 4, 8, 11, 17, 29, 37, 40, 44, 62, 74, 86, 100, 124, 148, 151, 169, 181, 201, 219, 251, 287, 311, 323, 329, 371, 375, 399, 429, 501, 533, 536, 584, 638, 686, 698, 736, 796, 852, 870, 912, 1008, 1052, 1088, 1112, 1184, 1232, 1244, 1252, 1270, 1342, 1384, 1438, 1450, 1522

Offset: 1

Ilya Gutkovskiy, Jul 16 2019

nonn

Partial sums of A048250.

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

Cf. A008683, A024916, A024924, A048250, A064608, A275205.

Table[Sum[MoebiusMu[k]^2 k Floor[n/k], {k, 1, n}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[1/(1 - x) Sum[MoebiusMu[k]^2 k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Accumulate[Table[Total[Select[Divisors[n], SquareFreeQ]], {n, 1, 100}]] (* Vaclav Kotesovec, Jul 16 2019 *)

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k)^2 * k * x^k/(1 - x^k).

a(n) ~ n^2/2. - Vaclav Kotesovec, Jul 16 2019

cp's OEIS Frontend

A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

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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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A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

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A069212 a(n) = Sum_{k=1..n} 3^omega(k).

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A328331 a(n) is the least k such that the average number of unitary divisors of {1..k} is >= n.

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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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A368673 Number of squarefree numbers less than n that do not divide n.

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