A082695 -id:A082695 - cp's OEIS frontend

A359280 Powerful numbers that are neither prime powers nor powers of squarefree composites.

72, 108, 144, 200, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3600, 3872, 3888, 3969

Offset: 1

Michael De Vlieger, Aug 01 2023

nonn

Numbers k such that omega(k) > 1 and for prime power factors p^e | k, multiplicities e > 1, yet the multiplicities are not equal.
Subset of A286708, which in turn is a subset of A361098, itself a subset of A126706, the sequence of numbers neither squarefree nor prime powers.
Since A001694 = Union({1}, A246547, A286708), this sequence is a subset of A001694.

			Let b(n) = A286708(n).
b(1) = 36 is not in the sequence since rad(36) = A007947(36) = 6, and 36 = 6^2.
b(2) = a(1) = 72 since 72 is not a perfect power of rad(72).
b(3) = 100 = rad(100)^2 = 10^2, so it is not in the sequence.
b(4) = a(2) = 108, since 108 is not a perfect power of rad(108) = 6.
b(5) = a(3) = 144, since 144 is not a perfect power of rad(144) = 6.
b(6) = 196 is not in the sequence since 196 = rad(196)^2 = 14^2, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot A001694(ym + x) at (x, y) for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and both prime powers and those in A303606 in white.
Michael De Vlieger, Plot A286708(n) at (x, y) for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and those in A303606 in white.

Cf. A001694, A007947, A082695, A126706, A286708, A303606, A361098.

nn = 5000; s = Rest@ Select[Union@ Flatten@Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not@*PrimePowerQ]; Select[s, !SameQ @@ FactorInteger[#][[All, -1]] &]

from math import isqrt
from sympy import mobius, integer_nthroot
def A359280(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j = isqrt(x)
        c, l = n+x+3-(y:=x.bit_length())+squarefreepi(j)+sum(squarefreepi(integer_nthroot(x, k)[0]) for k in range(4, y)), 0
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Feb 09 2025

This sequence is A286708 \ A303606.

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) - 1 = 0.094962568855... . - Amiram Eldar, Dec 09 2023

A382422 The product of exponents in the prime factorization of the biquadratefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 25 2025

Differs from A375766 and A375768 at n = 1, 31, 34, 35, 38, 39, ... .

All the terms are 3-smooth numbers (A003586).

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

Cf. A003586, A005361, A013662, A046100, A082695, A382423, A382424.

Cf. A375766, A375768.

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; biqFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 4; s /@ Select[Range[100], biqFreeQ]

list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 4, print1(vecprod(e), ", "))); }

a(n) = A005361(A046100(n)).
a(n) = 2^A382423(n) * 3^A382424(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 3/p^4) = 1.57226906210272200398... .
In general, the asymptotic mean of the product of exponents in the prime factorization of the k-free numbers (numbers that are not divisible by a k-th power other than 1), for k >= 2, is zeta(k) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + ... + 1/p^(k-1) - (k-1)/p^k). For k = 2 (squarefree numbers), the mean is 1 since the sequence contains only 1's. The limit when k->oo is zeta(2)*zeta(3)/zeta(6) (A082695).

A070243 a(n) = Card{ k, phi(k) <= n }.

Original entry on oeis.org

2, 5, 5, 9, 9, 13, 13, 18, 18, 20, 20, 26, 26, 26, 26, 32, 32, 36, 36, 41, 41, 43, 43, 53, 53, 53, 53, 55, 55, 57, 57, 64, 64, 64, 64, 72, 72, 72, 72, 81, 81, 85, 85, 88, 88, 90, 90, 101, 101, 101, 101, 103, 103, 105, 105, 108, 108, 110, 110, 119, 119, 119, 119, 127, 127

Offset: 1

Benoit Cloitre, May 11 2002

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-118.
Gérald Tenenbaum and Jie Wu, Exercices corrigés de théorie analytique et probabiliste des nombres, Collection SMF, Cours spécialisés, Numero 2, pp. 78-79.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, Journal of High Energy Physics, Vol. 2018 (2018), Article number 138; arXiv preprint, arXiv:1801.04542 [hep-th], 2018.

Partial sums of A014197.

Cf. A000010, A082695.

for(n=1,150,print1(sum(i=1,100*n,if(sign(eulerphi(i)-n)+1,0,1)+if((eulerphi(i)-n),0,1)),","))

list(nmax) = my(s = 0); for(n = 1, nmax, s += invphiNum(n); print1(s, ", ")); \\ Amiram Eldar, Dec 23 2024, using Max Alekseyev's invphi.gp

Limit_{n->oo} a(n)/n = zeta(2)*zeta(3)/zeta(6) = 1.943596436820759205057... = A082695.
From Benoit Cloitre, Apr 12 2003: (Start)
a(n) = Sum_{k=1..n} A014197(k).
a(n) = (zeta(2)*zeta(3)/zeta(6))*n + O(n*exp(-c*sqrt(log(n)))) for a suitable constant c > 0. (End)

A183094 a(n) = number of powerful divisors d (excluding 1) of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 3, 0, 0, 0, 2, 0

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

a(n) = number of divisors d of n from set A001694(m) - powerful numbers for m >=2.

			For n = 12, set of such divisors is {4}; a(12) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000
D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc. 34 (1972), 79-80.

Cf. A000005, A001694, A005361, A082695, A183095.

f:=  n -> convert(map(t->t[2], ifactors(n)[2]),`*`) - 1; # Robert Israel, Jul 14 2014

powerfulQ[n_] := Min[ Last@# & /@ FactorInteger[n]] > 1; f[n_] := Length@ Select[ Divisors@ n, powerfulQ]; Array[f, 105] (* Robert G. Wilson v, Jul 14 2014 *)

a(n) = A000005(n) - A183095(n) = A005361(n) - 1.
a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = k-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.9435964368... . - Amiram Eldar, Jul 30 2022

A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

Original entry on oeis.org

-1, 0, -1, 0, -1, 1, 1, 1, 1, 5, 19, 17, 29, 13, 21, 13, 47, 181, 503, 593, 533, 121, 1259, 1457, 6889, 7549, 7109, 7769, 52403, 59333, 11497, 6095, 29089, 61643, 59333, 63953, 62413, 7277, 21061, 2777, 10877, 11647, 3809, 3963, 1438, 271, 3064, 51439, 7217, 7493

Offset: 1

Benoit Cloitre, Feb 01 2013

			Fractions begin with -1, 0, -1/2, 0, -1/4, 1/4, 1/12, 1/3, 1/6, 5/12, 19/60, 17/30, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013) Article 13.6.3.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000010, A028415, A211178 (denominators).

Cf. A001620, A082695, A085609.

Numerator @ Accumulate[Table[(-1)^k/EulerPhi[k], {k, 1, 50}]] (* Amiram Eldar, Nov 20 2020 *)

a(n)=numerator(sum(k=1,n,(-1)^k/eulerphi(k)))

a(n)/A211178(n) = c*log(n) + O(1) with a suitable constant c (see ref).
The constant above is c = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 20 2020
More accurately, a(n)/A211178(n) ~ (A/3) * (log(n) + gamma - B - 8*log(2)/3) + O(log(n)^(5/3)/n), where A = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and B = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Bordellès and Cloitre, 2013; Tóth, 2017). - Amiram Eldar, Oct 14 2022

More terms from Amiram Eldar, Nov 20 2020

A211178 Denominator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 12, 3, 6, 12, 60, 30, 60, 20, 40, 20, 80, 240, 720, 720, 720, 144, 1584, 1584, 7920, 7920, 7920, 7920, 55440, 55440, 11088, 5544, 27720, 55440, 55440, 55440, 55440, 6160, 18480, 2310, 9240, 9240, 3080, 3080, 1155, 210, 2415, 38640, 5520, 5520

Offset: 1

Benoit Cloitre, Feb 01 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013) Article #13.6.3.

Cf. A000010, A082695, A211177 (numerators).

Denominator @ Accumulate[Table[(-1)^k/EulerPhi[k], {k, 1, 50}]] (* Amiram Eldar, Nov 20 2020 *)

a(n)=denominator(sum(k=1, n, (-1)^k/eulerphi(k)))

A211177(n)/a(n) = c*log(n) + O(1) with a suitable constant c (see ref).

The constant above is c = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 20 2020

More terms from Amiram Eldar, Nov 20 2020

A068885 Numerator of Sum_{k=1..n} k/phi(k).

Original entry on oeis.org

1, 3, 9, 13, 31, 43, 143, 167, 185, 215, 1141, 1321, 231, 763, 3277, 3517, 7289, 8009, 24787, 26587, 27847, 29431, 332021, 355781, 365681, 382841, 394721, 413201, 2949827, 3157727, 643003, 665179, 3417371, 3535181, 3616031, 3782351, 1279777, 3956371, 4046461

Offset: 1

N. J. A. Sloane, Jun 28 2002

			1, 3, 9/2, 13/2, 31/4, 43/4, 143/12, 167/12, 185/12, ...

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section I.27, page 29.

Amiram Eldar, Table of n, a(n) for n = 1..1000
R. Sitaramachandrarao, On an error term of Landau - II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.

Cf. A069947 (denominators), A000010, A028415, A048049, A082695.

Numerator @ Accumulate @ Table[k/EulerPhi[k], {k, 1, 40}] (* Amiram Eldar, Sep 18 2022 *)

a(n) = numerator(sum(k=1, n, k/eulerphi(k))); \\ Michel Marcus, Sep 18 2022

a(n)/A069947(n) ~ c * n - log(n)/2 + O(log(n)^(2/3)), where c = zeta(2)*zeta(3)/zeta(6) (A082695) (Sitaramachandrarao, 1985). - Amiram Eldar, Sep 18 2022

A079551 a(n) = Sum_{primes p <= n} d(p-1), where d() = A000005.

Original entry on oeis.org

0, 0, 1, 3, 3, 6, 6, 10, 10, 10, 10, 14, 14, 20, 20, 20, 20, 25, 25, 31, 31, 31, 31, 35, 35, 35, 35, 35, 35, 41, 41, 49, 49, 49, 49, 49, 49, 58, 58, 58, 58, 66, 66, 74, 74, 74, 74, 78, 78, 78, 78, 78, 78, 84, 84, 84, 84, 84, 84, 88, 88, 100, 100, 100, 100, 100, 100, 108, 108, 108, 108

Offset: 0

N. J. A. Sloane, Jan 24 2003

nonn

Yuri V. Linnik, The dispersion method in binary additive problems, American Mathematical Society, 1963, chapter VIII.
József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, section II.11, p. 49.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Enrico Bombieri, John B. Friedlander, and Henryk Iwaniec, Primes in arithmetic progressions to large moduli, Acta Mathematica, Vol. 156, No. 1 (1986), pp. 203-251.
Heini Halberstam, Footnote to the Titchmarsh-Linnik divisor problem, Proceedings of the American Mathematical Society, Vol. 18, No. 1 (1967), pp. 187-188.
Yurii Vladimirovich Linnik, New versions and new uses of the dispersion methods in binary additive problems, Doklady Akademii Nauk SSSR, Vol. 137, No. 6. (1961), pp. 1299-1302 (in Russian).
Gaetano Rodriquez, Sul problema dei divisori di Titchmarsh, Bollettino dell'Unione Matematica Italiana, Vol. 20, No. 3 (1965), pp. 358-366.
E. C. Titchmarsh, A divisor problem, Rendiconti del Circolo Matematico di Palermo (1884-1940), December 1930, Volume 54, Issue 1, pp. 414-429.

Cf. A000005, A079552, A082695.

Row sums of triangle A143540. - Gary W. Adamson, Aug 23 2008

a[n_] := Sum[DivisorSigma[0, p-1], {p, Select[Range[n], PrimeQ]}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 26 2015 *)

a(n) = sum(p=1, n, if (isprime(p), numdiv(p-1))); \\ Michel Marcus, Aug 03 2018

Several asymptotic estimates are known: see Sándor et al.

a(n) ~ (zeta(2)*zeta(3)/zeta(6)) * n. - Amiram Eldar, Jul 22 2019

A322965 Numerator of Sum_{d | n} 1/rad(d).

Original entry on oeis.org

1, 3, 4, 2, 6, 2, 8, 5, 5, 9, 12, 8, 14, 12, 8, 3, 18, 5, 20, 12, 32, 18, 24, 10, 7, 21, 2, 16, 30, 12, 32, 7, 16, 27, 48, 10, 38, 30, 56, 3, 42, 16, 44, 24, 2, 36, 48, 4, 9, 21, 24, 28, 54, 3, 72, 20, 80, 45, 60, 16, 62, 48, 40, 4, 84, 24, 68, 36, 32, 72, 72, 25, 74, 57, 28

Offset: 1

David S. Metzler, Dec 31 2018

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. The sequence a(n) lists the numerators of the fractions f(n) in lowest terms.

If p is prime, then a(p^k) = p+k if p does not divide k, 1 + k/p if it does. In particular, a(p^p) = 2. - Robert Israel, Jan 25 2019

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3 and a(12) = 8. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007947 (radical), A322966 (denominators), A008473 (unreduced numerators, i.e., f(n)*rad(n)), A082695.

Numbers n where f(n) increases to a record: A322447.

rad:= n -> convert(numtheory:-factorset(n),`*`):
f:= proc(n) numer(add(1/rad(d),d=numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 25 2019

Array[Numerator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]]] &] &, 71] (* Michael De Vlieger, Jan 19 2019 *)

rad(n) = factorback(factor(n)[, 1]); \\ A007947
a(n) = numerator(sumdiv(n, d, 1/rad(d))); \\ Michel Marcus, Jan 10 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A322966(k) = zeta(2)*zeta(3)/zeta(6) (A082695). - Amiram Eldar, Dec 09 2023

More terms from Michel Marcus, Jan 19 2019

A339925 Decimal expansion of 105*zeta(3)/Pi^4.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 23 2020

			1.295730957880506136704713575049842291458572334511870477351090459267...

H. Riesel and R. C. Vaughan, On sums of primes, Ark. Mat., Volume 21, Number 1-2 (1983), 45-74 (p. 47).

Cf. A002117, A001615, A005597, A082695.

RealDigits[N[105 Zeta[3]/Pi^4, 105]][[1]]

prodeulerrat(1+1/(p*(p-1)),1,3) \\ Hugo Pfoertner, Dec 23 2020

Equals Product_{p>=3} 1+1/(p*(p-1)) where p are successive odd primes.
Equals A082695*2/3.
Equals Sum_{k>=1} A001615(k)/k^4. - Amiram Eldar, Jan 25 2024

cp's OEIS Frontend

A359280 Powerful numbers that are neither prime powers nor powers of squarefree composites.

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A382422 The product of exponents in the prime factorization of the biquadratefree numbers.

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A070243 a(n) = Card{ k, phi(k) <= n }.

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A183094 a(n) = number of powerful divisors d (excluding 1) of n.

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A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

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A211178 Denominator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

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A068885 Numerator of Sum_{k=1..n} k/phi(k).

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