A085966 -id:A085966 - cp's OEIS frontend

A077761 Decimal expansion of Mertens's constant, which is the limit of (Sum_{i=1..k} 1/prime(i)) - log(log(prime(k))) as k goes to infinity, where prime(i) is the i-th prime number.

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

Offset: 0

T. D. Noe, Nov 14 2002

Graham, Knuth & Patashnik incorrectly give this constant as 0.261972128. - Robert G. Wilson v, Dec 02 2005 [This was corrected in the second edition (1994). - T. D. Noe, Mar 11 2017]
Also the average deviation of the number of distinct prime factors: sum_{n < x} omega(n) = x log log x + B_1 x + O(x) where B_1 is this constant, see (e.g.) Hardy & Wright. - Charles R Greathouse IV, Mar 05 2021
Named after the Polish mathematician Franz Mertens (1840-1927). Sometimes called Meissel-Mertens constant, after Mertens and the German astronomer Ernst Meissel (1826-1895). - Amiram Eldar, Jun 16 2021

			0.26149721284764278375542683860869585905156664826119920619206421392...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2004, pp. 94-98
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, A Foundation For Computer Science, Addison-Wesley, Reading, MA, 1989, p. 23.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed. (1975). Oxford, England: Oxford University Press. See 22.10, "The number of prime factors of n".
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.28, p. 257.

Eduard Baumann, Table of n, a(n) for n = 0..9999 (first 5001 digits from Robert G. Wilson v), Dec 03 2024.
Christian Axler, New estimates for some functions defined over primes, Integers, Vol. 18 (2018), Article #A52.
Chris Caldwell, The Prime Pages, There are infinitely many primes, but, how big of an infinity?
Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. - From _N. J. A. Sloane_, Jan 26 2013
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, Vol. 45 (2018), pp. 227-251.
Pierre Dusart, On the divergence of the sum of prime reciprocals, WSEAS Transactions on Math. (2023) Vol.22, 508-513.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 203.
Philippe Flajolet and Ilan Vardi, Zeta function expansions of some classical constants.
Tengiz O. Gogoberidze, Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor, arXiv:2407.12047 [math.GM], 2024. See p. 3.
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expos. Math. 15 (1997) 467-478.
Peter Lindqvist and Jaak Peetre, On a number theoretic sum considered by Meissel : a historical observation, Nieuw Archief voor Wiskunde (Serie 4) 1997 Vol. 15 (3) pp. 175-179.
Ernst Meissel, Bericht über die Provinzial-Gewerbe-Schule zu Iserlohn 1866. Title page [courtesy Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften]
Ernst Meissel, Bericht über die Provinzial-Gewerbe-Schule zu Iserlohn 1866. Notiz No. 55 [courtesy Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften].
Ernst Meissel, Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen, Math. Ann. 2 (4) (1870) 636-642, EuDML
Pieter Moree, Mathematical constants.
Rikard Olofsson, Properties of the Beurling generalized primes, Journal of Number Theory (Volume 131), Issue 1, January 2011, Pages 45-58 (p.51).
Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, An explicit upper bound for Siegel zeros of imaginary quadratic fields, arXiv:2001.05782 [math.NT], 2020.
Xavier Gourdon and Pascal Sebah, Constants from number theory
Torsten Sillke, The Harmonic Numbers and Series.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; also on arXiv preprint, arXiv:math/1812.06566 [math.NT], 2018.
Mark B. Villarino, Mertens' proof of Mertens' Theorem, arXiv:math/0504289 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Mertens Constant.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Eric Weisstein's World of Mathematics, Harmonic Series of Primes.
Wikipedia, Meissel-Mertens constant.
Marek Wójtowicz, Another proof on the existence of Mertens's constant, Proc. Japan Acad. Ser. A Math. Sci., Vol. 87, No. 2 (2011), pp. 22-23.

Cf. A001620.

$MaxExtraPrecision = 400; RealDigits[ N[EulerGamma + NSum[(MoebiusMu[m]/m)*Log[N[Zeta[m], 120]], {m, 2, 1000}, Method -> "EulerMaclaurin", AccuracyGoal -> 120, NSumTerms -> 1000, PrecisionGoal -> 120, WorkingPrecision -> 120] , 120]][[1, 1 ;; 105]]
(* or, from version 7 up: *) digits = 105; M = EulerGamma - NSum[ PrimeZetaP[n] / n, {n, 2, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 3*digits]; RealDigits[M, 10, digits] // First (* Jean-François Alcover, Mar 16 2011, updated Sep 01 2015 *)

Equals A001620 - Sum_{n>=2} zeta_prime(n)/n where the zeta prime sequence is A085548, A085541, A085964, A085965, A085966 etc. [Sebah and Gourdon] - R. J. Mathar, Apr 29 2006
Equals gamma + Sum_{p prime} (log(1-1/p) + 1/p), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021
Equals lim_{k->oo} -k + Sum_{p prime} 1/(p*log(p)^(1/k)) conjectured by Meissel in 1866 and proven by Peter Lindqvist and Jaak Peetre in 1997 see links - Artur Jasinski, Mar 11 2025

A030516 Numbers with 7 divisors. 6th powers of primes.

Original entry on oeis.org

64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889, 594823321, 887503681, 2565726409, 4750104241, 6321363049, 10779215329, 22164361129, 42180533641, 51520374361, 90458382169, 128100283921

Offset: 1

Jeff Burch

These are the numbers p^6 with p prime. - Jorge Coveiro, Apr 13 2004

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

R. J. Mathar, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Subsequence of A046306.

Cf. A013664, A050997, A085966, A131993, A000005, A000040, A001248, A030514, A258603, A269404.

a030516 = (^ 6) . a000040
a030516_list = map (^ 6) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

[NthPrime(n)^6: n in [1..400]];  // Bruno Berselli, Jul 30 2011

Array[Prime[ # ]^6&,60] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)

for(n=1,1e3,print1(prime(n)^6", "));  \\ Bruno Berselli, Jul 30 2011

[p**6 for p in prime_range(100)]
# Zerinvary Lajos, May 16 2007

a(n) = A000040(n)^(7-1) = A000040(n)^6. - Omar E. Pol, May 06 2008
A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(6) = 0.0170700868... (A085966). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(6)/zeta(12) = 675675/(691*Pi^6) (A269404).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(6) = 945/Pi^6 = 1/A013664. (End)

A085965 Decimal expansion of the prime zeta function at 5.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017

			0.0357550174839242571328...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Jason Kimberley, Table of n, a(n) for n = 0..1702
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Eric Weisstein's World of Mathematics, Prime Zeta Function

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), this sequence (at 5), A085966 (at 6) to A085969 (at 9).

Cf. A013663, A050997, A242304.

R := RealField(106);
PrimeZeta := func;
[0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(5,69)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[5*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n=200]; While[s[n] != s[n-100], n = n+100]; s[n] (* Jean-François Alcover, Feb 14 2013, from 1st formula *)
RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,5) \\ Hugo Pfoertner, Feb 03 2020

P(5) = Sum_{p prime} 1/p^5 = Sum_{n>=1} mobius(n)*log(zeta(5*n))/n.
Equals 1/2^5 + A085994 + A086035. - R. J. Mathar, Jul 14 2012
Equals Sum_{k>=1} 1/A050997(k). - Amiram Eldar, Jul 27 2020

A154945 Decimal expansion of Sum_{p} 1/(p^2-1), summed over the primes p = A000040.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 17 2009

By geometric series expansion, the same as the sum over the prime zeta function at even arguments, P(2i), i=1,2,....

(Pi^2/6)*density of A190641, the numbers divisible by exactly one prime with exponent greater than 1. - Charles R Greathouse IV, Aug 02 2016

			0.551693297656999184439731023971343578131500377778628252230...

Jacques Grah, Comportement moyen du cardinal de certains ensembles de facteurs premiers, Monatsh. Math., Vol. 118 (1994), pp. 91-109, Corollary 6.1.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.
Index to constants which are prime zeta sums {0,1,1}

Cf. A000040, A000961, A056798, A084920, A085548, A085964, A085966, A085968, A152447, A154932, A190641.

digits = 105; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[delta1 = Sum[PrimeZetaP[2n], {n, 1, m}] , 10, digits][[1]]; rd[m0]; rd[m = 2m0];
While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[delta1, digits]]; rd[m] (* Jean-François Alcover, Sep 11 2015, updated Mar 16 2019 *)

eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumpos(n=1,primezeta(2*n)) \\ Charles R Greathouse IV, Aug 02 2016

sumeulerrat(1/(p^2-1)) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{k>=1} 1/A084920(k) = Sum_{i>=1} P(2i) = A085548+A085964+A085966+A085968+... = A152447+A085548-A154932.
Equals Sum_{k>=2} 1/A000961(k)^2 = Sum_{k>=2} 1/A056798(k). - Amiram Eldar, Sep 21 2020
Equals (A136141 + A179119)/2. - Artur Jasinski, Mar 31 2025

More digits from Jean-François Alcover, Sep 11 2015

A085967 Decimal expansion of the prime zeta function at 7.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017

			0.0082838328561335925351...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Jason Kimberley, Table of n, a(n) for n = 0..1902
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Eric Weisstein's World of Mathematics, Prime Zeta Function

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085966 (at 6), this sequence (at 7), A085968 (at 8), A085969 (at 9).

Cf. A013665, A092759.

R := RealField(106);
PrimeZeta := func;
[0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(7,47)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Join[{0, 0}, Sum[ MoebiusMu[k]*Log[Zeta[7*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104] & // First]; s[100]; s[n = 200]; While[ s[n] != s[n - 100], n = n + 100]; s[n] (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[ PrimeZetaP[ 7], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,7) \\ Hugo Pfoertner, Feb 03 2020

P(7) = Sum_{p prime} 1/p^7 = Sum_{n>=1} mobius(n)*log(zeta(7*n))/n.

Equals Sum_{k>=1} 1/A092759(k). - Amiram Eldar, Jul 27 2020

A162142 Numbers that are the cube of a product of two distinct primes (p^3*q^3).

Original entry on oeis.org

216, 1000, 2744, 3375, 9261, 10648, 17576, 35937, 39304, 42875, 54872, 59319, 97336, 132651, 166375, 185193, 195112, 238328, 274625, 328509, 405224, 456533, 551368, 614125, 636056, 658503, 753571, 804357, 830584, 857375, 1191016, 1367631, 1520875, 1643032

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Subset of A046306, of A000578, and of A007774. - R. J. Mathar, Jun 27 2009

			216=2^3*3^3. 1000=2^3*5^3. 2744=2^3*7^3.

Cf. A000578, A006881, A007774, A046306.

Cf. A085541, A085966.

fQ[n_]:=Last/@FactorInteger[n]=={3,3}; lst={};Do[If[fQ[n],AppendTo[lst, n]],{n,6*9!}];lst
With[{nn=30},Select[Union[(Times@@@Subsets[Prime[Range[nn]],{2}])^3],#<= (2Prime[ nn])^3&]](* Harvey P. Dale, May 27 2024 *)

from math import isqrt
from sympy import primepi, primerange
def A162142(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3 # Chai Wah Wu, Dec 09 2024

a(n) = (A006881(n))^3 = A000578(A006881(n)). - R. J. Mathar, Jun 27 2009

Sum_{n>=1} 1/a(n) = (P(3)^2 - P(6))/2 = (A085541^2 - A085966)/2 = 0.006735..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Definition rephrased by R. J. Mathar, Jun 27 2009

A162143 Numbers that are the squares of the product of three distinct primes.

Original entry on oeis.org

900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, 28900, 30276, 33124, 34596, 36100, 38025, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 74529, 79524, 81225, 81796, 84100, 96100, 101124, 103684, 119025, 125316, 127449

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Numbers that are the product of exactly 3 distinct squares of primes (p^2*q^2*r^2).

			900 = 2^2*3^2*5^2, 1764 = 2^2*3^2*7^2, 4356 = 2^2*3^2*11^2, ..

Cf. A007304, A006881, A050326, A162142, A085548, A085964, A085966.

h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A162143List := upto -> seq(n^2, n=select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025

fQ[n_]:=Last/@FactorInteger[n]=={2,2,2}; Select[Range[100000], f]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162143(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 29 2024

def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n*n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

a(n) = A007304(n)^2.
A050326(a(n)) = 8. - Reinhard Zumkeller, May 03 2013
Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 = (A085548^3 + 2*A085966 - 3*A085548*A085964)/6 = 0.0036962441..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Edited by N. J. A. Sloane, Jun 27 2009

A189988 Numbers with prime factorization p^2*q^4.

Original entry on oeis.org

144, 324, 400, 784, 1936, 2025, 2500, 2704, 3969, 4624, 5625, 5776, 8464, 9604, 9801, 13456, 13689, 15376, 21609, 21904, 23409, 26896, 29241, 29584, 30625, 35344, 42849, 44944, 55696, 58564, 59536, 60025, 68121, 71824, 75625, 77841

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

Numbers k such that tau(k^2)/tau(k) = 3 where tau(n) is the number of divisors of n (A000005). - Michel Marcus, Feb 09 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A178740.

Cf. A085548, A085964, A085966.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,4}; Select[Range[150000],f]
Module[{upto=80000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^4,#[[1]]^4 #[[2]]^2}&/@ Subsets[Prime[Range[Sqrt[upto/16]]],{2}]]],#<=upto&]] (* Harvey P. Dale, Dec 15 2017 *)

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/4), t=p^4;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189988(n):
    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): return n+x-sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(4) - P(6) = A085548 * A085964 - A085966 = 0.017749..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A054753(n)^2. - R. J. Mathar, May 05 2023

A189990 Numbers with prime factorization p^2*q^6.

Original entry on oeis.org

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896, 470596

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Subsequence of A137484.
Cf. A179645, A179664.
Cf. A065036, A085548, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,6}; Select[Range[800000],f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/6), t=p^6;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189990(n):
    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): return n+x-sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) = A085548 * A085966 - A085968 = 0.003658..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A065036(n)^2. - Chai Wah Wu, Mar 27 2025

A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

Original entry on oeis.org

0, 1, 1, 32, 1, 275, 1, 1024, 243, 3157, 1, 8800, 1, 16839, 3368, 32768, 1, 66825, 1, 101024, 17050, 161083, 1, 281600, 3125, 371325, 59049, 538848, 1, 867151, 1, 1048576, 161294, 1419889, 19932, 2138400, 1, 2476131, 371536, 3232768, 1, 4629701, 1, 5154656, 818424, 6436375, 1

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^5. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 275; a(6) = 6^5 * Sum_{p|6, p prime} 1/p^5 = 7776 * (1/2^5 + 1/3^5) = 275.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), this sequence (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059378, A085966.

Array[#^5*DivisorSum[#, 1/#^5 &, PrimeQ] &, 47] (* Stefano Spezia, Jul 15 2025 *)

a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^5) \\ David A. Corneth, Jul 15 2025

a(A000040(n)) = 1.
Dirichlet g.f.: zeta(s-5)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^5/n^s) Sum_{p|n} 1/p^5. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^5*(p*j)^(s-5)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-5) = zeta(s-5)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 03 2023
Sum_{k=1..n} a(k) ~ A085966 * n^6/6. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059378(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^5, where c = A010051.
a(p^k) = p^(5*k-5) for p prime and k>=1. (End)

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