A085541 -id:A085541 - cp's OEIS frontend

A030078 Cubes of primes.

8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091, 2571353, 2685619, 3307949

Offset: 1

Patrick De Geest

Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001
Intersection of A014612 and A000578. Intersection of A014612 and A030513. - Wesley Ivan Hurt, Sep 10 2013
Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020
There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021

			a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.

Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen über Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Eric Weisstein's World of Mathematics, Prime Power.
Wikipedia, p-group, Classification.
Index to sequences related to prime signature

Other sequences that are k-th powers of primes are: A000040 (k=1), A001248 (k=2), this sequence (k=3), A030514 (k=4), A050997 (k=5), A030516 (k=6), A092759 (k=7), A179645 (k=8), A179665 (k=9), A030629 (k=10), A079395 (k=11), A030631 (k=12), A138031 (k=13), A030635 (k=16), A138032 (k=17), A030637 (k=18).
Cf. A060800, A131991, A000578, subsequence of A046099.
Subsequence of A007422 and of A054397.
Cf. A036966, A085541, A088453, A157289, A258600.

a030078 = a000578 . a000040
a030078_list = map a000578 a000040_list -- Reinhard Zumkeller, May 26 2012

[p^3: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Array[Prime[ # ]^3&, 5! ] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^3 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime, primerange
def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
print(aupton(35)) # Michael S. Branicky, Aug 27 2021

[p**3 for p in prime_range(100)] # Zerinvary Lajos, May 15 2007

n such that A062799(n) = 3. - Benoit Cloitre, Apr 06 2002
a(n) = A000040(n)^3. - Omar E. Pol, Jul 27 2009
A064380(a(n)) = A000010(a(n)). - Vladimir Shevelev, Apr 19 2010
A003415(a(n)) = A079705(n). - Reinhard Zumkeller, Jun 26 2011
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
A000005(a(n)) = 4. - Wesley Ivan Hurt, Sep 10 2013
a(n) = A119959(n) * A008864(n) -1.- R. J. Mathar, Aug 13 2019
Sum_{n>=1} 1/a(n) = P(3) = 0.1747626392... (A085541). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) (A157289).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) (A088453). (End)

A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Jul 03 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

			0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.

Jason Kimberley, Table of n, a(n) for n = 0..1093
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Persi Diaconis, Frederick Mosteller, and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors-including the square free case, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Lajos Hajdu and Rob Tijdeman, Integers represented by Lucas sequences, arXiv:2408.04982 [math.NT], 2024. See p. 16.
Shanta Laishram and Florian Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.
Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, Polynomial upper bounds on the number of differing columns of Delta-modular integer programs, arXiv:2105.08160 [math.OC], 2021, see page 23.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Gerhard Niklasch and Pieter Moree, Some number-theoretical constants. [Cached copy]
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Hanson Smith, Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves, arXiv:1810.04809 [math.NT], 2018.
Hanson Smith, Ramification in Division Fields and Sporadic Points on Modular Curves, U. Conn. (2020).
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Prime Zeta Function.
Index to constants which are prime zeta sums {2,0,0}

Decimal expansion of the prime zeta function: this sequence (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.
Cf. A000720, A001248, A013661, A078437, A242301.

R := RealField(106);
PrimeZeta := func;
Reverse(IntegerToSequence(Floor(PrimeZeta(2,173)*10^105)));
// Jason Kimberley, Dec 30 2016

RealDigits[PrimeZetaP[2], 10, 105][[1]]  (* Jean-François Alcover, Jun 24 2011, updated May 06 2021 *)

recip2(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^2; ); print(v) }

eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)),9)
lm=lambertw(log(4)/eps())\log(4);
sum(k=1,lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013

sumeulerrat(1/p,2) \\ Hugo Pfoertner, Feb 03 2020

P(2) = Sum_{p prime} 1/p^2 = Sum_{n>=1} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010
Equals Sum_{k>=1} 1/A001248(k). - Amiram Eldar, Jul 27 2020
Equals Sum_{k>=2} pi(k)*(2*k+1)/(k^2*(k+1)^2), where pi(k) = A000720(k) (Shamos, 2011, p. 9). - Amiram Eldar, Mar 12 2024

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

Offset corrected by R. J. Mathar, Feb 05 2009

A003627 Primes of the form 3n-1.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane and Mira Bernstein

Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003
A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - Juri-Stepan Gerasimov, Jan 28 2010
Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
2 adjoined to A007528. - David A. Corneth, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Eisenstein Prime
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317.
Index to sequences related to decomposition of primes in quadratic fields

Primes of form 3n+1 give A002476.
These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.
Cf. A001359, A007528, A007645, A221717, A057145.
Subsequence of A034020.

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011

t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644.
Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

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.

Original entry on oeis.org

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

A136141 Decimal expansion of Sum_{p prime} 1/(p*(p-1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 09 2008

Excess of prime factors with multiplicity over distinct prime factors for random (large) integers. - Charles R Greathouse IV, Sep 06 2011
Sum of reciprocals of (proper) prime powers. The sum of reciprocals of all proper powers is A072102. - Charles R Greathouse IV, Apr 24 2012
Decimal expansion of the infinite sum of the reciprocals of the prime powers which are not prime (A246547). - Robert G. Wilson v, May 13 2019
See the second 'Applications' example under the Mathematica help file for the function PrimePowerQ. - Robert G. Wilson v, May 13 2019
It easy to prove that this constant < 1 (Sum_{p prime} 1/(p*(p-1)) < Sum_{k>=2} 1/(k*(k-1)) = 1). Luthar (1969) asks for a better upper bound. The solution shows that this constant is < 3/2 - log(2) = 0.80685... . - Amiram Eldar, Feb 14 2025

			Equals 1/2 + 1/(3*2) + 1/(5*4) + 1/(7*6) + ...
= 0.7731566690497951278643674598559423956187413360831860483110060673567...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants, p. 94.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 1002 terms from Jason Kimberley).
Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint 1991.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Stephan Ramon Garcia and Ethan Simpson Lee, Explicit conditional bounds for the residue of a Dedekind zeta-function at s = 1, arXiv:2506.17416 [math.NT], 2025. See p. 3.
R. S. Luthar, Problem E 2192, Elementary Problems, The American Mathematical Monthly, Vol. 76, No. 8 (1969), p. 938; An Upper Bound, Solution to Problem E 2192, by O. P. Lossers, ibid., Vol. 77, No. 7 (1970), pp. 769-770.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Tables 8 and 10.
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Index to constants which are prime zeta sums {1,1,0}

Cf. A152447 (over the semiprimes), A000040, A000720, A001248, A046660 (excess, see first comment), A072102, A077761, A083342, A179119, A246547.

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

R := RealField(105);
c := &+[R|(EulerPhi(n)-MoebiusMu(n))/n*Log(ZetaFunction(R,n)):n in[2..360]];
Reverse(IntegerToSequence(Floor(c*10^103))); // Jason Kimberley, Jan 12 2017

digits = 103; sp = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)

W(x)=solve(y=log(x)/2,max(1,log(x)),y*exp(y)-x)
eps()=2. >> (32*ceil(default(realprecision)/9.63))
primezeta(s)=my(t=s*log(2),iter=W(t/eps())\t);sum(k=1,iter, moebius(k)/k*log(abs(zeta(k*s))))
a(lim,e)={ \\ choose parameters to maximize speed and precision
    my(x,y=exp(W(lim)-.5));
    x=lim^e*(e*log(y))^e*(y*log(y))^-e*incgam(-e,e*log(y));
    forprime(p=2,lim,x+=1/((p*1.)^e*(p-1)));
    x+sum(n=2,e,primezeta(n))
}; \\ Charles R Greathouse IV, Sep 07 2011

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

Equals Sum_{n>=1} 1/A036689(n).
Equals Sum_{s>=2} P(s), where P is the prime zeta function. - Charles R Greathouse IV, Sep 06 2011
Equals A083342 - A077761, that is, Sum_{n>=2} ((EulerPhi(n) - MoebiusMu(n))/n) * log(zeta(n)). - Jean-François Alcover, Sep 02 2015
Equals 2 * Sum_{k>=2} pi(k)/(k^3-k), where pi(k) = A000720(k) (Shamos, 2011, p. 8). - Amiram Eldar, Mar 12 2024

More terms from D. S. McNeil, Sep 06 2011

More digits from Jean-François Alcover, Sep 02 2015

A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 21 2013

			0.33022992626420324101.. = 1/(2*3) +1/(3*4) +1/(5*6) + 1/(7*8) +... = sum_{n>=1} 1/ (A000040(n)*A008864(n)).

Jason Kimberley, Table of n, a(n) for n = 0..683
Index to constants which are prime zeta sums {1,0,1}

Cf. A136141 for 1/(p(p-1)), A085548 for 1/p^2.
Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A307379.

R:=RealField(103);
ExhaustSum :=
  function(
    k_min, term
  : IZ := func)
    c:=R!0; k:=k_min;
    repeat
      t:=term(k); c+:=t; k+:=1;
    until IZ(t,k-1);
    return c;
  end function;
RealField(101)!
ExhaustSum(2,
  func
    : IZ:=func
    )>);
// Jason Kimberley, Jan 20 2017

interface(quiet=true):
read("transforms") ;
Digits := 300 ;
ZetaM := proc(s,M)
    local v,p;
    v := Zeta(s) ;
    p := 2;
    while p <= M do
        v := v*(1-1/p^s) ;
        p := nextprime(p) ;
    end do:
    v ;
end proc:
Hurw := proc(a)
        local T,p,x,L,i,Le,pre,preT,v,t,M ;
    T := 40 ;
    preT := 0.0 ;
    while true do
            1/p/(p+a) ;
            subs(p=1/x,%) ;
            exp(%) ;
            t := taylor(%,x=0,T) ;
            L := [] ;
            for i from 1 to T-1 do
                    L := [op(L),evalf(coeftayl(t,x=0,i))] ;
            end do:
            Le := EULERi(L) ;
        M := -a ;
            v := 1.0 ;
            pre := 0.0 ;
            for i from 2 to nops(Le) do
                    pre := log(v) ;
                    v := v*evalf(ZetaM(i,M))^op(i,Le) ;
                    v := evalf(v) ;
            end do:
        pre := (log(v)+pre)/2. ;
        printf("%.105f\n",%) ;
        if abs(1.0-preT/pre)  < 10^(-Digits/3) then
            break;
        end if;
        preT := pre ;
        T := T+10 ;
    end do:
        pre ;
end proc:
A179119 := proc()
    Hurw(1) ;
end proc:
A179119() ;

digits = 101; S = NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 5]; RealDigits[S, 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)

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))))
sumalt(k=2,(-1)^k*primezeta(k)) \\ Charles R Greathouse IV, Aug 03 2016

sumeulerrat(1/(p*(p+1))) \\ Amiram Eldar, Mar 18 2021

P(2) - P(3) + P(4) - P(5) + ..., where P is the prime zeta function. - Charles R Greathouse IV, Aug 03 2016

A085964 Decimal expansion of the prime zeta function at 4.

Original entry on oeis.org

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

			0.0769931397642468449426...

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..1603
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), this sequence (at 4), A085965 (at 5) to A085969 (at 9).

Cf. A013662, A030514, A242303.

R := RealField(106);
PrimeZeta := func;
[0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(4,87)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[4*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 *)
RealDigits[ PrimeZetaP[ 4], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,4) \\ Hugo Pfoertner, Feb 03 2020

P(4) = Sum_{p prime} 1/p^4 = Sum_{n>=1} mobius(n)*log(zeta(4*n))/n
Equals A086034 + A085993 + 1/16. - R. J. Mathar, Jul 22 2010
Equals Sum_{k>=1} 1/A030514(k). - Amiram Eldar, Jul 27 2020

A085966 Decimal expansion of the prime zeta function at 6.

Original entry on oeis.org

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

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

			0.0170700868506365129541...

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..1802
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), A085965 (at 5), this sequence (at 6), A085967 (at 7) to A085969 (at 9).

Cf. A013664, A030516.

R := RealField(106);
PrimeZeta := func;
[0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(6,57)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[6*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 *)
RealDigits[ PrimeZetaP[ 6], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p,6) \\ Hugo Pfoertner, Feb 03 2020

P(6) = Sum_{p prime} 1/p^6 = Sum_{n>=1} mobius(n)*log(zeta(6*n))/n
Equals 1/2^6 + A085995 + A086036. - R. J. Mathar, Jul 14 2012
Equals Sum_{k>=1} 1/A030516(k). - Amiram Eldar, Jul 27 2020

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

A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436

Offset: 1

M. F. Hasler, Aug 27 2008

Also: numbers with prime signature {3,2}.
This is a subsequence of A114128. [Hasler]
Every a(n) is an Achilles number (A052486). They are minimal, meaning no proper divisor is an Achilles number. - Antonio Roldán, Dec 27 2011

			The first three terms of this sequence are 3^2 * 2^3 = 72, 2^2 * 3^3 = 108, 5^2 * 2^3 = 200.

T. D. Noe, Table of n, a(n) for n = 1..1000
Project Euler, Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string 200.
Index to sequences related to prime signature

Cf. A114128.

Cf. A085541, A085548, A085965.

f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)

for(n=1, 10^5, omega(n)==2 || next; vecsort(factor(n)[,2])==[2,3]~ && print1(n","))

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/3), t=p^3;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, primerange, integer_nthroot
def A143610(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**3)) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) = A085548 * A085541 - A085965 = 0.043280..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

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A179119 Decimal expansion of Sum_{p prime} 1/(p*(p+1)).

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