A124012 -id:A124012 - cp's OEIS frontend

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

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

A033286 a(n) = n * prime(n).

Original entry on oeis.org

2, 6, 15, 28, 55, 78, 119, 152, 207, 290, 341, 444, 533, 602, 705, 848, 1003, 1098, 1273, 1420, 1533, 1738, 1909, 2136, 2425, 2626, 2781, 2996, 3161, 3390, 3937, 4192, 4521, 4726, 5215, 5436, 5809, 6194, 6513, 6920, 7339, 7602, 8213, 8492, 8865, 9154, 9917

Offset: 1

N. J. A. Sloane

Does an n exist such that n*prime(n)/(n+prime(n)) is an integer? - Ctibor O. Zizka, Mar 04 2008. The answer to Zizka's question is easily seen to be No: such an integer k would be positive and less than prime(n), but then k*(n + prime(n)) = prime(n)*n would be impossible. - Robert Israel, Apr 20 2015
Sums of rows of the triangle in A005145. - Reinhard Zumkeller, Aug 05 2009
Complement of A171520(n). - Jaroslav Krizek, Dec 13 2009
Partial sums of A090942. - Omar E. Pol, Apr 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 1.
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.

Cf. A000040, A007504, A014689, A090942, A124012, A141042, A152535.

Cf. A005145 (primes repeated), A171520 (complement), A076146 (iterated).

a033286 n = a000040 n * n  -- Reinhard Zumkeller, Jul 24 2013

[ n*NthPrime(n): n in [1..47] ]; // Klaus Brockhaus, Sep 09 2009

A033286 := proc(n) n*ithprime(n) ; end proc:
seq(A033286(n),n=1..20) ; # R. J. Mathar, Mar 21 2011

Table[Prime[n]*n, {n, 38}] (* Alonso del Arte *)

ithprime(i)*i $ i = 1..47 // Zerinvary Lajos, Feb 26 2007

a(n)=n*prime(n) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = n * A000040(n) = n * A008578(n+1) = n * A158611(n+2). - Jaroslav Krizek, Aug 31 2009
a(n) = A007504(n) + A152535(n). - Omar E. Pol, Aug 09 2012
Sum_{n>=1} 1/a(n) = A124012. - Amiram Eldar, Oct 15 2020

Correction for change of offset in A158611 and A008578 in Aug 2009 from Jaroslav Krizek, Jan 27 2010

A324852 Number of distinct prime indices of n that divide n.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			60060 has 7 prime indices {1,1,2,3,4,5,6}, all of which divide 60060, and 6 of which are distinct, so a(60060) = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..65536

The version for all prime indices (counted with multiplicity) is A324848.
Positions of zeros are A324846.
Positions of ones are A323440.
Cf. A000720, A001222, A003963, A056239, A120383, A124012.
Cf. A324704, A324771, A324847, A324849, A324850, A324853, A324856.

a:= n-> add(`if`(irem(n, numtheory[pi](i[1]))=0, 1, 0), i=ifactors(n)[2]):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 19 2019

Table[Count[If[n==1,{},FactorInteger[n]],{p_,_}/;Divisible[n,PrimePi[p]]],{n,100}]

a(n) = {my(f = factor(n)[,1]); sum(k=1, #f, !(n % primepi(f[k])));} \\ Michel Marcus, Mar 19 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/(k*prime(k)) = 0.848969... (A124012). - Amiram Eldar, Jan 11 2025

The convergence is very slow, need to use the first 100000000 primes to obtain the correct value of the coefficient of 10^(-23).

The constant is in the interval [0.28417070547086825017714, 0.28417070547086825017743]; these safe limits are computed by accumulating in parallel the partial sum of the lower estimate 1/n^4 = Zeta(4). - R. J. Mathar, Feb 06 2015

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

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A033286 a(n) = n * prime(n).

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A324852 Number of distinct prime indices of n that divide n.

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A253634 Decimal expansion of Sum_{n>=1} 1/A033286(n)^2.

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