A085967 -id:A085967 - cp's OEIS frontend

A190472 Numbers with prime factorization p^3q^3r^4 where p, q, and r are distinct primes.

54000, 81000, 135000, 148176, 222264, 518616, 574992, 686000, 862488, 949104, 1423656, 1715000, 2122416, 2401000, 2662000, 2963088, 3162456, 3183624, 3472875, 4394000, 4444632, 5256144, 5788125, 6169176, 6655000, 7304528, 7884216

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of prime signatures, 2010.
Index to sequences related to prime signature.

Cf. A190470, A190471.

Cf. A085541, A085964, A085966, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,3,4}; Select[Range[5000000],f] (*and*) lst={}; Do[If[k!=n && k!=m && n!=m, AppendTo[lst, Prime[k]^3*Prime[n]^3*Prime[m]^4]], {n,25}, {m,25}, {k,25}]; Take[Union@lst,60]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\216)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;forprime(r=q+1, (lim\t2)^(1/3), if(p==r,next);listput(v,t2*r^3)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Sum_{n>=1} 1/a(n) = P(3)^2*P(4)/2 - P(4)*P(6)/2 - P(3)*P(7) + P(10) = 0.000064520760706206924448..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A179705 Numbers of the form p^7*q^3 where p and q are distinct primes.

Original entry on oeis.org

3456, 16000, 17496, 43904, 170368, 273375, 281216, 625000, 628864, 750141, 877952, 1557376, 2109375, 2910897, 3121792, 3813248, 4804839, 6483584, 6588344, 8821888, 10176896, 10744731, 13289344, 15000633, 19056256

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A085541, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,7}; Select[Range[100000],f]
With[{nn=25},Take[Union[#[[1]]^7 #[[2]]^3&/@(Flatten[{#,Reverse[ #]}&/@ Subsets[ Prime[Range[nn]],{2}],1])],nn]] (* Harvey P. Dale, Jan 01 2016 *)

list(lim)=my(v=List(),t);forprime(p=2, (lim\8)^(1/7), t=p^7;forprime(q=2, (lim\t)^(1/3), if(p==q, next);listput(v,t*q^3))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Sum_{n>=1} 1/a(n) = P(3)*P(7) - P(10) = A085541 * A085967 - P(10) = 0.000454..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A086037 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

			1.2818448599795268251026582166507935820606749563344794362656914682... * 10^-5

X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=1, s=7), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085996 (same for primes 4k+3), A343627 (for primes 3k+1), A343617 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085967 (PrimeZeta(7)), A002144 (primes of the form 4k+1).

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 1200; Join[{0, 0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*7]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

A086037_upto(N=100)={localprec(N+3); digits((PrimeZeta41(7)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(7) = Sum_{p in A002144} 1/p^7 where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m*log(DirichletBeta(7m)*zeta(7m)/zeta(14m)/(1+2^(-7m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A085996 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

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

			0.0004585144075337972668731121472822151533627221357444614502792647239732950115...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547, value P(m=4, n=3, s=7), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A086037 (analog for primes 4k+1), A085967 (PrimeZeta(7)), A002145 (primes 4k+3).

Cf. A085991 .. A085998 (Zeta_R(2..9)).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 275; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*7]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085996_upto(N=100)={localprec(N+3); digits((PrimeZeta43(7)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(7) = Sum_{primes p == 3 mod 4} 1/p^7
  = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*7))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A343627 Decimal expansion of the Prime Zeta modulo function P_{3,1}(7) = Sum 1/p^7 over primes p == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 23 2021

The Prime Zeta modulo function at 7 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^7 = 1/7^7 + 1/13^7 + 1/19^7 + 1/31^7 + ...

The complementary Sum_{primes in A003627} 1/p^7 is given by P_{3,2}(7) = A085967 - 1/3^7 - (this value here) = 0.0078253541130504928742517... = A343607.

			P_{3,1}(7) = 1.231372255481919674448947124444003936669057866...*10^-6

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p.21.
OEIS index to entries related to the (prime) zeta function.

Cf. A175645, A343624 - A343629 (P_{3,1}(3..9): same for 1/p^n, n = 3..9), A343607 (P_{3,2}(7): same for p==2 (mod 3)), A086037 (P_{4,1}(7): same for p==1 (mod 4)).

Cf. A085967 (PrimeZeta(7)), A002476 (primes of the form 3k+1).

With[{s=7}, Do[Print[N[1/2 * Sum[(MoebiusMu[2*n + 1]/(2*n + 1)) * Log[(Zeta[s + 2*n*s]*(Zeta[s + 2*n*s, 1/6] - Zeta[s + 2*n*s, 5/6])) / ((1 + 2^(s + 2*n*s))*(1 + 3^(s + 2*n*s)) * Zeta[2*(1 + 2*n)*s])], {n, 0, m}], 120]], {m, 100, 500, 100}]] (* adopted from Vaclav Kotesovec's code in A175645 *)

s=0; forprimestep(p=1, 1e8, 3, s+=1./p^7); s \\ For illustration: primes up to 10^N give 6N+2 (= 50 for N=8) correct digits.

A343627_upto(N=100)={localprec(N+5);digits((PrimeZeta31(7)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

A126226 Continued fraction of Product_{primes p} ((p-1)/p)^(1/p).

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 11, 1, 1, 4, 1, 9, 2, 2, 1, 1, 4, 4, 2, 2, 2, 1, 14, 1, 2, 2, 2, 7, 2, 2, 1, 1, 4, 2, 4, 1, 11, 7, 2, 8, 32, 2, 1, 293, 2, 145, 1, 2, 1, 21, 1, 1, 3, 1, 1, 8, 8, 5, 2, 3, 4, 3, 1, 3, 1, 1, 1, 1, 3, 2, 1, 3, 1, 2, 2, 1, 2, 19, 3, 2, 1, 15, 1, 2, 1, 2, 5, 3, 1, 1, 1, 38, 1, 10, 1, 2, 1, 80, 1

Offset: 0

Martin Fuller, Dec 20 2006

This might be interpreted as the expected value of phi(n)/n for very large n. - David W. Wilson, Dec 05 2006

			0.55986561693237348...

Cf. A124175, A085548, A085541, A085964, A085965, A085966, A085967, A085968, A085969.

contfrac(exp(-suminf(m=2,log(zeta(m))*sumdiv(m,k,if(k

A190114 Numbers with prime factorization p^2q^2r^5 where p, q, and r are distinct primes.

Original entry on oeis.org

7200, 14112, 24300, 34848, 39200, 47628, 48672, 83232, 96800, 103968, 112500, 117612, 135200, 152352, 164268, 189728, 231200, 242208, 264992, 276768, 280908, 288800, 297675, 350892, 394272, 423200, 453152, 484128, 514188, 532512, 566048

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of prime signatures, 2010.
Index to sequences related to prime signature.

Cf. A179665, A190011, A190012.

Cf. A085548, A085964, A085965, A085967, A085968, A085969.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,5};Select[Range[900000],f]
With[{upto=600000},Select[#[[1]]^2 #[[2]]^2 #[[3]]^5&/@ Flatten[ Permutations/@ Subsets[Prime[Range[Ceiling[Surd[upto,5]+1]]],{3}],1]// Union,#<=upto&]] (* Harvey P. Dale, Jul 29 2018 *)

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=q+1, sqrt(lim\t2), if(p==r,next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Sum_{n>=1} 1/a(n) = P(2)^2*P(5)/2 - P(2)*P(8)/2 - P(4)*P(5)/2 - P(2)*P(7) + P(9) = 0.00053812627050585644544..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0078253541130504928742517016707559206033079309751324433146804883394...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=7), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001015 (n^7), A085967 (PrimeZeta(7)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343627 (for primes 3k+1), A086037 (for primes 4k+1), A085996 (for primes 4k+3).

A343617_upto(N=100)={localprec(N+5); digits((PrimeZeta32(7)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(7) = Sum_{p in A003627} 1/p^7 = P(7) - 1/3^7 - P_{3,1}(7).

cp's OEIS Frontend

A190472 Numbers with prime factorization p^3*q^3*r^4 where p, q, and r are distinct primes.

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A179705 Numbers of the form p^7*q^3 where p and q are distinct primes.

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A086037 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+1.

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A085996 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+3.

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A343627 Decimal expansion of the Prime Zeta modulo function P_{3,1}(7) = Sum 1/p^7 over primes p == 1 (mod 3).

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A126226 Continued fraction of Product_{primes p} ((p-1)/p)^(1/p).

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A190114 Numbers with prime factorization p^2*q^2*r^5 where p, q, and r are distinct primes.

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A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

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A190472 Numbers with prime factorization p^3q^3r^4 where p, q, and r are distinct primes.

A190114 Numbers with prime factorization p^2q^2r^5 where p, q, and r are distinct primes.