A085966 -id:A085966 - cp's OEIS frontend

A190471 Numbers with prime factorization p^2q^4r^4 where p, q, and r are distinct primes.

32400, 63504, 90000, 156816, 202500, 219024, 345744, 374544, 467856, 490000, 685584, 777924, 960400, 1089936, 1210000, 1245456, 1690000, 1774224, 2108304, 2178576, 2396304, 2480625, 2862864, 2890000, 3610000, 3640464, 4112784, 4511376

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. A190469, A190470.

Cf. A085548, A085964, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,4,4}; Select[Range[3500000],f] (*and*) lst={}; Do[If[k!=n && k!=m && n!=m, AppendTo[lst, Prime[k]^2*Prime[n]^4*Prime[m]^4]], {n,33}, {m,33}, {k,33}]; Take[Union@lst,60]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\4)^(1/8), t1=p^4;forprime(q=p+1, (lim\t1)^(1/4), t2=t1*q^4;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Sum_{n>=1} 1/a(n) = P(2)*P(4)^2/2 - P(2)*P(8)/2 - P(4)*P(6) + P(10) = 0.00010139253539568059065..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

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

			0.0013808358869717391630318541280158226106013963275654296802648025785307522...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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 [math.NT], 2010-2015, value P(m=4, n=3, s=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A002145 (primes 4k+3), A001014 (n^6), A085966 (PrimeZeta(6)).

Cf. A085991 - A085998 (Zeta_R(2..9): same for 1/p^2, ..., 1/p^9), A086036 (same for primes 4k+1), A343626 (for primes 3k+1), A343616 (for primes 3k+2).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 250; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*6]]/(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 *)

A085995_upto(N=100)={localprec(N+3); digits((PrimeZeta43(6)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(6) = Sum_{p in A002145} 1/p^6 where A002145 = {primes p == 3 (mod 4)},
  = (1/2)*Sum_{n >= 0} möbius(2*n+1)*log(b((2*n+1)*6))/(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

A162144 Products of cubes of 3 distinct primes.

Original entry on oeis.org

27000, 74088, 287496, 343000, 474552, 1061208, 1157625, 1331000, 1481544, 2197000, 2628072, 3652264, 4492125, 4913000, 5268024, 6028568, 6434856, 6859000, 7414875, 10941048, 12167000, 12326391, 13481272, 14886936, 16581375, 17173512, 18821096

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Numbers of the form p^3*q^3*r^3 where p, q, r are three distinct primes.

The cubic analog of A085986 (squares of 2 distinct primes).

			27000 = 2^3*3^3*5^3. 74088 = 2^3*3^3*7^3. 287496 = 2^3*3^3*11^3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A006881, A085986, A162142, A162143.

Cf. A085541, A085966, A085969.

fQ[n_]:=Last/@FactorInteger[n]=={1,1,1}; Select[Range[1000], fQ]^3

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162144(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)**3 # Chai Wah Wu, Aug 30 2024

a(n) = (A007304(n))^3.
A000005(a(n)) = 64.
Sum_{n>=1} 1/a(n) = (P(3)^3 + 2*P(9) - 3*P(3)*P(6))/6 = (A085541^3 + 2*A085969 - 3*A085541*A085966)/6 = 0.0000661486..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Edited by R. J. Mathar, Aug 14 2009

A190469 Numbers with prime factorization p^2q^2r^6 where p, q, and r are distinct primes.

Original entry on oeis.org

14400, 28224, 69696, 72900, 78400, 97344, 142884, 166464, 193600, 207936, 270400, 304704, 352836, 379456, 462400, 484416, 492804, 529984, 553536, 562500, 577600, 788544, 842724, 846400, 893025, 906304, 968256, 1052676, 1065024, 1132096

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. A190466, A190467, A190468.

Cf. A085548, A085964, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,6}; Select[Range[1600000],f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/6), t1=p^6;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(6)/2 - P(2)*P(8)/2 - P(4)*P(6)/2 - P(2)*P(8) + P(10) = 0.00024535673248061231753..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A343626 Decimal expansion of the Prime Zeta modulo function P_{3,1}(6) = Sum 1/p^6 over primes p == 1 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 23 2021

The Prime Zeta modulo function at 6 for primes of the form 3k+1 is Sum_{primes in A002476} 1/p^6 = 1/7^6 + 1/13^6 + 1/19^6 + 1/31^6 + ...

The complementary Sum_{primes in A003627} 1/p^6 is given by P_{3,2}(6) = A085966 - 1/3^6 - (this value here) = 0.015689614727130461563527666... = A343606.

			P_{3,1}(6) = 8.7300110231981670120427791452319495610797645391837...*10^-8

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), A343606 (P_{3,2}(6): same for p==2 (mod 3)), A086036 (P_{4,1}(6): same for p==1 (mod 4)).

Cf. A085966 (PrimeZeta(6)), A002476 (primes of the form 3k+1).

With[{s=6}, 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^6); s \\ For illustration: primes up to 10^N give 5N+2 (= 42 for N=8) correct digits.

A343626_upto(N=100)={localprec(N+5);digits((PrimeZeta31(6)+1)\.1^N)[^1]} \\ cf. A175644 for PrimeZeta31

A131653 Decimal expansion of the sum of the reciprocals of squared 3-almost primes.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 10 2007, Mar 07 2008

zeta(2) = A013661 is 1 plus a sum over inverse squares of k-almost primes, k=1 to infinity, where A085548 represents k=1, A117543 represents k=2 and this constant here represents k=3.

			0.038516192982694640912837922628060395438900167478381571937...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009.

Cf. A000290, A013661, A014612, A085548, A085964, A085966, A117543, A085548.

RealDigits[PrimeZetaP[2]^3/6 + PrimeZetaP[6]/3 + PrimeZetaP[2]*PrimeZetaP[4]/2, 10, 120, -1][[1]] (* Amiram Eldar, Jun 25 2023 *)

Equals A085548^3 / 6 + A085966 / 3 + A085548 * A085964 / 2.

Equals Sum_{i>=1} 1/A000290(A014612(i)).

a(104) corrected by Amiram Eldar, Jun 25 2023

A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

Original entry on oeis.org

72, 108, 128, 200, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000

Offset: 1

V. Raman, Sep 07 2012

nonn

Terms of A216427 that are not 5th powers of squarefree numbers (A113850) and not 10th powers of primes (A030629). - Amiram Eldar, Feb 07 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A143610.
Subsequence of A216427. - Zak Seidov, Jan 03 2014
Cf. A002117, A013661, A013663, A013664, A013668, A030629, A085966, A113850.

With[{upto=4000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^3,#[[2]]^2 #[[1]]^3}& /@ Subsets[Range[2,Surd[upto,2]],{2}]]],#<=upto&]](* Harvey P. Dale, Jan 04 2014 *)
pMx = 25; mx = 2^3 pMx^2; t = Flatten[Table[If[a != b, a^2 b^3, 0], {a, 2, mx^(1/2)}, {b, 2, mx^(1/3)}]]; Union[Select[t, 0 < # <= mx &]] (* T. D. Noe, Jan 02 2014 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4,3), for(a=2, sqrtint(lim\b^3), if(a!=b, listput(v, a^2*b^3)))); Set(v) \\ Charles R Greathouse IV, Jan 02 2014

from math import isqrt
from sympy import integer_nthroot, mobius, primepi
def A216426(n):
    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
    def f(x):
        j, b, a, d = isqrt(x), integer_nthroot(x,6)[0], integer_nthroot(x,5)[0], integer_nthroot(x,10)[0]
        l, c = 0, n+x-2+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(d+1, b+1))+primepi(d)+sum(mobius(k)*(a//k**2+j//k**3) for k in range(1, d+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = sum(mobius(k)*((k2-1)//k**2) for k in range(1, isqrt(k2-1)+1))
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Sum_{n>=1} 1/a(n) = 2 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - zeta(5)/zeta(10) - P(6) - P(10) = 0.09117811499514578262..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

Name corrected by Charles R Greathouse IV, Jan 02 2014

A272190 Either 6th power of a prime, or product of the square of two different primes.

Original entry on oeis.org

36, 64, 100, 196, 225, 441, 484, 676, 729, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129, 15625, 16641, 17689, 17956, 19881

Offset: 1

Paolo P. Lava, Apr 22 2016

Numbers such that the sum of the number of divisors of their aliquot parts is three times the number of their divisors.

			36 = 2^2 * 3^2;  64 = 2^6.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Paolo P. Lava)

Cf. A030516, A080258, A085986, A272191.

Cf. A085548, A085964, A085966.

with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);
if 3*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);

Select[Range[20000], MemberQ[{{6}, {2, 2}}, FactorInteger[#][[;; , 2]]] &] (* Amiram Eldar, Oct 03 2023 *)

isok(n) = 3*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ Michel Marcus, Apr 22 2016

is(n) = {my(e = factor(n)[, 2]~); e == [6] || e == [2, 2];} \\ Amiram Eldar, Oct 03 2023

Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 + P(6) = (A085548^2 - A085964)/2 + A085966 = 0.080837..., where P is the prime zeta function. - Amiram Eldar, Oct 03 2023

A343616 Decimal expansion of P_{3,2}(6) = Sum 1/p^6 over primes == 2 (mod 3).

Original entry on oeis.org

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

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.

			0.015689614727130461563527666152209091814208675553077763366153188676457...

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=6), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001014 (n^6), A085966 (PrimeZeta(6)), A021733 (1/3^6).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343626 (for primes 3k+1), A086036 (for primes 4k+1), A085995 (for primes 4k+3).

A343616_upto(N=100)={localprec(N+5); digits((PrimeZeta32(6)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(6) = Sum_{p in A003627} 1/p^6 = P(6) - 1/3^6 - P_{3,1}(6).

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

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A190471 Numbers with prime factorization p^2*q^4*r^4 where p, q, and r are distinct primes.

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A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

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A162144 Products of cubes of 3 distinct primes.

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A190469 Numbers with prime factorization p^2*q^2*r^6 where p, q, and r are distinct primes.

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A343626 Decimal expansion of the Prime Zeta modulo function P_{3,1}(6) = Sum 1/p^6 over primes p == 1 (mod 3).

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A131653 Decimal expansion of the sum of the reciprocals of squared 3-almost primes.

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A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

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A190471 Numbers with prime factorization p^2q^4r^4 where p, q, and r are distinct primes.

A190469 Numbers with prime factorization p^2q^2r^6 where p, q, and r are distinct primes.