A085966 -id:A085966 - cp's OEIS frontend

A267697 Numbers with 7 odd divisors.

729, 1458, 2916, 5832, 11664, 15625, 23328, 31250, 46656, 62500, 93312, 117649, 125000, 186624, 235298, 250000, 373248, 470596, 500000, 746496, 941192, 1000000, 1492992, 1771561, 1882384, 2000000, 2985984, 3543122, 3764768, 4000000, 4826809, 5971968, 7086244, 7529536, 8000000, 9653618

Offset: 1

Omar E. Pol, Apr 03 2016

Positive integers that have exactly seven odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 7 subparts. - Omar E. Pol, Dec 28 2016
Numbers that can be formed in exactly 6 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018
Numbers of the form p^6 * 2^k where p is an odd prime. - David A. Corneth, Aug 14 2018

David A. Corneth, Table of n, a(n) for n = 1..10000

Column 7 of A266531.
Cf. A001227, A030516, A038547, A085966, A236104, A237593, A279387.
Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, this sequence, A267891, A267892, A267893.

isok(n) = sumdiv(n, d, (d%2)) == 7; \\ Michel Marcus, Apr 03 2016

upto(n) = {my(res = List()); forprime(p = 3, sqrtnint(n, 6), listput(res, p^6)); q = #res; for(i = 1, q, odd = res[i]; for(j = 1, logint(n \ odd, 2), listput(res, odd <<= 1))); listsort(res); res} \\ David A. Corneth, Aug 14 2018

from sympy import integer_log, primerange, integer_nthroot
def A267697(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 int(n+x-sum(integer_log(x//p**6,2)[0]+1 for p in primerange(3,integer_nthroot(x,6)[0]+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A001227(a(n)) = 7.

Sum_{n>=1} 1/a(n) = 2 * P(6) - 1/32 = 0.00289017370127..., where P(6) is the value of the prime zeta function at 6 (A085966). - Amiram Eldar, Sep 16 2024

More terms from Michel Marcus, Apr 03 2016

A382235 Decimal expansion of the multiple prime zeta value primezetamult(3, 3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

Prime zeta analog of A258987.

			0.00673594662213544672456...

Artur Jasinski, Complete list of multiple prime zeta values up to weight 6.
Index to constants which are prime multiple zeta values (3,3)

Cf. A258987, A382234 (2,2), A382236 (2,2,).

kk = {0, 0}; kkk = RealDigits[(PrimeZetaP[3]^2 - PrimeZetaP[6])/2, 10, 105][[1]]; Flatten[AppendTo[kk, kkk]]

Equals (A085541^2 - A085966)/2 .

Equals Sum_{p,q prime p>q} 1/(p^3*q^3).

A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.

Original entry on oeis.org

1728, 5832, 8000, 21952, 85184, 91125, 125000, 140608, 250047, 314432, 421875, 438976, 778688, 941192, 970299, 1560896, 1601613, 1906624, 3176523, 3241792, 3581577, 4410944, 5000211, 5088448, 5359375, 6644672

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. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693.

Cf. A085541, A085966, A085969.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,6}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\8)^(1/6), t=p^6;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

from sympy import primepi, integer_nthroot, primerange
def A179694(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(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(3)*P(6) - P(9) = A085541 * A085966 - A085969 = 0.000978..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A054753(n)^3. - R. J. Mathar, May 05 2023

A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.

Original entry on oeis.org

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

Offset: 0

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

			6.4250963664773791101819138043576598984545546978815052898566258438984520...*10^-5

Jean-François Alcover, Table of n, a(n) for n = 0..1006
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=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A085995 (same for primes 4k+3), A343626 (for primes 3k+1), A343616 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085966 (PrimeZeta(6)), A002144 (primes of the form 4k+1).

digits = 1003; m0 = 50; dm = 10; dd = 10; Clear[f, g];
b[s_] := (1 + 2^-s)^-1 DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits + dd]&;
f[n_] := f[n] = (1/2) MoebiusMu[2n + 1]*Log[b[(2n + 1)*6]]/(2n + 1);
g[m_] := g[m] = Sum[f[n], {n, 0, m}]; g[m = m0]; g[m += dm];
While[Abs[g[m] - g[m - dm]] < 10^(-digits - dd), Print[m]; m += dm];
Join[{0, 0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

A086036_upto(N=100)={localprec(N+3); digits((PrimeZeta41(6)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Zeta_Q(6) = Sum_{p in A002144} 1/p^6 where A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(6m)*zeta(6m)/zeta(12m)/(1+2^(-6m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Edited by M. F. Hasler, Apr 26 2021

A190106 Numbers with prime factorization p^2q^3r^3 where p, q, and r are distinct primes.

Original entry on oeis.org

5400, 9000, 10584, 13500, 24696, 26136, 36504, 37044, 49000, 62424, 68600, 77976, 95832, 114264, 121000, 143748, 158184, 165375, 169000, 171500, 181656, 207576, 231525, 237276, 266200, 289000, 295704, 332024, 353736, 361000, 363096

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. A179691, A179698, A179746, A189991.

Cf. A085548, A085541, A085965, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,3};Select[Range[500000],f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\4)^(1/6), t1=p^3;forprime(q=p+1, (lim\t1)^(1/3), t2=t1*q^3;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A190106(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((t:=primepi(s:=isqrt(y:=integer_nthroot(x//r**2,3)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(2)*P(3)^2/2 - P(2)*P(6)/2 - P(3)*P(5) + P(8) = 0.00085907862422456410530..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A190115 Numbers with prime factorization p^2q^3r^4 where p, q, and r are distinct primes.

Original entry on oeis.org

10800, 16200, 18000, 21168, 31752, 40500, 45000, 49392, 52272, 67500, 73008, 78408, 98000, 109512, 111132, 124848, 137200, 155952, 172872, 187272, 191664, 228528, 233928, 242000, 245000, 259308, 316368, 338000, 342792, 363312, 415152

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. A093770, A190012, A190014.

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

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,4};Select[Range[900000],f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\72)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A190115(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//(r**4*q**3))) for r in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(integer_nthroot(x//r**4,3)[0]+1))+sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,9)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(2)*P(3)*P(4) - P(2)*P(7) - P(3)*P(6) - P(4)*P(5) + 2*P(9) = 0.00061171477910848082277..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A190382 Numbers with prime factorization p^2q^2r^2*s^3 where p, q, r, and s are distinct primes.

Original entry on oeis.org

88200, 132300, 217800, 220500, 304200, 308700, 326700, 426888, 456300, 520200, 544500, 596232, 640332, 649800, 760500, 780300, 894348, 952200, 974700, 1019592, 1185800, 1197900, 1273608, 1300500, 1428300, 1472328, 1494108, 1513800

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 09 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. A190319, A190320.

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

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,2,3};Select[Range[2000000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\900, 3), t1=p^3; forprime(q=2,sqrtint(lim\(36*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(4*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,sqrtint(lim\t3), if(s==p || s==q || s==r, next); listput(v, t3*s^2))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Sum_{n>=1} 1/a(n) = (P(2)^3*P(3) - 3*P(2)^2*P(5) - 3*P(2)*P(3)*P(4) + 6*P(2)*P(7) + 2*P(3)*P(6) + 3*P(4)*P(5) - 6*P(9))/6 = 0.00010783911499432484110..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A190464 Numbers with prime factorization p^4*q^6.

Original entry on oeis.org

5184, 11664, 40000, 153664, 250000, 455625, 937024, 1265625, 1750329, 1827904, 1882384, 5345344, 8340544, 9529569, 10673289, 17909824, 20820969, 28344976, 37515625, 45265984, 59105344, 60886809, 73530625, 77228944, 95004009, 119946304, 143496441, 180848704, 204004089, 218803264

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2011

nonn

A subsequence of A175745 (Numbers with 35 divisors).

First different term in A175745 is 17179869184(=2^34).

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

Cf. A030629, A179692, A179699, A179705, A175745.

Cf. A085964, A085966.

f[n_]:=Sort[Last/@FactorInteger[n]]=={4,6}; Select[Range[50000000],f] (*and*) lst={};Do[Do[If[n!=m,AppendTo[lst,Prime[n]^6*Prime[m]^4]], {n,50}],{m,50}]; Take[Union@lst,50]

list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/6), t=p^6;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Sum_{n>=1} 1/a(n) = P(4)*P(6) - P(10) = A085964 * A085966 - P(10) = 0.000320..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

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

Original entry on oeis.org

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

A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

Original entry on oeis.org

32, 72, 108, 128, 200, 243, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1024, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3125, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4096, 4232, 4500, 4563, 4608

Offset: 1

V. Raman, Sep 07 2012

nonn

Powerful numbers (A001694) that are not squares of cubefree numbers (A004709), cubes of squarefree numbers (A062838), or 6th powers of primes (A030516). - Amiram Eldar, Feb 07 2023

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

Cf. A143610, A216426.

Cf. A001694, A002117, A004709, A013661, A013664, A062838, A085966.

With[{max = 5000}, Union[Table[i^2*j^3, {j, 2, max^(1/3)}, {i, 2, Sqrt[max/j^3]}] // Flatten]] (* Amiram Eldar, Feb 07 2023 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4, 3), for(a=2, sqrtint(lim\b^3), listput(v, a^2*b^3))); Set(v) \\ Charles R Greathouse IV, Jan 03 2014

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A216427(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+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
    def f(x):
        j, b = isqrt(x), integer_nthroot(x,6)[0]
        l, c = 0, n+x-1+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(1, b+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            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) = 1 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - P(6) = 0.12806919584708298724..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

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A267697 Numbers with 7 odd divisors.

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A382235 Decimal expansion of the multiple prime zeta value primezetamult(3, 3).

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A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.

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A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.

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A190106 Numbers with prime factorization p^2*q^3*r^3 where p, q, and r are distinct primes.

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A190115 Numbers with prime factorization p^2*q^3*r^4 where p, q, and r are distinct primes.

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A190382 Numbers with prime factorization p^2*q^2*r^2*s^3 where p, q, r, and s are distinct primes.

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A190106 Numbers with prime factorization p^2q^3r^3 where p, q, and r are distinct primes.

A190115 Numbers with prime factorization p^2q^3r^4 where p, q, and r are distinct primes.

A190382 Numbers with prime factorization p^2q^2r^2*s^3 where p, q, r, and s are distinct primes.

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