A085967 -id:A085967 - cp's OEIS frontend

A092759 a(n) = prime(n)^7.

128, 2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, 17249876309, 27512614111, 94931877133, 194754273881, 271818611107, 506623120463, 1174711139837, 2488651484819, 3142742836021, 6060711605323

Offset: 1

Jorge Coveiro, Apr 13 2004

Seventh powers of prime numbers. - Wesley Ivan Hurt, Mar 27 2014

			a(1) = 128 since the seventh power of the first prime is 2^7 = 128. - _Wesley Ivan Hurt_, Mar 27 2014

Subsequence of A030626.

Cf. A085967, A013665, A013672.

[p^7: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A092759:=n->ithprime(n)^7; seq(A092759(n), n=1..30); # Wesley Ivan Hurt, Mar 27 2014

Array[Prime[ # ]^7 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Prime[n]^7, {n, 30}] (* Wesley Ivan Hurt, Mar 27 2014 *)

a(n)=prime(n)^7 \\ Charles R Greathouse IV, Jul 20 2011

a(n) = A086874(n-1), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n)^7 = A001015(A000040(n)). - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = P(7) = 0.0082838328... (A085967). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(7)/zeta(14) = A013665/A013672.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(7) = 1/A013665. (End)

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

A085968 Decimal expansion of the prime zeta function at 8.

Original entry on oeis.org

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

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

			0.0040614053665178305605...

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..1971
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) to A085967 (at 7), this sequence (at 8), A085969 (at 9).

Cf. A013666, A179645.

R := RealField(106);
PrimeZeta := func;
[0,0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(8,43)*10^105)));
// Jason Kimberley, Dec 30 2016

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[8*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[ 8], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

sumeulerrat(1/p, 8) \\ Hugo Pfoertner, Feb 03 2020

P(8) = Sum_{p prime} 1/p^8 = Sum_{n>=1} mobius(n)*log(zeta(8*n))/n.

Equals Sum_{k>=1} 1/A179645(k). - Amiram Eldar, Jul 27 2020

A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

Original entry on oeis.org

288, 800, 972, 1568, 3872, 5408, 6075, 9248, 11552, 11907, 12500, 16928, 26912, 28125, 29403, 30752, 41067, 43808, 53792, 59168, 67228, 70227, 70688, 87723, 89888, 111392, 119072, 128547, 143648, 151263, 153125, 161312, 170528, 199712

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

nonn

288=2^5*3^2, 800=2^5*5^2,..

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

Cf. A030636, A046308, A007774.

Cf. A085548, A085965, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,5}; Select[Range[200000], f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\4)^(1/5),t=p^5;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 A189988(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**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(5) - P(7) = A085548 * A085965 - A085967 = 0.007886..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).

Original entry on oeis.org

432, 648, 2000, 5000, 5488, 10125, 16875, 19208, 21296, 27783, 35152, 64827, 78608, 107811, 109744, 117128, 177957, 194672, 214375, 228488, 300125, 390224, 395307, 397953, 476656, 555579, 668168, 771147, 810448, 831875

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

nonn

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

Cf. A046308, A030638, A007774.

Cf. A085541, A085964, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,4}; Select[Range[10^6], f]
With[{nn=40},Select[Flatten[{#[[1]]^4 #[[2]]^3,#[[1]]^3 #[[2]]^4}&/@ Subsets[ Prime[Range[nn]],{2}]]//Union,#<=16nn^3&]] (* Harvey P. Dale, Nov 15 2020 *)

list(lim)=my(v=List(),t);forprime(p=2,(lim\8)^(1/4),t=p^4;forprime(q=2,(lim\t)^(1/3),if(p==q,next);listput(v,t*q^3)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, integer_nthroot, primerange
def A179666(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**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,7)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(3)*P(4) - P(7) = A085541 * A085964 - A085967 = 0.005171..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

Original entry on oeis.org

1152, 3200, 6272, 8748, 15488, 21632, 36992, 46208, 54675, 67712, 107163, 107648, 123008, 175232, 215168, 236672, 264627, 282752, 312500, 359552, 369603, 445568, 476288, 574592, 632043, 645248, 682112, 703125, 789507, 798848, 881792, 1013888

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Numbers with same prime signature.
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.

Cf. A085548, A085967, A085969.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 2]
      do od; k
    end:
seq (a(n), n=1..32);  # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,7}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/7), t=p^7;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, integer_nthroot, primerange
def A179689(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**7)) for p in primerange(integer_nthroot(x,7)[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(2)*P(7) - P(9) = A085548 * A085967 - A085969 = 0.001741..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Title edited by Daniel Forgues, Jan 22 2011

A179695 Numbers of the form p^3q^2r^2 where p, q, and r are distinct primes.

Original entry on oeis.org

1800, 2700, 3528, 4500, 5292, 8712, 9800, 12168, 12348, 13068, 18252, 20808, 24200, 24500, 25992, 31212, 33075, 33800, 34300, 38088, 38988, 47432, 47916, 55125, 57132, 57800, 60500, 60552, 66248, 69192, 72200, 77175, 79092, 81675, 84500

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

Subsequence of A225228. - Reinhard Zumkeller, May 03 2013

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. A050326, A225228.

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

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,3}; Select[Range[10^5], f]
f[n_]:={Times@@(n^{2,2,3}),Times@@(n^{2,3,2}),Times@@(n^{3,2,2})}; Module[ {nn=20},Select[Flatten[f/@Subsets[Prime[Range[nn]],{3}]],#<= 72*Prime[ nn]^2&]]//Union (* Harvey P. Dale, Jul 05 2019 *)

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/3), t1=p^3;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 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179695(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:=isqrt(x//r**3))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,7)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

A050326(a(n)) = 5. - Reinhard Zumkeller, May 03 2013

Sum_{n>=1} 1/a(n) = P(2)^2*P(3)/2 - P(3)*P(4)/2 - P(2)*P(5) + P(7) = 0.0032578591481263202818..., 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

A190470 Numbers with prime factorization p^2q^3r^5 where p, q, and r are distinct primes.

Original entry on oeis.org

21600, 36000, 42336, 48600, 95256, 98784, 104544, 121500, 146016, 196000, 225000, 235224, 249696, 274400, 311904, 328536, 333396, 337500, 383328, 457056, 484000, 561816, 632736, 676000, 701784, 726624, 830304, 1028376, 1064800, 1156000

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. A190467, A190468, A190469.

Cf. A085548, A085541, A085965, A085967, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,5}; Select[Range[2500000],f] (*and*) lst={}; Do[If[k!=n && k!=m && n!=m, AppendTo[lst, Prime[k]^2*Prime[n]^3*Prime[m]^5]], {n,20}, {m,20}, {k,20}]; Take[Union@lst,60]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\72)^(1/5), t1=p^5;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

Sum_{n>=1} 1/a(n) = P(2)*P(3)*P(5) - P(2)*P(8) - P(3)*P(7) - P(5)^2 + 2*P(10) = 0.00025025315357155375895..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

cp's OEIS Frontend

A092759 a(n) = prime(n)^7.

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A085966 Decimal expansion of the prime zeta function at 6.

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A085968 Decimal expansion of the prime zeta function at 8.

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A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

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A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).

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A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

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A179695 Numbers of the form p^3*q^2*r^2 where p, q, and r are distinct primes.

A179695 Numbers of the form p^3q^2r^2 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.

A190470 Numbers with prime factorization p^2q^3r^5 where p, q, and r are distinct primes.