A179698 -id:A179698 - cp's OEIS frontend

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

2592, 3888, 20000, 50000, 76832, 151875, 253125, 268912, 468512, 583443, 913952, 1361367, 2576816, 2672672, 3557763, 4170272, 5940688, 6940323, 7503125, 8954912, 10504375, 13045131, 20295603, 22632992, 22717712, 29552672, 30074733

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

Subsequence of A046312 and of A137493. - R. J. Mathar, Jul 27 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of 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, A179694, A179695, A179696, A179698, A179699, A179700.

Cf. A085964, A085965, A085969.

fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by Robert G. Wilson v, Aug 26 2010 *)
lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* Robert G. Wilson v, Aug 26 2010 *)
Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* Harvey P. Dale, Jan 01 2012 *)

list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, integer_nthroot, primerange
def A179702(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**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Edited and extended by Ray Chandler and R. J. Mathar, Jul 26 2010

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

A175749 Numbers with 40 divisors.

Original entry on oeis.org

1680, 2160, 2640, 3024, 3120, 3240, 3696, 4080, 4368, 4536, 4560, 4752, 5520, 5616, 5670, 5712, 6000, 6160, 6384, 6864, 6960, 7128, 7280, 7344, 7440, 7680, 7728, 8208, 8424, 8880, 8910, 8976, 9520, 9744, 9840, 9936, 10032, 10320, 10368, 10416, 10530, 10608

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the forms p^39, p^19*q^1, p^9*q^3, p^7*q^4, p^9*q^1*r^1, p^4*q^3*r^1 (A179698), and p^4*q^1*r^1*s^1 (A179693), where p, q, r, and s are distinct primes.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A175747, A175748.

Select[Range[20000],DivisorSigma[0,#]==40&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==40 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=40.

A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

Original entry on oeis.org

1, 60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726, 732, 735

Offset: 1

Amiram Eldar, Aug 06 2022

nonn

Numbers k such that A350386(k) = A350387(k).

A085987 is a subsequence. Terms that are not in A085987 are 1, 2160, 3024, ...

			60 is a term since A350386(60) = A350387(60) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A350386, A350387.
Subsequence of A028260.
Subsequences: A085987, A179698, A190109, A190110.
Similar sequences: A048109, A187039, A348097.

f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[1000], q]

isok(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};

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A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

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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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A175749 Numbers with 40 divisors.

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A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

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Author

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Programs

Mathematica

PARI

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