cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A190110 Numbers with prime factorization p*q*r*s*t^4 (where p, q, r, s, t are distinct primes).

Original entry on oeis.org

18480, 21840, 28560, 31920, 34320, 38640, 44880, 48048, 48720, 50160, 52080, 53040, 59280, 60720, 62160, 62370, 62832, 68880, 70224, 71760, 72240, 73710, 74256, 76560, 77520, 78960, 80080, 81840, 82992, 85008, 89040, 90480, 93840, 96390, 96720, 97680, 99120
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Keywords

Comments

That is, numbers with prime signature {1,1,1,1,4}.

Examples

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^4)*3*5*7*11 = 18480;
a(2) = (2^4)*3*5*7*13 = 21840;
a(3) = (2^4)*3*5*7*17 = 28560;
a(4) = (2^4)*3*5*7*19 = 31920.
(End)

Links

Crossrefs

Cf. A179672, A190107, A190108, A190109.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,1,4};Select[Range[150000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4); forprime(p1=2,sqrtnint(lim\210, 4), t1=p1^4; forprime(p2=2,lim\(30*t1), if(p2==p1, next); t2=p2*t1; forprime(p3=2,lim\(6*t2), if(p3==p1 || p3==p2, next); t3=p3*t2; forprime(p4=2,lim\(2*t3), if(p4==p1 || p4==p2 || p4==p3, next); t4=p4*t3; forprime(p5=2,lim\t4, if(p5==p1 || p5==p2 || p5==p3 || p5==p4, next); listput(v, t4*p5)))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Extensions

Name edited by Petros Hadjicostas, Oct 26 2019

A190111 Numbers with prime factorization p*q*r*s^2*t^3 (where p, q, r, s, t are distinct primes).

Original entry on oeis.org

27720, 32760, 41580, 42840, 46200, 47880, 49140, 51480, 54600, 57960, 64260, 64680, 67320, 71400, 71820, 72072, 73080, 75240, 76440, 77220, 78120, 79560, 79800, 85800, 86940, 88920, 91080, 93240, 94248, 96600, 99960, 100980, 101640, 103320
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Keywords

Examples

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^2)*5*7*11 = 27720;
a(2) = (2^3)*(3^2)*5*7*13 = 32760;
a(3) = (2^2)*(3^3)*5*7*11 = 41580;
a(4) = (2^3)*(3^2)*5*7*17 = 42840;
a(5) = (2^3)*3*(5^2)*7*11 = 46200.
(End)

Links

Crossrefs

Cf. A179645, A189990, A190109, A190110.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,2,3};Select[Range[150000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4); forprime(p=2,sqrtnint(lim\420, 3), t1=p^3; forprime(q=2,sqrtint(lim\(30*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(6*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\(2*t3), if(s==p || s==q || s==r, next); t4=s*t3; forprime(t=2,lim\t4, if(t==p || t==q || t==r || t==s, next); listput(v, t4*t)))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

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

Views

Author

Amiram Eldar, Aug 06 2022

Keywords

Comments

Numbers k such that A350386(k) = A350387(k).
A085987 is a subsequence. Terms that are not in A085987 are 1, 2160, 3024, ...

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[1000], q]
  • PARI
    isok(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};

A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Subsequence of A176297 and A375072, and first differs from them at n = 20: A176297(20) = A375072(20) = 216 = 2^3 * 3^3 is not a term of this sequence.
The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p+p^2+p^3) = 0.089602607198058453295... .

Links

Crossrefs

Subsequence of A046100, A176297, A375072 and A375145.
Subsequences: A030078, A048109, A065036, A143610, A163569, A179670, A179695, A179700, A189975, A189984, A190109, A190378, A190382.
Cf. A002117.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, MemberQ[e, 3] && Count[e, _?(# < 3 &)] == Length[e] - 1]; Select[Range[600], q]
  • PARI
    isok(k) = {my(e = factor(k)[, 2]~); select(x -> x > 2, e) == [3];}

A190377 Numbers with prime factorization p^2*q^2*r^2*s^2 where p, q, r, and s are distinct primes.

Original entry on oeis.org

44100, 108900, 152100, 213444, 260100, 298116, 324900, 476100, 509796, 592900, 636804, 736164, 756900, 828100, 864900, 933156, 1232100, 1258884, 1334025, 1416100, 1483524, 1512900, 1572516, 1664100, 1695204, 1758276, 1768900, 1863225
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

Links

Crossrefs

Cf. A190108, A190109.
Cf. A046386, A085548, A085964, A085966, A085968.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,2,2};Select[Range[3000000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\900), t1=p^2; 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
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A190377(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(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = (P(2)^4 - 6*P(2)^2*P(4) + 8*P(2)*P(6) + 3*P(4)^2 - 6*P(8))/24 = 0.00010511750849230980748..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024
a(n) = A046386(n)^2. - Chai Wah Wu, Mar 27 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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