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 1-6 of 6 results.

A190378 Numbers with prime factorization p*q*r*s*t*u^3 (where p, q, r, s, t, u are distinct primes).

Original entry on oeis.org

120120, 157080, 175560, 185640, 207480, 212520, 251160, 267960, 270270, 271320, 286440, 291720, 316680, 326040, 328440, 338520, 341880, 353430, 367080, 378840, 394680, 395010, 397320, 404040, 408408, 414120, 417690, 426360, 434280, 442680
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

Examples

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*3*5*7*11*13 = 120120;
a(2) = (2^3)*3*5*7*11*17 = 157080,
a(3) = (2^3)*3*5*7*11*19 = 175560;
a(4) = (2^3)*3*5*7*13*17 = 185640;
a(5) = (2^3)*3*5*7*13*19 = 207480;
a(6) = (2^3)*3*5*7*11*23 = 212520;
a(7) = (2^3)*3*5*7*13*23 = 251160;
a(8) = (2^3)*3*5*7*11*29 = 267960;
a(9) = 2*(3^3)*5*7*11*13 = 270270.
(End)

Links

Crossrefs

Cf. A190110, A190111.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,1,1,3};Select[Range[1000000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4,t5); forprime(p=2,sqrtnint(lim\2310, 3), t1=p^3; forprime(q=2,lim\(210*t1), if(q==p, next); t2=q*t1; forprime(r=2,lim\(30*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\(6*t3), if(s==p || s==q || s==r, next); t4=s*t3; forprime(t=2,lim\(2*t4), if(t==p || t==q || t==r || t==s, next); t5=t*t4; forprime(u=2,lim\t5, if(u==p || u==q || u==r || u==s || u==t, next); listput(v, t5*u))))))); 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};

A190379 Numbers with prime factorization pqr^2s^2t^2.

Original entry on oeis.org

69300, 81900, 97020, 107100, 114660, 119700, 128700, 144900, 149940, 152460, 161700, 167580, 168300, 182700, 188100, 191100, 195300, 198900, 202860, 212940, 222300, 227700, 233100, 242550, 249900, 252252, 254100, 255780, 258300, 269100
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

Links

Crossrefs

Cf. A190110, A190111.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,2,2};Select[Range[600000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4); forprime(p=2,sqrtint(lim\1260), t1=p^2; forprime(q=2,sqrtint(lim\(60*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(6*t2)), if(r==p || r==q, next); t3=r^2*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

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

Links

Crossrefs

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

Programs

  • Mathematica
    q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]
  • PARI
    isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

Links

Crossrefs

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

Programs

  • Mathematica
    q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]
  • PARI
    isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));
Showing 1-6 of 6 results.

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