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.

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

Links

Crossrefs

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

Programs

  • Mathematica
    q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[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] > 1 && (#e == 1 || e[2] == 1));

A190474 Numbers with prime factorization pqr^2s^6.

Original entry on oeis.org

20160, 31680, 33600, 37440, 44352, 47040, 48960, 52416, 52800, 54720, 62400, 66240, 68544, 76608, 81600, 82368, 83520, 89280, 91200, 92736, 102060, 103488, 106560, 107712, 110400, 116160, 116928, 118080, 120384, 122304, 123200, 123840
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Links

Crossrefs

Cf. A190472, A190473.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,6}; Select[Range[300000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\60, 6), t1=p^6; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

Original entry on oeis.org

1, 8, 9, 72, 13440, 21120, 24960, 29568, 32640, 34944, 36480, 44160, 45696, 49280, 51072, 54912, 55680, 58240, 59520, 61824, 71040, 71808, 76160, 77952, 78720, 80256, 82560, 83328, 84864, 85120, 90240, 91520, 94848, 97152, 99456, 101760, 103040, 110208, 113280, 114816, 115584, 117120, 119680
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Comments

Numbers k such that A000005(k)*A007947(k) = k.
Fixed points of A304409.
All terms are refactorable numbers (A033950).

Examples

			13440 is a term because 13440 = 2^7*3*5*7 = 2*(7 + 1) * 3*(1 + 1) * 5*(1 + 1) * 7*(1 + 1).

Links

Crossrefs

Cf. A000005, A000026, A007947, A008478, A033950, A036762, A048768, A078779, A190473, A304409.

Programs

  • Mathematica
    a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Select[Range[120000], a[#] == # &]
  • PARI
    isok(k) = {my(f = factor(k)); numdiv(f) * vecprod(f[, 1]) == k;} \\ Amiram Eldar, Jan 31 2025

A190475 Numbers with prime factorization pqr^3s^5.

Original entry on oeis.org

30240, 47520, 56160, 66528, 68040, 73440, 78624, 82080, 84000, 99360, 102816, 106920, 114912, 123552, 125280, 126360, 132000, 133920, 139104, 149688, 156000, 159840, 161568, 164640, 165240, 175392, 176904, 177120, 180576, 184680, 185760
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Links

Crossrefs

Cf. A190472, A190473, A190474.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3,5}; Select[Range[400000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\120, 5), t1=p^5; forprime(q=2,sqrtnint(lim\(6*t1), 3), if(q==p, next); t2=q^3*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

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 1-5 of 5 results.

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