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));

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];}

A190380 Numbers with prime factorization pqrst^2u^2.

Original entry on oeis.org

180180, 235620, 263340, 278460, 300300, 311220, 318780, 376740, 392700, 401940, 406980, 420420, 429660, 437580, 438900, 450450, 464100, 475020, 489060, 492660, 507780, 512820, 518700, 531300, 549780, 550620, 568260, 589050, 592020, 595980
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

Links

Crossrefs

Cf. A190378, A190379.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,1,2,2};Select[Range[1000000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4,t5); forprime(p=2,sqrtint(lim\4620), t1=p^2; forprime(q=2,sqrtint(lim\(210*t1)), if(q==p, next); t2=q^2*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

A190381 Numbers with prime factorization pqrstuv^2.

Original entry on oeis.org

1021020, 1141140, 1381380, 1492260, 1531530, 1711710, 1741740, 1763580, 1806420, 1861860, 2018940, 2072070, 2134860, 2222220, 2238390, 2277660, 2386020, 2434740, 2462460, 2545620, 2552550, 2582580, 2612610, 2645370, 2691780, 2709630
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

Links

Crossrefs

Cf. A190378, A190380.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,1,1,1,2};Select[Range[3000000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3,t4,t5,t6); forprime(p1=2,sqrtint(lim\30030), t1=p1^2; forprime(p2=2,lim\(2310*t1), if(p2==p1, next); t2=p2*t1; forprime(p3=2,lim\(210*t2), if(p3==p1 || p3==p2, next); t3=p3*t2; forprime(p4=2,lim\(30*t3), if(p4==p1 || p4==p2 || p4==p3, next); t4=p4*t3; forprime(p5=2,lim\(6*t4), if(p5==p1 || p5==p2 || p5==p3 || p5==p4, next); t5=p5*t4; forprime(p6=2,lim\(2*t5), if(p6==p1 || p6==p2 || p6==p3 || p6==p4 || p6==p5, next); t6=p6*t5; forprime(p7=2,lim\t6, if(p7==p1 || p7==p2 || p7==p3 || p7==p4 || p7==p5 || p7==p6, next); listput(v, t6*p7)))))))); 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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