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 1-4 of 4 results.

A190473 Numbers with prime factorization pqrs^7.

Original entry on oeis.org

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
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Links

Crossrefs

Cf. A190470, A190471, A190472.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,7}; Select[Range[150000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\30, 7), t1=p^7; forprime(q=2,lim\(6*t1), if(q==p, next); t2=q*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

A190472 Numbers with prime factorization p^3*q^3*r^4 where p, q, and r are distinct primes.

Original entry on oeis.org

54000, 81000, 135000, 148176, 222264, 518616, 574992, 686000, 862488, 949104, 1423656, 1715000, 2122416, 2401000, 2662000, 2963088, 3162456, 3183624, 3472875, 4394000, 4444632, 5256144, 5788125, 6169176, 6655000, 7304528, 7884216
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Links

Crossrefs

Cf. A190470, A190471.
Cf. A085541, A085964, A085966, A085967.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={3,3,4}; Select[Range[5000000],f] (*and*) lst={}; Do[If[k!=n && k!=m && n!=m, AppendTo[lst, Prime[k]^3*Prime[n]^3*Prime[m]^4]], {n,25}, {m,25}, {k,25}]; Take[Union@lst,60]
  • PARI
    list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\216)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;forprime(r=q+1, (lim\t2)^(1/3), if(p==r,next);listput(v,t2*r^3)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Formula

Sum_{n>=1} 1/a(n) = P(3)^2*P(4)/2 - P(4)*P(6)/2 - P(3)*P(7) + P(10) = 0.000064520760706206924448..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A190471 Numbers with prime factorization p^2*q^4*r^4 where p, q, and r are distinct primes.

Original entry on oeis.org

32400, 63504, 90000, 156816, 202500, 219024, 345744, 374544, 467856, 490000, 685584, 777924, 960400, 1089936, 1210000, 1245456, 1690000, 1774224, 2108304, 2178576, 2396304, 2480625, 2862864, 2890000, 3610000, 3640464, 4112784, 4511376
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Links

Crossrefs

Cf. A190469, A190470.
Cf. A085548, A085964, A085966, A085968.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={2,4,4}; Select[Range[3500000],f] (*and*) lst={}; Do[If[k!=n && k!=m && n!=m, AppendTo[lst, Prime[k]^2*Prime[n]^4*Prime[m]^4]], {n,33}, {m,33}, {k,33}]; Take[Union@lst,60]
  • PARI
    list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\4)^(1/8), t1=p^4;forprime(q=p+1, (lim\t1)^(1/4), t2=t1*q^4;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(4)^2/2 - P(2)*P(8)/2 - P(4)*P(6) + P(10) = 0.00010139253539568059065..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

Original entry on oeis.org

4, 9, 25, 49, 72, 108, 121, 144, 169, 200, 288, 289, 324, 361, 392, 400, 500, 529, 576, 675, 784, 800, 841, 961, 968, 972, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1600, 1681, 1849, 1936, 2025, 2209, 2304, 2312, 2500, 2704, 2809, 2888, 2916, 3087, 3136, 3200
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).

Links

Crossrefs

Subsequence of A001694, A377816 and A377818.
Subsequences: A001248, A143610, A179646, A179689, A179699, A189988, A189990, A190106, A190115, A190470, A190471.
Cf. A036966, A065483.

Programs

  • Mathematica
    With[{max = 3200}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], 2] == 1 &]]
  • PARI
    isok(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x==2), e) == 1);

Formula

Sum_{n>=1} 1/a(n) = Sum_{p prime}((p-1)/(p^3-p^2+1)) * Product_{p prime} (1 + 1/(p^2*(p-1))) = 0.53045141423939736076... .
Showing 1-4 of 4 results.

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