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 1-3 of 3 results.

A369563 Powerful numbers whose prime factors are all of the form 4*k + 1.

Original entry on oeis.org

1, 25, 125, 169, 289, 625, 841, 1369, 1681, 2197, 2809, 3125, 3721, 4225, 4913, 5329, 7225, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 21025, 21125, 22201, 24389, 24649, 28561, 29929, 32761, 34225, 36125, 37249, 38809, 42025, 48841, 50653, 52441, 54289, 54925
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2024

Keywords

Comments

Closed under multiplication.

Links

Crossrefs

Intersection of A001694 and A004613.
Subsequence: A146945.
Similar sequence: A352492, A369564, A369565, A369566.
Cf. A002144, A008846, A175647, A334424.

Programs

  • Mathematica
    q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 1 && Last[#] > 1 &]; Select[Range[50000], q]
  • PARI
    is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 1 || f[i, 2] == 1, return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) = A175647 * A334424 = 1.0654356335... .

A369564 Powerful numbers whose prime factors are all of the form 4*k + 3.

Original entry on oeis.org

1, 9, 27, 49, 81, 121, 243, 343, 361, 441, 529, 729, 961, 1089, 1323, 1331, 1849, 2187, 2209, 2401, 3087, 3249, 3267, 3481, 3969, 4489, 4761, 5041, 5929, 6241, 6561, 6859, 6889, 8649, 9261, 9747, 9801, 10609, 11449, 11907, 11979, 12167, 14283, 14641, 16129, 16641
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2024

Keywords

Comments

Closed under multiplication.

Links

Crossrefs

Intersection of A001694 and A004614.
Similar sequence: A352492, A369563, A369565, A369566.
Cf. A002145, A013661, A175647, A334426.

Programs

  • Mathematica
    q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 3 && Last[#] > 1 &]; Select[Range[20000], q]
  • PARI
    is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 3 || f[i, 2] == 1, return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 3 (mod 4)} (1 + 1/(p*(p-1))) = 3*A013661*A334426/(4*A175647) = 1.2161513254... .

A369565 Powerful numbers whose prime factors are all of the form 3*k + 1.

Original entry on oeis.org

1, 49, 169, 343, 361, 961, 1369, 1849, 2197, 2401, 3721, 4489, 5329, 6241, 6859, 8281, 9409, 10609, 11881, 16129, 16807, 17689, 19321, 22801, 24649, 26569, 28561, 29791, 32761, 37249, 39601, 44521, 47089, 49729, 50653, 52441, 57967, 58081, 61009, 67081, 73441
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2024

Keywords

Comments

Closed under multiplication.

Links

Crossrefs

Intersection of A001694 and A004611.
Similar sequence: A352492, A369563, A369564, A369566.
Cf. A002476, A175646, A334477.

Programs

  • Mathematica
    q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 3] == 1 && Last[#] > 1 &]; Select[Range[75000], q]
  • PARI
    is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%3 != 1 || f[i, 2] == 1, return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 1 (mod 3)} (1 + 1/(p*(p-1))) = A175646 * A334477 = 1.0377399555...
Showing 1-3 of 3 results.

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

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