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.

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...

A369566 Powerful numbers whose prime factors are all of the form 3*k + 2.

Original entry on oeis.org

1, 4, 8, 16, 25, 32, 64, 100, 121, 125, 128, 200, 256, 289, 400, 484, 500, 512, 529, 625, 800, 841, 968, 1000, 1024, 1156, 1331, 1600, 1681, 1936, 2000, 2048, 2116, 2209, 2312, 2500, 2809, 3025, 3125, 3200, 3364, 3481, 3872, 4000, 4096, 4232, 4624, 4913, 5000
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2024

Keywords

Comments

Closed under multiplication.

Links

Crossrefs

Intersection of A001694 and A004612.
Similar sequence: A352492, A369563, A369564, A369565.
Cf. A003627, A333240, A334479.

Programs

  • Mathematica
    q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 3] == 2 && Last[#] > 1 &]; Select[Range[5000], q]
  • PARI
    is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%3 != 2 || f[i, 2] == 1, return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 2 (mod 3)} (1 + 1/(p*(p-1))) = (9/8) * A333240 * A334479 = 1.6053538210...

A371010 Powerful numbers that are the sum of 2 squares.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 72, 81, 100, 121, 125, 128, 144, 169, 196, 200, 225, 256, 288, 289, 324, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 648, 676, 729, 784, 800, 841, 900, 961, 968, 1000, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1352, 1369
Offset: 1

Views

Author

Amiram Eldar, Mar 08 2024

Keywords

Comments

Each term can be decomposed in a unique way as 2^m * i * j^2 where m >= 2, i is a powerful number whose prime factors are all of the form 4*k + 1 (A369563), and j is a number whose prime factors are all of the form 4*k + 3 (A004614).

Links

Crossrefs

Intersection of A001481 and A001694.
A371011 is a subsequence.
Cf. A004614, A020893, A369563.

Programs

  • Mathematica
    Select[Range[1500], SquaresR[2, #] > 0 && (# == 1 || Min[FactorInteger[#][[;; , 2]]] > 1) &]
  • PARI
    is(n) = {my(f=factor(n)); for(i=1, #f~, if(f[i, 2] == 1 || (f[i, 2]%2 && f[i, 1]%4 == 3), return(0))); 1;}

Formula

The number of terms that do not exceed x is ~ c * sqrt(x), where c = (6/Pi^2) * (1 + 1/(3*(sqrt(2)-1))) * Product_{primes p == 1 (mod 4)} (1 + 1/((sqrt(p)-1)*(p+1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2-1)) = 1.58769... (Jakimczuk, 2024, Theorem 4.7, p. 50).
Sum_{n>=1} 1/a(n) = (3/2) * Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2-1)) = (3*Pi^2/16) * A334424 = 1.86676402705119927669... .
Showing 1-4 of 4 results.

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

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