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.

A356453 Numbers k such that 2*k^2 is not in A014567; complement of A356449.

Original entry on oeis.org

3, 6, 9, 10, 12, 15, 18, 21, 24, 27, 28, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 110, 111, 114, 117, 120, 123, 126, 129, 130, 132, 133, 135, 136, 138, 140, 141, 144, 147, 150, 153, 154, 155
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Numbers k such that k and sigma(2*k^2) are not coprime, sigma = A000203.
Includes all multiples of 3 since 3 divides sigma(m) if m is twice a square (cf. A065766).

Examples

			10 is a term since 10 and sigma(2*10^2) = 465 have a common factor 5.

Links

Crossrefs

Cf. A014567, A000203, A356448, A356449, A356452, A356454, A356456 (all multiples of 3 removed), A065766.

Programs

  • Mathematica
    Select[Range[155],GCD[#, DivisorSigma[1,2#^2]]>1 &] (* Stefano Spezia, Aug 07 2024 *)
  • PARI
    isA356453(n) = gcd(n, sigma(2*n^2))>1

A356448 Even numbers k such that k^2 is in A014567.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 72, 74, 76, 80, 82, 86, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 110, 116, 118, 120, 122, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Even numbers k such that k and sigma(k^2) are coprime, sigma = A000203.

Examples

			12 is a term since 12 and sigma(12^2) = 403 are coprime.

Links

Crossrefs

Cf. A014567, A000203, A356382, A356449, A356451, A356452 (complement in the even numbers).

Programs

  • PARI
    isA356448(n) = !(n%2) && gcd(n,sigma(n^2))==1

Formula

a(n) = 2*A356451(n).

A356451 Numbers k such that 4*k^2 is in A014567.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Numbers k such that k and sigma(4*k^2) are coprime, sigma = A000203.

Examples

			12 is a term since 12 and sigma(4*12^2) = 1651 are coprime.

Links

Crossrefs

Cf. A014567, A000203, A356382, A356448, A356449, A356454 (complement).

Programs

  • PARI
    isA356450(n) = gcd(n,sigma(4*n^2))==1

Formula

a(n) = A356448(n)/2.

A356382 Even terms in A014567.

Original entry on oeis.org

2, 4, 8, 16, 32, 36, 50, 64, 98, 100, 128, 144, 242, 256, 324, 338, 392, 400, 484, 512, 576, 578, 676, 722, 784, 800, 900, 968, 1024, 1058, 1156, 1250, 1296, 1352, 1444, 1600, 1682, 1922, 1936, 2048, 2116, 2304, 2312, 2450, 2500, 2704, 2738, 2888, 2916, 3136, 3362, 3364
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Even numbers k such that k and sigma(k) are coprime, sigma = A000203.
Each term is an even square or twice a square.
No term can be of the form 18*k^2 since sigma(m) is divisible by 3 if m is twice a square (cf. A065766).

Examples

			3362 is a term since 3362 and sigma(3362) = 5169 are coprime.
3364 is a term since 3364 and sigma(3364) = 6097 are coprime.

Links

Crossrefs

Cf. A014567, A000203, A065766.
Subsequence of A088827. Includes A000079 as a subsequence.
Equals {A356448(n)^2} U {2*A356449(n)^2} = {2*A356449(n)^2} U {4*A356451(n)^2}.

Programs

  • PARI
    isA356382(n) = !(n%2) && gcd(n, sigma(n))==1
Showing 1-4 of 4 results.

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