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 1-6 of 6 results.

A356452 Even numbers k such that k^2 is not in A014567; complement of A356448 in the even numbers.

Original entry on oeis.org

14, 42, 70, 78, 84, 98, 112, 114, 124, 126, 154, 156, 168, 182, 186, 198, 210, 222, 228, 234, 238, 252, 258, 266, 294, 308, 310, 312, 322, 336, 342, 350, 366, 372, 378, 390, 396, 402, 406, 418, 420, 434, 438, 444, 456, 462, 468, 474, 490, 504, 516, 518, 532, 546, 550, 558
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Even numbers k such that k and sigma(k^2) are not coprime, sigma = A000203.

Examples

			14 is a term since 14 and sigma(14^2) = 399 have a common factor 7.

Links

Crossrefs

Cf. A014567, A000203, A356448, A356453, A356454.

Programs

  • Mathematica
    Select[2 Range[300],!CoprimeQ[#,DivisorSigma[1,#^2]]&] (* Harvey P. Dale, Mar 09 2023 *)
  • PARI
    isA356452(n) = !(n%2) && gcd(n, sigma(n^2))>1

Formula

a(n) = 2*A356454(n).

A356449 Numbers k such that 2*k^2 is in A014567.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106, 107, 109, 112, 113
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Numbers k such that k and sigma(2*k^2) are coprime, sigma = A000203.
No term can be a multiple of 3 since 3 divides sigma(m) if m is twice a square (cf. A065766).

Examples

			20 is a term since 20 and sigma(2*20^2) = 1953 are coprime.

Links

Crossrefs

Cf. A014567, A000203, A356382, A356448, A356451, A356453 (complement), A065766.

Programs

  • PARI
    isA356449(n) = gcd(n,sigma(2*n^2))==1

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.

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

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

A356454 Numbers k such that 4*k^2 is not in A014567; complement of A356451.

Original entry on oeis.org

7, 21, 35, 39, 42, 49, 56, 57, 62, 63, 77, 78, 84, 91, 93, 99, 105, 111, 114, 117, 119, 126, 129, 133, 147, 154, 155, 156, 161, 168, 171, 175, 183, 186, 189, 195, 198, 201, 203, 209, 210, 217, 219, 222, 228, 231, 234, 237, 245, 252, 258, 259, 266, 273, 275, 279, 280, 285
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Numbers k such that k and sigma(4*k^2) are not coprime, sigma = A000203.

Examples

			7 is a term since 7 and sigma(4*7^2) = 399 have a common factor 7.

Links

Crossrefs

Cf. A014567, A000203, A356448, A356452, A356453.

Programs

  • PARI
    isA356454(n) = gcd(n, sigma(4*n^2))>1

Formula

a(n) = A356452(n)/2.
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.