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.

A065768 Numbers that are sums of divisors of the odd squares; Intersection of A065764 and A065766, written in ascending order and duplicates removed.

Original entry on oeis.org

1, 13, 31, 57, 121, 133, 183, 307, 381, 403, 553, 741, 781, 871, 993, 1093, 1407, 1723, 1729, 1767, 1893, 2257, 2379, 2801, 2863, 3541, 3751, 3783, 3991, 4123, 4557, 4953, 5113, 5403, 5673, 6321, 6897, 6973, 7189, 7581, 8011, 9507, 9517, 9841, 10153
Offset: 1

Views

Author

Labos Elemer, Nov 19 2001

Keywords

Comments

Terms are the sum of the odd divisors (A000593) of the odd squares (A016754), written in ascending order. Subsequence of the odd terms of A274790. - Timothy L. Tiffin, Feb 12 2022
Equally, the sum of divisors (A000203) as only odd divisors are present in odd squares. - Antti Karttunen, Dec 22 2024

Links

Crossrefs

Cf. A000203, A000593, A016754, A065764, A065766, A274790.
Sequence A379223 sorted into ascending order, with duplicates removed.

Programs

  • Mathematica
    f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[2, e_] := (4^(e + 1) - 1)/3; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; seq[max_] := Intersection[Select[Array[s1, max], # < max^2 &], Select[Array[s2, max], # < max^2 &]]; seq[101] (* Amiram Eldar, Aug 24 2024 *)

Extensions

Old definition clarified and Timothy L. Tiffin's comment adopted as a new primary definition - Antti Karttunen, Dec 22 2024

A065765 Sum of divisors of twice square numbers.

Original entry on oeis.org

3, 15, 39, 63, 93, 195, 171, 255, 363, 465, 399, 819, 549, 855, 1209, 1023, 921, 1815, 1143, 1953, 2223, 1995, 1659, 3315, 2343, 2745, 3279, 3591, 2613, 6045, 2979, 4095, 5187, 4605, 5301, 7623, 4221, 5715, 7137, 7905, 5169, 11115, 5679, 8379, 11253
Offset: 1

Views

Author

Labos Elemer, Nov 19 2001

Keywords

Comments

The sum of divisors of 2m^2 [twice a square] always gives remainder r = 3 modulo 6. See A097022.

Links

Crossrefs

Cf. A000203, A000290, A002117, A028982, A065766, A097022.

Programs

Formula

a(n) = sigma(2*n^2) = A000203(2*A000290(n)).
a(n) = 3*A065766(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 12*zeta(3)/Pi^2 = 1.461525... . - Amiram Eldar, Oct 28 2022

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

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

A356456 Numbers k not divisible by 3 such that 2*k^2 is not in A014567.

Original entry on oeis.org

10, 28, 40, 50, 70, 77, 110, 130, 133, 136, 140, 154, 155, 160, 161, 170, 176, 190, 196, 200, 209, 224, 230, 250, 259, 266, 275, 280, 290, 308, 310, 322, 350, 364, 370, 371, 377, 385, 410, 416, 418, 430, 440, 469, 470, 476, 490, 496, 518, 520, 530, 532, 539, 550, 553, 590
Offset: 1

Views

Author

Jianing Song, Aug 07 2022

Keywords

Comments

Numbers k not divisible by 3 such that k and sigma(2*k^2) are not coprime, sigma = A000203.
Note that if m is twice a square, then sigma(m) is divisible by 3 (cf. A065766).

Examples

			28 is a term since 28 and sigma(2*28^2) = 3591 have a common factor 7, and 28 is not divisible by 3.

Links

Crossrefs

Cf. A014567, A000203, A065766.
Equals A356453 \ A008585.

Programs

  • PARI
    isA356456(n) = (n%3) && gcd(n, sigma(2*n^2))>1
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.