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.

A351758 a(n) = Sum_{p|n, p prime} Sum_{d|n} lcm(d,p).

Original entry on oeis.org

0, 4, 6, 8, 10, 34, 14, 16, 15, 54, 22, 74, 26, 74, 76, 32, 34, 97, 38, 118, 104, 114, 46, 154, 35, 134, 42, 162, 58, 324, 62, 64, 160, 174, 164, 209, 74, 194, 188, 246, 82, 440, 86, 250, 220, 234, 94, 314, 63, 229, 244, 294, 106, 286, 252, 338, 272, 294, 118, 724, 122, 314
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 18 2022

Keywords

Examples

			a(6) = 34; a(6) = Sum_{p|6, p prime} Sum_{d|6} lcm(d,p) = Sum_{p|6, p prime} (lcm(1,p) + lcm(2,p) + lcm(3,p) + lcm(6,p)) = (lcm(1,2) + lcm(2,2) + lcm(3,2) + lcm(6,2)) + (lcm(1,3) + lcm(2,3) + lcm(3,3) + lcm(6,3)) = (2+2+6+6) + (3+6+3+6) = 34.

Links

Crossrefs

Cf. A351711.

Programs

  • PARI
    a(n) = my(f=factor(n), d=divisors(f)); sum(k=1, #f~, sum(j=1, #d, lcm(d[j], f[k,1]))); \\ Michel Marcus, Feb 19 2022

Formula

a(p) = 2*p, p prime.

A351844 a(n) = Sum_{d|n} Sum_{p|n, p prime} n^gcd(d,p).

Original entry on oeis.org

0, 6, 30, 36, 3130, 528, 823550, 200, 1467, 200240, 285311670622, 5820, 302875106592266, 210827456, 1525560, 1040, 827240261886336764194, 24390, 1978419655660313589123998, 9601700, 3602195688, 1168636602823712, 20880467999847912034355032910590, 58896, 19531275
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 21 2022

Keywords

Examples

			a(6) = 528; a(6) = Sum_{d|6} Sum_{p|6, p prime} 6^gcd(d,p) = Sum_{d|6} (6^gcd(d,2) + 6^gcd(d,3)) = (6^gcd(1,2) + 6^gcd(2,2) + 6^gcd(3,2) + 6^gcd(6,2)) + (6^gcd(1,3) + 6^gcd(2,3) + 6^gcd(3,3) + 6^gcd(6,3)) = 528.

Crossrefs

Cf. A351711, A351845.

A351746 a(n) = Sum_{p|n, p prime} (p-1) * tau(n/p).

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 6, 3, 4, 10, 10, 10, 12, 14, 12, 4, 16, 11, 18, 16, 16, 22, 22, 14, 8, 26, 6, 22, 28, 28, 30, 5, 24, 34, 20, 18, 36, 38, 28, 22, 40, 36, 42, 34, 20, 46, 46, 18, 12, 19, 36, 40, 52, 16, 28, 30, 40, 58, 58, 44, 60, 62, 26, 6, 32, 52, 66, 52, 48, 44, 70, 25, 72, 74
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 17 2022

Keywords

Examples

			a(12) = 10; a(12) = Sum_{p|12, p prime} (p-1) * tau(12/p) = (2-1)*tau(12/2) + (3-1)*tau(12/3) = tau(6) + 2*tau(4) = 4 + 2*3 = 10.

Crossrefs

Cf. A000005 (tau), A001221 (omega), A248577, A351711.

Programs

  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, (f[k,1]-1)*numdiv(n/f[k,1])); \\ Michel Marcus, Feb 18 2022

Formula

a(n) = A351711(n) - A248577(n).
Showing 1-3 of 3 results.

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

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