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.

A128456 Quotients A128452(p+1)/p for prime p = A000040(n).

Original entry on oeis.org

2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, 758771382833029, 12409, 71233, 18438666190697, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843
Offset: 1

Views

Author

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

Keywords

Comments

a(n) coincides with A128357(n) from n = 2 up to n = 6.

Crossrefs

Cf. A128452, A128357, A128356, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Formula

a(n) = A128452(A000040(n)+1)/A000040(n).
a(n) = A020639(((p+1)^p - 1)/p^2), i.e., the smallest prime factor of ((p+1)^p - 1)/p^2, where p = A000040(n).

Extensions

Terms a(14) onward from Max Alekseyev, May 05 2010

A137666 Largest prime factor of A137664(n) = (p + 1)^p - 1 for p = prime(n).

Original entry on oeis.org

2, 7, 311, 337, 266981089, 29914249171, 7563707819165039903, 192696104561, 58769065453824529, 847499019384726257346113954958447091, 18158209813151, 138233050898929517126243814850350442620694127
Offset: 1

Views

Author

Alexander Adamchuk, Feb 04 2008

Keywords

Comments

a(n) is also the largest prime factor of A137665(n) = A137664(n)/prime(n)^2. p^2 divides A137664(n) = (p + 1)^p - 1, p = prime(n). Least prime factors of A137664(n) are listed in A128456.
a(n) = A128456(n) = A137665(n) = ((p + 1)^p - 1)/p^2 for n = {1,2,3,7,595,...} corresponding to p = prime(n) = {2,3,5,17,4357,...} = A127837.

Links

Crossrefs

Cf. A128452, A128456, A128356, A128357, A137664, A137665, A128466, A127837.

Programs

  • Mathematica
    FactorInteger[#][[-1,1]]&/@((#+1)^#-1&/@Prime[Range[12]]) (* Harvey P. Dale, Apr 07 2018 *)

A137664 a(n) = (p+1)^p - 1 where p = prime(n).

Original entry on oeis.org

8, 63, 7775, 2097151, 743008370687, 793714773254143, 2185911559738696531967, 5242879999999999999999999, 55572324035428505185378394701823, 6863037736488299999999999999999999999999999
Offset: 1

Views

Author

Alexander Adamchuk, Feb 04 2008

Keywords

Comments

p^2 divides a(n) = (p+1)^p - 1, p = prime(n).
Quotients a(n)/prime(n)^2 are listed in A137665(n) = {2, 7, 311, 42799, 6140565047, 4696537119847, 7563707819165039903, ...}.
Least prime factors of A137665(n) = a(n)/prime(n)^2 are listed in A128456(n) = {2, 7, 311, 127, 23, 157, 7563707819165039903, ...}.
Largest prime factors A137665(n) = a(n)/prime(n)^2 are listed in A137666(n) = {2, 7, 311, 337, 266981089, 29914249171, 7563707819165039903, ...}.

Crossrefs

Cf. A128452, A128456, A128356, A128357, A137665, A137666.

Programs

  • Mathematica
    Table[ (Prime[n] + 1)^Prime[n] - 1, {n,1,15} ]
(#+1)^#-1&/@Prime[Range[10]] (* Harvey P. Dale, Jan 10 2025 *)

Formula

a(n) = (prime(n) + 1)^prime(n) - 1.

A137665 Quotients ((p+1)^p - 1)/p^2 for p = prime(n).

Original entry on oeis.org

2, 7, 311, 42799, 6140565047, 4696537119847, 7563707819165039903, 14523213296398891966759, 105051652240885643072548950287, 8160568057655529131985731272294887039239, 47525417447024678661670292427038339608998847, 20681861558186805237407813095538883147812221153173966103
Offset: 1

Views

Author

Alexander Adamchuk, Feb 04 2008

Keywords

Comments

p^2 divides a(n) = (p+1)^p - 1, p = prime(n). (p+1)^p - 1 = A137664(n) = {8, 63, 7775, 2097151, 743008370687, 793714773254143, 2185911559738696531967, ...}.
Least prime factors of a(n) are listed in A128456(n) = {2, 7, 311, 127, 23, 157, 7563707819165039903, ...}.
Largest prime factors a(n) are listed in A137666.
a(n) is prime for n = {1, 2, 3, 7, 595, ...} corresponding to p = prime(n) = {2, 3, 5, 17, 4357, ...} = A127837.
Primes in this sequence are A128466.

Crossrefs

Cf. A128452, A128456, A128356, A128466, A127837, A128357, A060073, A137664, A137666.

Programs

  • Mathematica
    Table[ ((Prime[n] + 1)^Prime[n] - 1)/Prime[n]^2, {n,1,15} ]
  • PARI
    a(n) = my(p=prime(n)); polcyclo(p,p+1)/p \\ Hugo Pfoertner, Jul 21 2024

Formula

a(n) = ((prime(n) + 1)^prime(n) - 1)/prime(n)^2;
a(n) = A137664(n)/prime(n)^2.
Showing 1-4 of 4 results.

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