A127104 -id:A127104 - cp's OEIS frontend

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

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

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

nonn

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

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.

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).

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

A127837 Numbers k such that ((k+1)^k-1)/k^2 is a prime.

Original entry on oeis.org

2, 3, 5, 17, 4357

Offset: 1

Farideh Firoozbakht and Alexander Adamchuk, Mar 13 2007

All terms are primes. Corresponding primes of the form ((k+1)^k-1)/k^2 are listed in A128466 = 2, 7, 311, 7563707819165039903, ... .
It seems that if p is in the sequence then the first three numbers k such that k^2 divides (p+1)^k-1 are: 1, p & ((p+1)^p-1)/p. 2 is in the sequence and the first three terms of A127103 are : 1, 2 & ((2+1)^2-1)/2; 3 is in the sequence and the first three terms of A127104 are : 1, 3 & ((3+1)^3-1)/3; 5 is in the sequence and the first three terms of A127106 are : 1, 5 & ((5+1)^5-1)/5.
No other terms below 20000. - Max Alekseyev, Apr 25 2007

			4357 is in the sequence because (4358^4357-1)/4357^2 is prime.

Cf. A128466, A037205, A060072, A060073, A058128, A128456, A127103, A127104, A127106, A128398.

A128452 Least number k > n such that k^2 divides n^k - 1.

Original entry on oeis.org

4, 21, 6, 1555, 8, 889, 10, 111, 12, 253, 14, 2041, 16, 21, 18, 128583032925805678351, 20, 1432001198261, 22, 39, 24, 1081, 26, 55, 28, 171, 30, 279241, 32, 9641, 34, 1191, 36, 55, 38, 950123, 40, 1641, 42, 33661, 44, 32627169461820247, 46, 63, 48, 583223, 50

Offset: 3

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

nonn

For prime p, p divides a(p+1). Quotients a(p+1)/p for prime p = A000040(n) are listed in A128456(n) which coincides with A128357(n) for n from 2 to 6.

a(n) divides n^(n-1) - 1.

Cf. A128456, A128357, A128356, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

a(2n-1) = 2n.

More terms from Alexander Adamchuk, Mar 09 2007

Terms a(22) onward from Max Alekseyev, May 05 2010

A333500 A(n,k) is the n-th number m such that m^2 divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 21, 20, 0, 6, 1, 5, 4, 903, 220, 0, 7, 1, 2, 1555, 6, 2667, 1220, 0, 8, 1, 7, 3, 9673655, 12, 7077, 2420, 0, 9, 1, 2, 889, 4, 187159211791705, 42, 113799, 5060, 0, 10, 1, 3, 4, 2359, 6, 776119592182705, 52, 114681, 13420, 0, 11

Offset: 1

Seiichi Manyama, Mar 24 2020

			Square array A(n,k) begins:
  1, 1,    1,    1,  1,               1, ...
  2, 0,    2,    3,  2,               5, ...
  3, 0,    4,   21,  4,            1555, ...
  4, 0,   20,  903,  6,         9673655, ...
  5, 0,  220, 2667, 12, 187159211791705, ...
  6, 0, 1220, 7077, 42, 776119592182705, ...

Columns k=1-24 give: A000027, A063524, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.
Main diagonal gives A333502.
Cf. A333432.

A292330 Numbers k such that k^2 divides 4^k + 1.

Original entry on oeis.org

1, 5, 205, 168305, 2084645, 37217545, 1711493545, 2483072545, 2763736405, 8866165745, 30555604445, 55328770405, 169592124685, 378465215105, 423977184745, 2038602559445, 3815777985545, 7279122076645, 25250364710105, 28104435502445, 45424920502505, 55625535217765, 90160039460905

Offset: 1

Max Alekseyev, Sep 14 2017

nonn

Cf. A052539. Subsequence of A015950.

Cf. A127104, A128678.

A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

Original entry on oeis.org

1, 5, 9972894583, 449853889404077636694265177903207995382439448590987815041588427345865911961016023550064137351211162870609

Offset: 2

Alexander Adamchuk, May 04 2010

nonn

The next term has 1204 digits.
r = (n^(n-1) - 1)/(n-1) = A060072(n) is the (n-1)-digit repunit in base n.
r^2 divides n^r - 1 for all bases n > 1.

			a(10) = (10^111111111 - 1)/111111111^2.

Cf. A060072, A127100, A127101, A127102, A127103, A127104, A127105, A127106, A127107, A127092.

Table[(n^((n^(n - 1) - 1)/(n - 1)) - 1)/((n^(n - 1) - 1)/(n - 1))^2, {n, 2, 6}]

a(n) = (n^((n^(n-1) - 1)/(n-1)) - 1)/((n^(n-1) - 1)/(n-1))^2.

a(n) = (n^A060072(n) - 1)/A060072(n)^2.

A307217 Semiprimes pq such that 2^(p+q) == 1 (mod pq).

Original entry on oeis.org

9, 15, 35, 119, 5543, 74447, 90859, 110767, 222179, 389993, 1526849, 2927297, 3626699, 4559939, 24017531, 137051711, 160832099, 229731743, 627699239, 880021141, 1001124539, 1041287603, 1104903617, 1592658611, 1717999139, 8843679683, 15575602979, 15614760199, 20374337479

Offset: 1

Thomas Ordowski, Mar 29 2019

nonn

For k > 9, these are semiprimes k such that 2^(k+1) == 1 (mod k): semiprimes in A187787.

In this sequence, only 9 is a perfect square. - Jinyuan Wang, Mar 30 2019

Cf. A001358, A046315, A127104, A187787, A208728.

isok(k) = (bigomega(k)==2) && (Mod(2, k)^(k+1) == 1); \\ (for k > 9) Michel Marcus, Mar 29 2019

use ntheory ":all"; forsemiprimes { print "$\n" if powmod(2, vecsum(factor($)), $) == 1 } 4, 1e7; # _Daniel Suteu, Mar 30 2019

a(7)-a(18) from Amiram Eldar, Mar 29 2019

a(19)-a(29) from Daniel Suteu, Mar 29 2019

cp's OEIS Frontend

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

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Extensions

A127837 Numbers k such that ((k+1)^k-1)/k^2 is a prime.

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A128452 Least number k > n such that k^2 divides n^k - 1.

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A333500 A(n,k) is the n-th number m such that m^2 divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A292330 Numbers k such that k^2 divides 4^k + 1.

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A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

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Mathematica

Formula

A307217 Semiprimes p*q such that 2^(p+q) == 1 (mod p*q).

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PARI

Perl

Extensions

A307217 Semiprimes pq such that 2^(p+q) == 1 (mod pq).