A128456 -id:A128456 - cp's OEIS frontend

A128357 Quotients A128356(n)/prime(n).

10, 7, 311, 127, 23, 157, 343927, 7805561, 47, 9629, 311, 25679, 821, 1470086279, 12409, 71233, 1232333, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843, 1880415721, 263

Offset: 1

Alexander Adamchuk, Mar 02 2007, Mar 09 2007

A128356 = {20, 21, 1555, 889, 253, 2041, 5846759, ...} = Least number k>1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n). Most listed terms are primes, except a(7) = 20231*17 and a(8) = 410819*19. a(15) = 12409. a(16) = 71233.

Note that all prime listed terms of {a(n)} coincide with terms of A128456 = {2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, ...} = least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n).

Cf. A128356 (least number k > 1 (that is not a power of prime p) such that k divides (p+1)^k-1, where p = prime(n)).
Cf. A014960, A128360, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945.
Cf. A128456 (least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n)).

Terms a(14) onwards from Max Alekseyev, Feb 08 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

A128466 Primes of the form ((k+1)^k - 1)/k^2 = A060073(k+1).

Original entry on oeis.org

2, 7, 311, 7563707819165039903

Offset: 1

Alexander Adamchuk, Mar 09 2007

nonn

Corresponding numbers k are listed in A127837.
Terms are the primes in A060073.
Next term has 15850 = 1 + floor((4357*log(4358) - 2*log(4357))/log(10)) digits and is too large to include. - M. F. Hasler, May 22 2007

Cf. A127837, A037205, A060072, A060073, A058128, A128456

Select[Table[((n+1)^n-1)/n^2,{n,500}],PrimeQ]  (* Harvey P. Dale, Apr 30 2011 *)

A128466(n)=A060073(A127837(n)+1) /* see there. --- or: */ forprime(p=1,10^5,if(ispseudoprime(n=((p+1)^p-1)/p^2),print1(n,", "))); \\ M. F. Hasler, May 22 2007

a(n) = ((A127837(n) + 1)^A127837(n) - 1) / A127837(n)^2.

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

Alexander Adamchuk, Feb 04 2008

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.

Amiram Eldar, Table of n, a(n) for n = 1..31 (terms 1..26 from Jens Kruse Andersen)

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

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

Alexander Adamchuk, Feb 04 2008

nonn

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, ...}.

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

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

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

Alexander Adamchuk, Feb 04 2008

nonn

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.

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

Table[ ((Prime[n] + 1)^Prime[n] - 1)/Prime[n]^2, {n,1,15} ]

a(n) = my(p=prime(n)); polcyclo(p,p+1)/p \\ Hugo Pfoertner, Jul 21 2024

a(n) = ((prime(n) + 1)^prime(n) - 1)/prime(n)^2;

a(n) = A137664(n)/prime(n)^2.

cp's OEIS Frontend

A128357 Quotients A128356(n)/prime(n).

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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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A128466 Primes of the form ((k+1)^k - 1)/k^2 = A060073(k+1).

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

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

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

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