A130058 -id:A130058 - cp's OEIS frontend

A130059 Primes p such that k=p*23^2 divides 3^(k-1) - 2^(k-1); or primes in A130058.

67, 89, 199, 331, 617, 881, 5281, 35839, 22270249, 24939553, 395297101, 2414250301, 40834167001, 184879309516177, 207091473814443440700193, 30576308069075829315234744136241, 175651822579831731574054050278935909201, 109606420475170539243380866438311892933511638772789857

Offset: 1

Alexander Adamchuk, May 04 2007

Prime divisors of (3^528 - 2^528) / 23^2 that are congruent to 1 modulo 11.

Cf. A073631, A001047, A130058

Definition clarified by Max Alekseyev, Mar 09 2009

A073631 Nonprimes k such that k divides 3^(k-1) - 2^(k-1).

Original entry on oeis.org

1, 65, 133, 529, 793, 1105, 1649, 1729, 2059, 2321, 2465, 2701, 2821, 4187, 5185, 6305, 6541, 6601, 6697, 6817, 7471, 7613, 8113, 8911, 10585, 10963, 11521, 13213, 13333, 13427, 14701, 14981, 15841, 18721, 19171, 19201, 19909, 21349, 21667, 22177, 26065

Offset: 1

Benoit Cloitre, Aug 29 2002

Terms 1,65,2059,6305,19171,... are also in A001047
All primes p>3 divide 3^(p-1) - 2^(p-1). It appears that a(1) = 1 and a(4) = 529 = 23^2 are the only perfect squares in a(n). Most terms of a(n) are squarefree. First 50 nonsquarefree terms of a(n) are the multiples of 23^2. Conjecture: All nonsquarefree terms of a(n) are the multiples of 23^2. Numbers n such that k=n*23^2 divides 3^(k-1) - 2^(k-1) are listed in A130058 = {1, 67, 89, 133, 199, 331, 617, 793, 881, 5281, 8911, 1419, 13333,...}. - Alexander Adamchuk, May 04 2007
Contains all Carmichael numbers (A002997) that are not divisible by 3. - Robert Israel, May 19 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Cf. A001047 (3^n - 2^n), A002997.
Cf. A038876, A097934 (primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A130059, A130058 (numbers n such that k=n*23^2 divides 3^(k-1) - 2^(k-1)).

[n: n in [1..3*10^4] | not IsPrime(n) and IsDivisibleBy(3^(n-1)-2^(n-1), n)]; // Vincenzo Librandi, May 20 2015

1,op(select(n -> (3 &^ (n-1) - 2 &^ (n-1) mod n = 0 and not isprime(n)), [seq(2*i+1,i=1..10000)])); # Robert Israel, May 19 2015

Select[Range[3 10^4], ! PrimeQ[#] && Mod[3^(# - 1) - 2^(# - 1), #] == 0 &] (* Vincenzo Librandi, May 20 2015 *)
Select[Range[3*10^4], PowerMod[3, # - 1, #] == PowerMod[2, # - 1, #] && !PrimeQ[#] &] (* Amiram Eldar, Mar 27 2021 *)

isok(n) = ! isprime(n) && !((3^(n-1)-2^(n-1)) % n); \\ Michel Marcus, Nov 28 2013

Term 14701 added and more terms from Michel Marcus, Nov 28 2013

A130062 Nonprime numbers k such that k divides 3^((k+1)/2) - 2^((k+1)/2) - 1.

Original entry on oeis.org

1, 21, 49, 105, 1729, 2465, 2877, 7305, 10585, 15841, 31021, 31621, 32041, 41041, 46657, 52633, 54145, 75361, 83333, 115921, 126217, 162401, 172081, 211141, 282133, 284649, 294409, 334153, 383161, 399001, 417241, 449065, 488881, 530881

Offset: 1

Alexander Adamchuk, May 05 2007

nonn

The perfect squares in listed terms are a(1) = 1, a(3) = 49 = 7^2, a(13) = 32041 = 179^2 and a(29) = 383161 = 619^2.

Note that primes {7,179,619} are the terms of A130060 or primes in A127074.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A097934 (primes p that divide 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A038876 (primes p such that 6 is a square mod p).
Cf. A127071, A127072, A127073, A127074.
Cf. A130058, A130059, A130061, A130063, A130060, A127074.

Select[ 2*Range[100000]-1, !PrimeQ[ # ] && Mod[ PowerMod[3,(#+1)/2,# ] - PowerMod[2,(#+1)/2,# ] - 1, # ] == 0 & ]

More terms from Ryan Propper, Jan 07 2008

A130060 Primes p such that p^2 divides 3^p - 2^p - 1; or primes in A127074(n).

Original entry on oeis.org

2, 3, 7, 179, 619, 17807

Offset: 1

Alexander Adamchuk, May 05 2007

The prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p, where p = Prime[n], are listed in A127071. - Alexander Adamchuk, Jan 31 2008

a(7) > 10^9. [From D. S. McNeil, Mar 16 2009]

Cf. A127071, A127072, A127073, A127074 = numbers n such that n^2 divides 3^n - 2^n - 1. Cf. A130058, A130059, A130061, A130062, A130063.

Do[ n=Prime[k]; f=PowerMod[3,n,n^2] - PowerMod[2,n,n^2] - 1; If[ IntegerQ[ f/n^2 ], Print[n] ], {k,1,100000} ]

2 more terms found by Ryan Propper, Jan 01 2008.

Incorrect a(7), a(8) removed by D. S. McNeil, Mar 16 2009. (The old version was 2,3,7,179,619,17807,135433,1376257.)

A130061 Numbers k that divide 3^((k-1)/2) - 2^((k-1)/2) - 1.

Original entry on oeis.org

1, 3, 35, 147, 195, 219, 291, 399, 579, 583, 723, 939, 1011, 1023, 1227, 1299, 1371, 1443, 1731, 1803, 2019, 2307, 2499, 2811, 3003, 3027, 3099, 3387, 3459, 3603, 3747, 3891, 3963, 4467, 4623, 4827, 4971, 5187, 5259, 5331, 5403, 5619, 5979, 6051, 6267

Offset: 1

Alexander Adamchuk, May 05 2007

nonn

It appears that all terms are composite except a(1) = 1 and a(2) = 3. Most listed terms are divisible by 3, except {1, 35, 583, 70643, ...}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A097934 (primes p that divide 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A038876 (primes p such that 6 is a square mod p).
Cf. A127071, A127072, A127073, A127074.
Cf. A130058, A130059, A130060, A130062, A130063.

Select[ Range[10000], Mod[ PowerMod[3,(#-1)/2,# ] - PowerMod[2,(#-1)/2,# ] -1, # ]==0&]

A130063 Primes p such that p divides 3^((p+1)/2) - 2^((p+1)/2) - 1.

Original entry on oeis.org

23, 47, 71, 73, 97, 167, 191, 193, 239, 241, 263, 311, 313, 337, 359, 383, 409, 431, 433, 457, 479, 503, 577, 599, 601, 647, 673, 719, 743, 769, 839, 863, 887, 911, 937, 983, 1009, 1031, 1033, 1103, 1129, 1151, 1153, 1201, 1223, 1249, 1297, 1319, 1321, 1367

Offset: 1

Alexander Adamchuk, May 05 2007

Primes = 1 or 23 mod 24. Hence, together with 2, primes such that (2/p) = 1 = (3/p) where (k/p) is the Legendre symbol. - Charles R Greathouse IV, Apr 06 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A097934 = Primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2).
Subsequence of A038876.
Cf. A127071, A127072, A127073, A127074. Cf. A130058, A130059, A130060, A130061, A130062.

Select[ Range[2000], PrimeQ[ # ]&&Mod[ PowerMod[3,(#+1)/2,# ] - PowerMod[2,(#+1)/2,# ] - 1, # ]==0&]
Select[Prime[Range[250]],Divisible[3^((#+1)/2)-2^((#+1)/2)-1,#]&] (* Harvey P. Dale, Mar 21 2021 *)

is(n)=(n+1)%24<3 && isprime(n) \\ Charles R Greathouse IV, Apr 06 2012

cp's OEIS Frontend

A130059 Primes p such that k=p*23^2 divides 3^(k-1) - 2^(k-1); or primes in A130058.

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A073631 Nonprimes k such that k divides 3^(k-1) - 2^(k-1).

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A130062 Nonprime numbers k such that k divides 3^((k+1)/2) - 2^((k+1)/2) - 1.

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A130060 Primes p such that p^2 divides 3^p - 2^p - 1; or primes in A127074(n).

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A130061 Numbers k that divide 3^((k-1)/2) - 2^((k-1)/2) - 1.

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Mathematica

A130063 Primes p such that p divides 3^((p+1)/2) - 2^((p+1)/2) - 1.

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Mathematica

PARI