A130060 -id:A130060 - cp's OEIS frontend

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

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

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

A260507 Primes p such that (2^p+1)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

2, 7, 179, 619, 17807

Offset: 1

Felix Fröhlich, Jul 27 2015

A000040(n) such that A260531(n) = 1.
Is this a subsequence of A130060?
a(6) > 10325801 if it exists.
a(6) > 3037000499 if it exists. - Hiroaki Yamanouchi, Aug 20 2015

			2^7 + 1 = 129 and 129^6 == 1 (mod 7^2), so 7 is a term of the sequence.

Cf. A098640, A127074, A130060, A130062, A260531.

Select[Prime@ Range@ 120, Mod[(2^# + 1)^(# - 1), #^2] == 1 &] (* Michael De Vlieger, Jul 29 2015 *)

forprime(p=2, , if(Mod(2^p+1, p^2)^(p-1)==1, print1(p, ", ")))

cp's OEIS Frontend

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

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

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

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A260507 Primes p such that (2^p+1)^(p-1) == 1 (mod p^2).

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