A161897 -id:A161897 - cp's OEIS frontend

A284660 Terms of A161897 that are not in A005385.

3083, 4931, 6563, 9923, 166667, 865643, 1306667, 2266883, 3367367, 3906563, 4128959, 5493179, 5591039, 6040187, 9122963, 9402179, 9871403, 10174343, 13081379, 13756403, 14924003, 16550243, 24165287, 29492747, 32140859, 34633427, 38425643, 42249587, 42258779, 43014659, 45067523, 52678643

Offset: 1

Robert Israel, Mar 31 2017

nonn

Primes p such that q = (p-1)/2 is composite, 3^q == 1 (mod p) and 3^(q-1) == 1 (mod p-1).

All terms == 5 (mod 6).

			p = 3083 is in the sequence because it is prime, q = (3083-1)/2 = 23*67 is composite, 3^q == 1 (mod p) and 3^(q-1) == 1 mod (p-1).

Robert Israel, Table of n, a(n) for n = 1..212

Cf. A005385, A161897.

filter:= p -> isprime(p) and not isprime((p-1)/2) and
   3&^((p-3)/2)  mod (p-1) = 1 and
   3 &^((p-1)/2) mod p = 1;
select(filter, [seq(p, p=5..10^7, 6)]); # Robert Israel, Mar 31 2017

A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1541, 1559

Offset: 1

Reikku Kulon, Jun 21 2009

Near superset of the Sophie Germain primes (A005384), excluding 2 and 3: 2n + 1 is prime. Nearly all members of this sequence are also prime, but four members less than 10000 are composite: 1541 = 23 * 67, 2465 = 5 * 17 * 29, 3281 = 17 * 193, and 4961 = 11^2 * 41.
The congruence of n modulo 4 is evenly distributed between 1 and 3. n is congruent to 5 (mod 6) for all n less than two billion.
This sequence has roughly twice the density of the sequence (A158034) corresponding to the Diophantine equation
f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)),
and contains most members of that sequence. Those it does not contain are composite and often congruent to 3 (mod 6).
Composite terms appear to predominantly belong to A262051. - Bill McEachen, Aug 29 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A161897 A000040, A002515, A005384, A158034, A158035, A158036, A145918, A002943.

a161896 n = a161896_list !! (n-1)
a161896_list = [x | x <- [1..],
                    (9^x - 3*3^x - 4*x) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

is(n)=my(m=2*n*(2*n+1),t=Mod(3,m)^n); t^2-3*t==4*n \\ Charles R Greathouse IV, Nov 25 2014

A162587 Prime numbers p = 8n + 7 for which k = (7 * (256^n - 16^n) + n^2) / (n * p) is an integer.

Original entry on oeis.org

23, 31, 47, 71, 79, 103, 127, 151, 167, 199, 223, 263, 367, 439, 487, 607, 631, 647, 727, 823, 887, 967, 1031, 1087, 1303, 1327, 1367, 1447, 1543, 1567, 1607, 1879, 1951, 2207, 2311, 2503, 2647, 2671, 2887, 3079, 3271, 3463, 3527, 3607, 3727, 3847, 3967

Offset: 1

Reikku Kulon, Jul 07 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A162586
Cf. A158034, A158035
Cf. A161896, A161897

Transpose[Select[Table[{n,8n+7},{n,500}],PrimeQ[Last[#]]&&IntegerQ[(7*(256^First[ #]-16^First[#])+First[#]^2)/Times@@#]&]][[2]] (* Harvey P. Dale, Sep 13 2014 *)

A162586 Integers n for which k = (7 * (256^n - 16^n) + n^2) / (n * (8n + 7)) is an integer.

Original entry on oeis.org

2, 3, 5, 8, 9, 12, 15, 18, 20, 24, 27, 32, 45, 54, 60, 75, 78, 80, 90, 102, 110, 120, 128, 135, 162, 165, 170, 180, 192, 195, 200, 234, 243, 275, 288, 312, 330, 333, 360, 384, 408, 432, 440, 450, 465, 480, 495, 500, 513, 540, 612, 620, 624, 675, 684, 702, 729

Offset: 1

Reikku Kulon, Jul 07 2009

8n + 7 is almost always prime. It is first composite for 8 * 372060 + 7 = 2976487 = 863 * 3449.

Cf. A162587
Cf. A158034, A158035
Cf. A161896, A161897

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A284660 Terms of A161897 that are not in A005385.

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A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

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A162587 Prime numbers p = 8n + 7 for which k = (7 * (256^n - 16^n) + n^2) / (n * p) is an integer.

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A162586 Integers n for which k = (7 * (256^n - 16^n) + n^2) / (n * (8n + 7)) is an integer.

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