A158036 -id:A158036 - cp's OEIS frontend

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

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

A161897 Prime numbers p for which k = (3^p - 3 * 3^((p + 1) / 2) - 6p + 6) / (3p^2 - 3p) is an integer.

Original entry on oeis.org

11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903, 2963, 2999

Offset: 1

Reikku Kulon, Jun 21 2009

Superset of the inverse Sophie Germain primes (A005385): (p - 1) / 2 is almost always prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A161896, A000040, A005385, A158034, A158035, A158036, A145918.

filter:= p -> isprime(p) and
   (3&^p - 3 * 3&^((p + 1) / 2) - 6*p + 6) mod (3*p^2-3*p) = 0:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Mar 31 2017

A160556 Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000.

Original entry on oeis.org

2, 8, 14, 17, 26, 29, 32, 38, 41, 47, 50, 59, 62, 64, 65, 68, 74, 77, 83, 89, 95, 98, 101, 104, 110, 119, 122, 128, 131, 134, 137, 140, 143, 149, 152, 155, 161, 164, 167, 173, 179, 182, 185, 188, 194, 197, 200, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233

Offset: 1

Reikku Kulon, May 19 2009

For these equations (not exclusively), the sequences of 2n + 1 are dominated by primes.
When b = 2, there are 105 solutions with n less than 10000, and in this case, the sequence of n is also dominated by primes: only five of these are composite. The average difference between successive composite terms is near the magnitude of n. No composite values of 2n + 1 have been found. n and 2n + 1 account for roughly 3% of primes less than 20 billion. For other bases, n is almost always composite, and 2n + 1 is almost always prime.
The next most productive values of b less than 1000 are 509 (41 solutions) and 824 (40 solutions).
Bases that produce a greater or equal number of solutions than smaller bases, except 2, often have ones digit 4 or 9. Values of n associated with composite 2n + 1 are often divisible by 5.

Cf. A158034, A158035, A158036

Cf. A160557

A160557 Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000, n is never divisible by 5, and 2n + 1 is prime.

Original entry on oeis.org

2, 32, 41, 101, 161, 185, 206, 215, 230, 251, 290, 311, 326, 335, 356, 395, 416, 446, 461, 521, 566, 611, 626, 641, 656, 740, 860, 866, 926, 941, 956, 965, 1025, 1055, 1076, 1091, 1130, 1151, 1241, 1256, 1271, 1286, 1361, 1370, 1385, 1391, 1436, 1451, 1466

Offset: 1

Reikku Kulon, May 19 2009

nonn

When b = 2, there are 105 solutions less than 10000, and in this case, the sequence of n is dominated by primes: only five of these are composite. The average difference between successive composite terms is near the magnitude of n. n and 2n + 1 account for roughly 3% of primes less than 20 billion. For other bases, n is almost always composite.

There are 31 solutions when b = 1286.

Cf. A158034, A158035, A158036

Cf. A160556

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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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A161897 Prime numbers p for which k = (3^p - 3 * 3^((p + 1) / 2) - 6p + 6) / (3p^2 - 3p) is an integer.

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A160556 Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000.

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A160557 Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000, n is never divisible by 5, and 2n + 1 is prime.

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