A267985 -id:A267985 - cp's OEIS frontend

A267984 Numbers congruent to {17, 23} mod 30.

17, 23, 47, 53, 77, 83, 107, 113, 137, 143, 167, 173, 197, 203, 227, 233, 257, 263, 287, 293, 317, 323, 347, 353, 377, 383, 407, 413, 437, 443, 467, 473, 497, 503, 527, 533, 557, 563, 587, 593, 617, 623, 647, 653, 677, 683, 707, 713, 737, 743, 767, 773

Offset: 1

Arkadiusz Wesolowski, Jan 23 2016

Union of A128468 and A128473.

For all k >= 1 the numbers 2^k + a(n) and a(n)*2^k + 1 do not form a pair of primes, where n is any positive integer.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A128468, A128473, A267985.

[n: n in [0..773] | n mod 30 in {17, 23}];

LinearRecurrence[{1, 1, -1}, {17, 23, 47}, 52]

Vec(x*(17 + 6*x + 7*x^2)/((1 + x)*(1 - x)^2) + O(x^53))

a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.
G.f.: x*(17 + 6*x + 7*x^2)/((1 + x)*(1 - x)^2).
a(n) = a(n-2) + 30.
a(n) = 10*(3*n - 2) - a(n-1).
From Colin Barker, Jan 24 2016: (Start)
a(n) = (30*n - 9*(-1)^n - 5)/2 for n>0.
a(n) = 15*n - 7 for n>0 and even.
a(n) = 15*n + 2 for n odd.
(End)
E.g.f.: 7 + ((30*x - 5)*exp(x) - 9*exp(-x))/2. - David Lovler, Sep 10 2022

A296716 Numbers congruent to {7, 11, 13, 29} mod 30.

Original entry on oeis.org

7, 11, 13, 29, 37, 41, 43, 59, 67, 71, 73, 89, 97, 101, 103, 119, 127, 131, 133, 149, 157, 161, 163, 179, 187, 191, 193, 209, 217, 221, 223, 239, 247, 251, 253, 269, 277, 281, 283, 299, 307, 311, 313, 329, 337, 341, 343, 359, 367, 371, 373, 389, 397, 401, 403

Offset: 1

Arkadiusz Wesolowski, Dec 19 2017

For any m >= 0, if F(m) = 2^(2^m) + 1 has a factor of the form b = a(n)*2^k + 1 with k >= m + 2 and n >= 1, then the integer F(m)/b is congruent to 13 or 19 mod 30.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A191062, A267985.

[n: n in [0..403] | n mod 30 in {7, 11, 13, 29}];

LinearRecurrence[{1, 0, 0, 1, -1}, {7, 11, 13, 29, 37}, 60]

Vec(x*(7 + 4*x + 2*x^2 + 16*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2 + O(x^55)))

a(n) = a(n-1) + a(n-4) - a(n-5), n >= 6.
a(n) = a(n-4) + 30.
G.f.: x*(7 + 4*x + 2*x^2 + 16*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).
a(n) = (-15 + 5*(-1)^n + (3+9*i)*(-i)^n + (3-9*i)*i^n + 30*n) / 4 where i=sqrt(-1). - Colin Barker, Dec 19 2017
E.g.f.: (5*(e^(-x) + (6*x - 3)*e^x) + 6*cos(x) + 18*sin(x))/4. - Iain Fox, Dec 19 2017

A267943 Numbers n such that 2^n - 3 and 3*2^n - 1 are both prime.

Original entry on oeis.org

3, 4, 6, 94

Offset: 1

Arkadiusz Wesolowski, Jan 22 2016

The intersection of A002235 and A050414 is not empty (3 does not belong to A267985).

			a(3) = 6 because 2^6 - 3 = 61 and 3*2^6 - 1 = 191 are both prime.

Cf. A002235, A007505, A050414, A050415, A238694, A267985.

[n: n in [2..94] | IsPrime(2^n-3) and IsPrime(3*2^n-1)];

isok(n) = isprime(2^n-3) && isprime(3*2^n-1);

A002235 INTERSECT A050414.

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A267984 Numbers congruent to {17, 23} mod 30.

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A296716 Numbers congruent to {7, 11, 13, 29} mod 30.

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A267943 Numbers n such that 2^n - 3 and 3*2^n - 1 are both prime.

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