A165780 -id:A165780 - cp's OEIS frontend

A165778 Numbers k such that |2^k - 57| is prime.

2, 4, 6, 7, 8, 10, 16, 19, 22, 28, 43, 46, 56, 58, 62, 67, 74, 82, 140, 160, 316, 346, 376, 454, 458, 487, 580, 607, 1018, 1579, 1739, 1870, 2006, 3014, 3056, 6962, 7075, 7852, 8207, 9190, 11854, 14816, 23308, 29222, 33808, 40618, 47408, 50843, 58312, 98554

Offset: 1

M. F. Hasler, Oct 11 2009

nonn

If p = 2^k-57 is prime, then 2^(k-1)*p is in A101260, i.e., a solution to sigma(x)-2x = 56 = 2^3*(2^3-1) = 2*A000396(2).

			a(3) = 6 since 2^6-57 = 7 is prime.
For exponents a(1) = 2 and a(2) = 4, we get 2^a(n)-57 = -53 and -41 which are negative, but which are prime in absolute value.

Cf. A096818, A165779, A165780.

[n: n in [1..1100] | IsPrime(2^n-57)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 57]}, {n, 0, 100}], PrimeQ[#[[2]]] &][[All,1]] (* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-57)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

from sympy import isprime, nextprime
def afind(limit):
    k, pow2 = 1, 2
    for k in range(1, limit+1):
        if isprime(abs(pow2-57)):
            print(k, end=", ")
        k += 1
        pow2 *= 2
afind(2100) # Michael S. Branicky, Dec 27 2021

a(36)-a(42) from Altug Alkan, Apr 08 2016
a(43)-a(44) from Michael S. Branicky, Dec 27 2021
a(45)-a(49) from Michael S. Branicky, May 14 2023
a(50) from Michael S. Branicky, Sep 25 2024

A165779 Numbers k such that |2^k-993| is prime.

Original entry on oeis.org

1, 4, 6, 10, 14, 17, 26, 29, 54, 62, 77, 121, 344, 476, 1012, 1717, 1954, 2929, 2993, 3014, 3304, 4704, 8882, 24042, 43572, 45722, 54913, 57893, 72566, 74473, 82092, 117302

Offset: 1

M. F. Hasler, Oct 11 2009

If p = 2^k-993 is prime, then 2^(k-1)*p is a solution to sigma(x)-2x = 992 = 2^5*(2^5-1) = 2*A000396(3).

			a(4) = 10 since 2^10-993 = 31 is prime.
For exponents a(1) = 1, a(2) = 4 and a(3) = 6, we get 2^a(n)-993 = -991, -977 and -929 which are negative, but which are prime in absolute value.

Cf. A000396, A096818, A165778, A165780.

[n: n in [1..1100] |IsPrime(2^n-993)]; // Vincenzo Librandi, Apr 09 2016

Select[Table[{n, Abs[2^n - 993]}, {n,0,100}], PrimeQ[#[[2]]] &][[All, 1]] (* G. C. Greubel, Apr 08 2016 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(abs(2^n-993)), print1(n, ", "))); \\ Altug Alkan, Apr 08 2016

from sympy import isprime, nextprime
def afind(limit):
    k, pow2 = 1, 2
    for k in range(1, limit+1):
        if isprime(abs(pow2-993)):
            print(k, end=", ")
        k += 1
        pow2 *= 2
afind(2000) # Michael S. Branicky, Dec 26 2021

a(23) from Altug Alkan, Apr 08 2016
a(24) from Michael S. Branicky, Dec 26 2021
a(25)-a(26) from Michael S. Branicky, Apr 06 2023
a(27)-a(32) from Michael S. Branicky, Sep 25 2024

cp's OEIS Frontend

A165778 Numbers k such that |2^k - 57| is prime.

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