A057732 -id:A057732 - cp's OEIS frontend

A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.

1, 2, 6, 12, 18, 30

Offset: 1

Juri-Stepan Gerasimov, Mar 04 2014

Intersection of A057732 and A002253. - Joerg Arndt, Mar 04 2014
By checking primality of 2^n+3 for values n in A002253, it follows a(7) > 7033641. - Giovanni Resta, Mar 08 2014
Exponents of second Fermat prime pairs. - Juri-Stepan Gerasimov, Mar 08 2014
From Juri-Stepan Gerasimov, Mar 04 2014: (Start)
If prime pair {2^n + (2k+1), (2k+1)*2^n + 1} is called a Fermat prime pair, then numbers n such that 2^n + (2k + 1) and (2k + 1)*2^n + 1 are both prime:
for k = 0: 0, 1, 2, 4, 8, 16, ...   the exponents first Fermat prime pairs;
for k = 1: 1, 2, 6, 12, 18, 30, ... the exponents second Fermat prime pairs;
for k = 2: 1, 3, ...                the exponents third Fermat prime pairs;
for k = 3: 2, 4, 6, 20, 174, ...    the exponents fourth Fermat prime pairs;
for k = 4: 1, 2, 3, 6, 7, ...       the exponents fifth Fermat prime pairs;
for k = 5: 1, 3, 5, 7, ...          the exponents sixth Fermat prime pairs;
for k = 6: 2, 8, 20, ...            the exponents seventh Fermat prime pairs;
for k = 7: 1, 2, 4, 10, 12, ...     the exponents eighth Fermat prime pairs;
for k = 8:
for k = 9: 6, ...                   the exponents tenth Fermat prime pairs;
for k = 10: 1, 4, 5, 7, 16, ...     the exponents eleventh Fermat prime pairs;
for k = 11:
for k = 12: 2, 4, 6, 10, 20, 22, ...the exponents thirteenth Fermat prime pairs;
for k = 13: 2, 4, 16, 40, 44, ...   the exponents fourteenth Fermat prime pairs;
for k = 14: 1, 3, 5, 27, 43, ...    the exponents fifteenth Fermat prime pairs.
Semiprimes of the form (2^m+2k+1)*((2k+1)*2^m+1): 4, 9, 25, 35, 77, 91, 209, 289, 319, 481, 527, 533, 901, 989, ...
(End)

			a(1) = 1 because 2^1 + 3 = 5 and 3*2^1 + 1 = 7 are both prime,
a(2) = 2 because 2^2 + 3 = 7 and 3^2^2 + 1 = 13 are both prime,
a(3) = 6 because 2^6 + 3 = 67 and 3*2^6 + 1 = 193 are both prime.

Cf. A002253, A019434, A039687, A057732, A057733, A238554, A239038, A267984.

[n: n in [0..30] | IsPrime(2^n+3) and IsPrime(3*2^n+1)]; // Arkadiusz Wesolowski, Jan 23 2016

Select[Range[30],AllTrue[{2^#+3,3*2^#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 08 2015 *)

isok(n) = isprime(2^n + 3) && isprime(3*2^n + 1); \\ Michel Marcus, Mar 04 2014

A298563 Numbers k such that k - 2 | sigma(k).

Original entry on oeis.org

1, 3, 5, 6, 14, 44, 110, 152, 884, 2144, 8384, 18632, 116624, 8394752, 15370304, 73995392, 536920064, 2147581952, 34360131584, 27034175140420610, 36028797421617152, 576460753914036224

Offset: 1

Zdenek Cervenka, Jan 21 2018

Similar to A055708.
Sequence includes every number of the form 2^(j-1)*(2^j+3) such that 2^j+3 is prime (i.e., j is a term in A057732); terms of this form are 5, 14, 44, 152, 2144, 8384, 8394752, 536920064, 2147581952, 34360131584, ... - Jon E. Schoenfield, Jan 22 2018
Superset of A125246. - Giovanni Resta, Jan 23 2018
Contains 2 times odd terms of A191363. Also, if m is a term of A056006 and q := (sigma(m) + 2)/m is coprime to m, them q*m is a term. - Max Alekseyev, May 25 2025

			For k=44, sigma(k)/(k-2) = sigma(44)/(44-2) = 84/42 = 2, so 44 belongs to the sequence;
for k=110, sigma(k)/(k-2) = sigma(110)/(110-2) = 216/108 = 2, so 110 is also a term.

Cf. A055708, A057732, A125246.

[n: n in [3..10^7]| DivisorSigma(1, n) mod (n-2) eq 0]; // Vincenzo Librandi, Jan 22 2018

Select[Range[10^6], Divisible[DivisorSigma[1, #], # - 2] &] (* Michael De Vlieger, Jan 21 2018 *)

isok(k) = (k!=2) && !(sigma(k) % (k-2)); \\ Michel Marcus, Jan 22 2018

a(17)-a(18) from Robert G. Wilson v, Jan 21 2018
a(19) from Giovanni Resta, Jan 23 2018
a(20)-a(22) from Max Alekseyev, May 27 2025

A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 5, 4, 1, 6, 11, 6, 3, 2, 7, 47, 8, 5, 4, 1, 12, 53, 10, 7, 8, 13, 2, 15, 141, 16, 9, 20, 21, 6, 3, 16, 143, 18, 15, 38, 33, 30, 7, 1, 18, 191, 20, 23, 64, 81, 162, 39, 3, 4, 28, 273, 28, 29, 80, 129, 654, 79, 5, 12, 2

Offset: 2

Jean-Marc Rebert, Mar 22 2023

			p = prime(2) = 3, m=1, u = {p + 2^k | 1 <= k <= m} = {5} contains one prime, and no lesser m satisfies this, so A(2,1) = 1.
Square array A(n,k) n > 1 and k >= 1 begins:
 1,     2,     3,     4,     6,     7,    12,    15,    16,    18, ...
 1,     3,     5,    11,    47,    53,   141,   143,   191,   273, ...
 2,     4,     6,     8,    10,    16,    18,    20,    28,    30, ...
 1,     3,     5,     7,     9,    15,    23,    29,    31,    55, ...
 2,     4,     8,    20,    38,    64,    80,   292,  1132,  4108, ...
 1,    13,    21,    33,    81,   129,   285,   297,   769,  3381, ...
 2,     6,    30,   162,   654,   714,  1370,  1662,  1722,  2810, ...
 3,     7,    39,    79,   359,   451,  1031,  1039, 11311, 30227, ...
 1,     3,     5,     7,     9,    13,    15,    17,    23,    27, ...

Cf. A057732 (1st row), A094076 (1st column).
Cf. A361679.
Cf. A019434 (primes 2^n+1), A057732 (2^n+3), A059242 (2^n+5), A057195 (2^n+7), A057196(2^n+9), A102633 (2^n+11), A102634 (2^n+13), A057197 (2^n+15), A057200 (2^n+17), A057221 (2^n+19), A057201 (2^n+21), A057203 (2^n+23).
Cf. A205558 and A231232 (with 2*k instead of 2^k).

A(n, k)= {my(nb=0, p=prime(n), m=1); while (nb

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A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.

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A298563 Numbers k such that k - 2 | sigma(k).

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A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

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