A057197 -id:A057197 - cp's OEIS frontend

A253773 Numbers k such that 4^k + 15 is prime.

1, 2, 3, 4, 5, 6, 8, 11, 13, 15, 16, 20, 21, 23, 36, 38, 90, 99, 155, 164, 171, 254, 255, 273, 404, 1386, 1941, 1970, 2420, 3759, 5559, 5776, 6369, 6429, 22061, 32330, 81780, 90248, 162933, 240920, 504584

Offset: 1

Vincenzo Librandi, Jan 12 2015

Half of even terms of A057197. - Michel Marcus, Aug 28 2015

			For k = 15, 4^15 + 15 = 1073741839 is prime.

Cf. A057197, A237418, A253772 (similar sequence).

Magma
```
[n: n in [0..1300] | IsPrime(4^n+15)];
```

Select[Range[10000], PrimeQ[4^# + 15] &]

is(n)=isprime(4^n + 15) \\ Anders Hellström, Aug 28 2015

a(31)-a(39) from A057197 data by Michel Marcus, Aug 28 2015
a(40) derived from A057197 by Robert Price, Sep 18 2015
a(41) from A057197 data by Elmo R. Oliveira, Dec 11 2023

A295196 Numbers n > 1 such that 2^(n-1) and (2n-m)2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.

Original entry on oeis.org

7, 23, 31, 47, 71, 79, 263, 271, 1031, 1039, 2063, 4111, 32783, 65543, 65551, 262151, 1048583, 4194319, 8388623, 67108879, 268435463, 1073741831, 1073741839, 4294967311

Offset: 1

Jonas Kaiser, Nov 16 2017

This definition arises from the conjecture that pseudoprime numbers (A001567) occur only at certain distances m from the next smaller number of the form 2^n. So, if we know that a certain distance does not appear with pseudoprime numbers, we are able to calculate these numbers using Fermat's little theorem and we know that it has to be prime. To "plot" the distance of pseudoprime numbers to 2^n use m = A001567(n) - 2^floor(log_2(A001567(n))). So, the first values of m which do not have a "safe prime number distance" (values with "safe prime number distance" are those values for m which pseudoprime numbers never have) should be m = 1, 49, 81, 85, 129, 133, 273, 275, 289, 321, ....
Conjecture 1: There are no composite numbers in this sequence and perhaps infinitely many primes.
Conjecture 2: For m = 7 this definition generates A104066 and for m = 15 this definition generates A144487 (A057197).
Conjecture 3: There are (infinitely many?) m for which this definition generates nothing but (infinitely many?) primes of the form p = 2^k + m.
It appears that this sequence is a subsequence of A139035.

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A057197, A104066, A139035, A144487, A244626, A294717, A293394.

twoDistableQ[n_] := MemberQ[Mod[(2n - {3, 7, 15}) PowerMod[2, (n - 1)/2 - Floor@ Log2@ n, n], n], 1]; p = 3; twoDistablesList = {}; While[p < 1000000000, If[twoDistableQ@ p, AppendTo[ twoDistablesList, p]]; p = NextPrime@ p]; twoDistablesList (* Robert G. Wilson v, Nov 17 2017 *)

a(n) = (n%2) && lift((Mod(2, n)^(n-1))==1) && (lift((Mod((2*n-3), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-7), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-15), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1))

is(n)=if(Mod(2,n)^(n-1)!=1, return(0)); my(m=Mod(2,n)^(n\2-logint(n,2))); ((2*n-3)*m==1 || (2*n-7)*m==1 || (2*n-15)*m==1) && n>1 \\ Charles R Greathouse IV, Nov 17 2017

a(17)-a(24) from Charles R Greathouse IV, Nov 17 2017

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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A253773 Numbers k such that 4^k + 15 is prime.

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A295196 Numbers n > 1 such that 2^(n-1) and (2n-m)2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.

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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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cp's OEIS Frontend

A253773 Numbers k such that 4^k + 15 is prime.

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Programs

Magma

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Extensions

A295196 Numbers n > 1 such that 2^(n-1) and (2*n-m)*2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.

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PARI

Extensions

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.

Views

Author

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Examples

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Programs

PARI

A295196 Numbers n > 1 such that 2^(n-1) and (2n-m)2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15.