A141453 -id:A141453 - cp's OEIS frontend

A210844 A pair of solutions of a congruence related to A141453.

3, 5, 9, 15, 33, 63, 255, 513, 16383, 131073, 262143, 1048575, 4294967295, 4611686018427387903, 1237940039285380274899124223, 324518553658426726783156020576255, 340282366920938463463374607431768211455

Offset: 1

Wolfdieter Lang, Mar 28 2012

nonn

See the comment on A141453. There r(a(n)) is the present a(n).
The next entry a(18) has 158 digits.
The sequence of exponents of 2 of the Fermat and Mersenne primes FM:=A141453 (including the prime 2) starts with k:=[0, 1, 2, 3, 4, 5, 7, 8, 13, 16, 17, 19, 31, 61, 89, 107, 127, 521,...], n>=1.
For the second k entry one can also take 2 instead of 1. Then a(2) should be replaced by 7.
a(n) and FM(n)*2^(k(n)+1) - a(n) are an incongruent pair of solutions of the congruence x^2 == 1 (mod FM(n)*2^(k(n)+1)), n>=1. For n>=3 there are all-together eight incongruent solutions. The trivial pair of positive solutions is always 1 and FM(n)*2^(k(n)+1) - 1. Two more pairs should therefore be found.

			From Wolfdieter Lang, Apr 10 2012 (Start)
a(1)=3 because 3^2 = 9 == 1 (mod 2*2^(0+1)) = 1 (mod 4). The incongruent companion solution is 4 - 3 = 1. This is the trivial pair of solutions.
a(2)=5 because 5^2 = 25 == 1 (mod 3*2^(1+1)) = 1 (mod 12). The incongruent companion solution is 12 - 5 = 7, obtained also by taking k(2)=2. The trivial pair of solutions is (1,11).
  1, 5, 7 and 11 are all the solutions of this congruence.
a(3)=9  because 9^2 = 81 == 1 (mod 5*2^(2+1)) = 1 (mod 40).
  The companion solution is 40 - 9 = 31. The trivial pair is (1,39). The missing two pairs are (11,29) and (19,21), and all eight incongruent solutions are 1, 9, 11, 19, 21, 29, 31 and 39.
(End)

Cf. A141453.

a(n) = sqrt(FM(n)*2^(k(n)+2) + 1), n>=1, with FM(n):=A141453(n) and the sequence k is given for n=1..18 in the comment section.

A285929 Numbers m such that 2^m + (-1)^m is prime.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 8, 13, 16, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667

Offset: 1

Juri-Stepan Gerasimov, Apr 28 2017

nonn

With 1, exponents of A141453 (see comment by Wolfdieter Lang, Mar 28 2012).
Numbers m such that (1 + k)^m + (-k)^m is prime:
0 (k = 0);
this sequence (k = 1);
A283653 (k = 2);
0, 3, 4, 7, 16, 17, ... (k = 3);
0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);
0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ... (k = 5);
0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);
0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);
0, 4, 7, 29, 31, 32, 67, ... (k = 8);
0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);
0, 3, 5, 19, 32, ... (k = 10);
0, 3, 7, 89, 101, ... (k = 11);
0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);
0, 3, 4, 11, 83, ... (k = 13);
0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);
0, 43, ... (k = 15);
0, 4, 5, 7, 79, ... (k = 16);
0, 2, 3, 8, 13, 71, ... (k = 17);
0, 1607, ... (k = 18);
...
Numbers m such that (1 + k)^m + (-k)^m is not an odd prime for k <= m: 0, 1, 15, 18, 53, 59, 106, 114, 124, 132, 133, 143, 177, 214, 232, 234, 240, 256, ...
Conjecture: if (1 + y)^x + (-y)^x is a prime number then x is zero, or an even power of two, or an odd prime number.
The above conjecture can be proved by considering algebraic factorizations of the polynomials involved. - Jeppe Stig Nielsen, Feb 19 2023
Appears to be essentially the same as A174269. - R. J. Mathar, May 21 2017

			4 is in this sequence because 2^4 + (-1)^4 = 17 is prime.
5 is in this sequence because 2^5 + (-1)^5 = 31 is prime.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..52

Supersequence of A000043.

Cf. A000668, A019434, A141453, A174269, A283653, A285886, A285887, A285888.

[m: m in [0..1000]| IsPrime(2^m + (-1)^m)];

Select[Range[0, 10^4], PrimeQ[2^# + (-1)^#] &] (* Michael De Vlieger, May 03 2017 *)

is(m)=ispseudoprime(2^m+(-1)^m) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = A174269(n) for n > 2. - Jeppe Stig Nielsen, Feb 19 2023

A335911 Numbers of the form q*(2^k), where k >= 0 and q is either a Fermat prime or a Mersenne prime; Numbers k for which A335885(k) = 1.

Original entry on oeis.org

3, 5, 6, 7, 10, 12, 14, 17, 20, 24, 28, 31, 34, 40, 48, 56, 62, 68, 80, 96, 112, 124, 127, 136, 160, 192, 224, 248, 254, 257, 272, 320, 384, 448, 496, 508, 514, 544, 640, 768, 896, 992, 1016, 1028, 1088, 1280, 1536, 1792, 1984, 2032, 2056, 2176, 2560, 3072, 3584, 3968, 4064, 4112, 4352, 5120, 6144, 7168, 7936, 8128, 8191

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers k such that A000265(k) is either in A000668 or in A019434.

Product of any two terms (whether distinct or not) can be found in A335912.

Row 1 of A335910.
Union of A334101 and A335431. Subsequence of A038550.
Cf. A000668, A000668, A335885, A335912.
Cf. A141453 (after its initial 2, gives the primes present in this sequence).

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335911(n) = (1==A335885(n));

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2, n-1));
isA335911(n) = (isA000668(A000265(n))||isA019434(A000265(n)));

A169818 Rank of n-th prime as defined in A177854.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 1, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3

Offset: 1

T. D. Noe, May 28 2010

nonn

This notion of rank is closely related to the Erdős-Selfridge classification of primes.

T. D. Noe, Table of n, a(n) for n=1..10000

For records see A177854.

rank[1]=0; rank[2]=0; rank[3]=1;
SetAttributes[rank,Listable];
rank[p_] := rank[p] = 1+Min[Max@@rank[First/@FactorInteger[p-1]], Max@@rank[First/@FactorInteger[p+1]]]; rank[Prime[Range[100]]]

a(A000720(A141453(n)))=1 n>1. [From R. J. Mathar, May 28 2010]

A378108 Primes p such that neither p-1 nor p+1 are in A126706.

Original entry on oeis.org

2, 3, 5, 7, 31, 257, 131071, 618970019642690137449562111, 162259276829213363391578010288127

Offset: 1

Michael De Vlieger, Nov 26 2024

Primes p < 11 are in the sequence since the smallest number in A126706 is 12.
Consider p > 11, odd primes; then both p-1 and p+1 are even. Let j and k be neighbors of p. One neighbor, j, is also divisible by 4, while the other neighbor k is not divisible by 2^m, m > 1. The latter statement implies k cannot be a perfect power q^m, p != q, m > 0, but q^m may divide k.
This sequence is that of primes where j = 2^m and k is squarefree.
Proper subset of A141453.
The neighbor k is also divisible by 3, since abs(p-k) = 1 and neither are divisible by 3. Therefore, 6 | k.

			17 = 2^4+1 is not in the sequence since 18 = 2 * 3^2.
31 = 2^5-1 is in the sequence since 30 = 2*3*5 is squarefree.
127 = 2^7-1 is not in the sequence because 126 = 2 * 3^2 * 7.
257 = 2^8+1 is in the sequence since 258 = 2*3*43 is squarefree.
8191 = 2^13-1 is not in the sequence because 8190 = 2 * 3^2 * 5 * 7 * 13.
65537 = 2^16+1 is not in the sequence since 65538 = 2 * 3^2 * ll * 331.
131071 = 2^17-1 is in the sequence since 131070 = 2 * 3 * 5 * 17 * 257, etc

Cf. A000040, A000079, A126706, A141453, A303554.

Reap[Do[Which[
  And[PrimeQ[# + 1], SquareFreeQ[(# + 2)/6]], Sow[# + 1],
  And[PrimeQ[# - 1], SquareFreeQ[(# - 2)/6]], Sow[# - 1] ] &[2^i],
{i, 0, 650}] ][[-1, 1]]

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A210844 A pair of solutions of a congruence related to A141453.

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A285929 Numbers m such that 2^m + (-1)^m is prime.

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A335911 Numbers of the form q*(2^k), where k >= 0 and q is either a Fermat prime or a Mersenne prime; Numbers k for which A335885(k) = 1.

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PARI

A169818 Rank of n-th prime as defined in A177854.

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A378108 Primes p such that neither p-1 nor p+1 are in A126706.

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Mathematica