A263644 -id:A263644 - cp's OEIS frontend

A263822 Conjectured number of terms in A263644 less than 10^n.

0, 0, 0, 0, 0, 0, 0, 14, 141

Offset: 1

Arkadiusz Wesolowski, Oct 27 2015

A263645 Primes that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

2, 52504261, 55414847, 79933129, 152485283, 166441831, 177702619, 197903207, 199013093, 220403959, 226794259, 230701763, 245215801, 266642731, 304921637, 321979283, 335035097, 355404353, 359018299, 369810769, 388048561, 412590797, 445661719, 506400173, 540426473

Offset: 1

Arkadiusz Wesolowski, Oct 22 2015

nonn

Primes p such that for all k > 0 the numbers p + 2^k and p - 2^k are nonprimes.

Except for 2, this sequence is the intersection of A065381 and A137715.

Cf. A065381, A137715, A153352, A180247, A263644.

A276495 Odd numbers not of the form p + 2^m with p prime and m >= 0 for which the smallest k in A067760 such that n + 2^k is prime increases.

Original entry on oeis.org

1, 127, 251, 1657, 1777, 1973, 3181, 21893, 31951, 50839, 67607, 138977

Offset: 1

Arkadiusz Wesolowski, Sep 05 2016

There exist de Polignac numbers n such that for all k >= 1 the numbers n + 2^k are composite. It is conjectured that 30666137 is the smallest such number.

a(13) >= 453143.

Cf. A006285, A067760, A263644.

A276496 gives the record values.

lst:=[]; c:=0; for n in [1..31951 by 2] do m:=-1; repeat m+:=1; a:=n-2^m; until a lt 1 or IsPrime(a); if a lt 1 then k:=0; repeat k+:=1; b:=n+2^k; until IsPrime(b); if k gt c then Append(~lst, n); c:=k; end if; end if; end for; lst;

A283619 a(n) = (conjectured) smallest positive integer k which is neither of the form p + n^x nor of the form p - n^x with x >= 0 and p prime, where gcd(k, n) = 1 and gcd(k^2-1, n-1) = 1.

Original entry on oeis.org

30666137, 3902132276156, 2473929, 1015214, 464437, 40743218950116, 47, 2344, 61863, 32660, 4367, 7974, 11, 2021170066180678, 92343, 784, 571, 2364594, 13, 20450, 136113, 2596, 176011, 262638, 3223, 512, 59217, 26, 18973, 6360528, 23, 11848, 99, 292226, 832573

Offset: 2

Arkadiusz Wesolowski, Mar 12 2017

nonn

The definition is similar to that for A123159, but considering "p + n^x" and "p - n^x".
What does "conjectured" mean? A positive integer k is a candidate if:
1) gcd(k, n) = 1,
2) gcd(k^2-1, n-1) = 1,
3) every term in the sequence k + n^x is divisible by one of the prime numbers of a covering set,
4) all numbers of the form k - n^x are composite, k > n^x + 1, x >= 0.
The main problem is to prove that the given terms are indeed correct.
A quick search showed that a(8) = 47, a(14) = 11, a(20) = 13, a(27) = 512, a(29) = 26, a(32) = 23, a(34) = 99.
This is an interesting sequence: it leads to new classes of numbers. For example, the integer 30666137 is probably the smallest number that is simultaneously a Polignac number and a Sierpinski number.

Cf. A076336, A123159, A263644, A283622.

cp's OEIS Frontend

A263822 Conjectured number of terms in A263644 less than 10^n.

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A263645 Primes that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

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A276495 Odd numbers not of the form p + 2^m with p prime and m >= 0 for which the smallest k in A067760 such that n + 2^k is prime increases.

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A283619 a(n) = (conjectured) smallest positive integer k which is neither of the form p + n^x nor of the form p - n^x with x >= 0 and p prime, where gcd(k, n) = 1 and gcd(k^2-1, n-1) = 1.

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