A263925 -id:A263925 - cp's OEIS frontend

A264050 a(n) = least m > 1 such that m + 2^n is prime.

3, 3, 3, 3, 5, 3, 3, 7, 9, 7, 5, 3, 17, 27, 3, 3, 29, 3, 21, 7, 17, 15, 9, 43, 35, 15, 29, 3, 11, 3, 11, 15, 17, 25, 53, 31, 9, 7, 23, 15, 27, 15, 29, 7, 59, 15, 5, 21, 69, 55, 21, 21, 5, 159, 3, 81, 9, 69, 131, 33, 15, 135, 29, 13, 131, 9, 3, 33, 29, 25, 11, 15, 29, 37, 33

Offset: 1

Alexei Kourbatov, Nov 02 2015

nonn

The definition is similar to Fortunate numbers (A005235) but uses 2^n instead of primorial A002110(n).
Terms a(n) are often but not always prime; sometimes they are prime powers or semiprimes or have a more general form.
An analog of Fortune's conjecture for this sequence would be "a(n) is either a prime power or a semiprime." But even this relaxed conjecture is disproved by, e.g., a(62)=135, a(93)=a(97)=105, a(99)=255.
By definition, a(n) >= A013597(n). The integers n such that a(n) > A013597(n) are those with A013597(n)=1, i.e., 1, 2, 4, 8, 16, and then? - Michel Marcus, Nov 06 2015

			a(56)=81 because m=81 is the least m > 1 such that m + 2^56 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000

Cf. A005235, A013597, A092131, A263925.

Table[m = 2; While[! PrimeQ[m + 2^n], m++]; m, {n, 75}] (* Michael De Vlieger, Nov 06 2015 *)

a(n)=my(m=2);while(!isprime(m+2^n),m++);m \\ Anders Hellström, Nov 02 2015

a(n)=nextprime(2^n+2)-2^n \\ Charles R Greathouse IV, Nov 02 2015

a(60) corrected by Charles R Greathouse IV, Nov 02 2015

A268607 a(n) is the least m > 1 such that 2^n - m is prime.

Original entry on oeis.org

2, 3, 3, 3, 3, 15, 5, 3, 3, 9, 3, 13, 3, 19, 15, 9, 5, 19, 3, 9, 3, 15, 3, 39, 5, 39, 57, 3, 35, 19, 5, 9, 41, 31, 5, 25, 45, 7, 87, 21, 11, 57, 17, 55, 21, 115, 59, 81, 27, 129, 47, 111, 33, 55, 5, 13, 27, 55, 93, 31, 57, 25, 59, 49, 5, 19, 23, 19, 35, 231, 93

Offset: 2

Alexei Kourbatov, Feb 08 2016

nonn

a(1) is not defined (there are no primes less than 2).

The definition is similar to Lesser Fortunate numbers (A055211) but uses 2^n instead of primorials A002110(n).

			a(7)=15 because m=15 is the least m > 1 such that 2^7 - m is prime.

Paolo Xausa, Table of n, a(n) for n = 2..1000

Cf. A005235, A013603, A055211, A264050, A263925, A268608.

Map[# - NextPrime[#-1, -1] &, 2^Range[2, 100]] (* Paolo Xausa, Mar 10 2025 *)

PARI
```
a(n)=2^n-precprime(2^n-2)
```

a(n) = A013603(n), if A013603(n) > 1. - Jason Yuen, Mar 10 2025

A268608 a(n) is the least m > 1 such that (prime(n)#)^n - m is prime.

Original entry on oeis.org

5, 7, 19, 23, 41, 163, 67, 257, 83, 109, 43, 359, 293, 647, 277, 1567, 983, 419, 1723, 83, 103, 3089, 719, 733, 1723, 457, 331, 2729, 3389, 1123, 863, 1123, 1871, 6211, 19717, 5323, 5749, 419, 887, 811, 617, 2851, 2531, 5023, 6883, 6661, 2879, 16433, 19471

Offset: 2

Alexei Kourbatov, Feb 08 2016

nonn

Here prime(n)# denotes the primorial A002110(n), i.e., the product of the first n primes. a(1) is not defined (there are no primes less than 2).
The definition is similar to Lesser Fortunate numbers (A055211) - but here primorials A002110(n) are raised to the n-th power.
Similar to Fortunate numbers (A005235) and Lesser Fortunate numbers (A055211), the first fifty terms are all prime. (Cf. A263925 where the 6th term is composite.)

			a(2)=5 because m=5 is the least m > 1 such that A002110(2)^2 - m is prime.

Cf. A002110, A005235, A055211, A264050, A263925, A268607.

a(n)=my(s=prod(i=1, n, prime(i))^n); s-precprime(s-2)

cp's OEIS Frontend

A264050 a(n) = least m > 1 such that m + 2^n is prime.

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A268607 a(n) is the least m > 1 such that 2^n - m is prime.

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A268608 a(n) is the least m > 1 such that (prime(n)#)^n - m is prime.

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