A175090 -id:A175090 - cp's OEIS frontend

A175089 Numbers m with result 0 under iterations of {r mod (max prime p <= r)} starting at r = m.

2, 3, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100

Offset: 1

Jaroslav Krizek, Jan 28 2010

nonn

Complement of A175088.
Union of A000040 (primes) and A175090. [Jaroslav Krizek, Feb 05 2010]
Numbers m such that A121559(m) = 0. - Michel Marcus, Aug 22 2014

			Iteration procedure for a(3) = 5: 5 mod 5 = 0.
Iteration procedure for a(5) = 9: 9 mod 7 = 2, 2 mod 2 = 0.

Cf. A007917 and A064722 (both for the iterations).

More terms from Michel Marcus, Aug 22 2014

A350496 Positive integers k such that if p is the largest prime less than k then k - p is prime.

Original entry on oeis.org

5, 7, 9, 10, 13, 15, 16, 19, 21, 22, 25, 26, 28, 31, 33, 34, 36, 39, 40, 43, 45, 46, 49, 50, 52, 55, 56, 58, 61, 63, 64, 66, 69, 70, 73, 75, 76, 78, 81, 82, 85, 86, 88, 91, 92, 94, 96, 99, 100, 103, 105, 106, 109, 111, 112, 115, 116, 118, 120, 124, 126, 129

Offset: 1

Ryan Bresler, Jan 01 2022

nonn

a(n) is prime only when n is the greater of a twin prime pair (A006512). All other terms are composite.

			9 is a term because the previous prime < 9 is 7, and 9 - 7 = 2, which is prime.

Cf. A175090, A006512.

Select[Range[2, 130], PrimeQ[# - NextPrime[#, -1]] &] (* Amiram Eldar, Jan 02 2022 *)

isok(k) = if (k>2, my(q=precprime(k-1)); isprime(k-q)); \\ Michel Marcus, Jan 02 2022

from sympy import isprime, prevprime
def ok(n): return n > 2 and isprime(n-prevprime(n))
print([k for k in range(249) if ok(k)]) # Michael S. Branicky, Jan 01 2022

A350472 Positive integers k such that if p is the next prime > k, and q is the previous prime < k, then p - k is prime and k - q is prime.

Original entry on oeis.org

5, 9, 15, 21, 26, 34, 39, 45, 50, 56, 64, 69, 76, 81, 86, 92, 94, 99, 105, 111, 116, 120, 124, 129, 134, 142, 144, 146, 154, 160, 165, 170, 176, 184, 186, 188, 195, 204, 206, 216, 218, 225, 231, 236, 244, 246, 248, 254, 260, 266, 274, 279, 286, 288, 290, 296

Offset: 1

Ryan Bresler, Jan 01 2022

nonn

a(n) can only be composite (excluding a(1) = 5).

			9 is a term because the next prime > 9 is 11 and the previous prime < 9 is 7, and 11 - 9 = 2 (which is prime) and 9 - 7 = 2 (which is also prime).

Intersection of A350496 and A350460.

Cf. A175090, A225461.

q:= n-> andmap(isprime, [nextprime(n)-n, n-prevprime(n)]):
select(q, [$3..400])[];  # Alois P. Heinz, Jan 01 2022

Select[Range[350], And @@ PrimeQ[{# - NextPrime[#, -1], NextPrime[#] - #}] &] (* Amiram Eldar, Jan 01 2022 *)

isok(k) = my(p=nextprime(k+1), q=precprime(k-1)); isprime(p-k) && isprime(k-q); \\ Michel Marcus, Jan 01 2022

from sympy import isprime, nextprime, prevprime
def ok(n):
    return n > 2 and isprime(nextprime(n) - n) and isprime(n - prevprime(n))
print([k for k in range(341) if ok(k)]) # Michael S. Branicky, Jan 01 2022

cp's OEIS Frontend

A175089 Numbers m with result 0 under iterations of {r mod (max prime p <= r)} starting at r = m.

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A350496 Positive integers k such that if p is the largest prime less than k then k - p is prime.

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A350472 Positive integers k such that if p is the next prime > k, and q is the previous prime < k, then p - k is prime and k - q is prime.

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Programs

Maple

Mathematica

PARI

Python