Ryan Bresler - cp's OEIS frontend

A352002 a(n) = prime(n)# + prime(n), where prime(n)# is the n-th primorial number A002110(n).

4, 9, 35, 217, 2321, 30043, 510527, 9699709, 223092893, 6469693259, 200560490161, 7420738134847, 304250263527251, 13082761331670073, 614889782588491457, 32589158477190044783, 1922760350154212639129, 117288381359406970983331, 7858321551080267055879157, 557940830126698960967415461

Offset: 1

Ryan Bresler, Feb 27 2022

nonn

			a(3) = prime(3)# + prime(3) = 2 * 3 * 5 + 5 = 35.

Cf. A000040, A002110.

a(n) = prime(n) + prod(k=1, n, prime(k)); \\ Michel Marcus, Feb 28 2022

from sympy import primorial, prime
def a(n): return primorial(n) + prime(n)
for n in range(1,50):
    print(a(n), end=", ") # Javier Rivera Romeu, Mar 01 2022

a(n) = A002110(n) + A000040(n). Also a(n) = prime(n) * (prime(n-1)# + 1).

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

A350460 Positive integers k such that if p is the next prime > k then p - k is prime.

Original entry on oeis.org

3, 5, 8, 9, 11, 14, 15, 17, 20, 21, 24, 26, 27, 29, 32, 34, 35, 38, 39, 41, 44, 45, 48, 50, 51, 54, 56, 57, 59, 62, 64, 65, 68, 69, 71, 74, 76, 77, 80, 81, 84, 86, 87, 90, 92, 94, 95, 98, 99, 101, 104, 105, 107, 110, 111, 114, 116, 120, 122, 124, 125, 128, 129

Offset: 1

Ryan Bresler, Jan 01 2022

nonn

a(n) is only prime when n is the lesser of a twin prime pair (A001359). All other terms are composite.

			3 is a term because the next prime > 3 is 5, and 5 - 3 = 2, which is prime.
14 is a term because the next prime > 14 is 17, and 17 - 14 = 3, which is prime.

Cf. A038711, A001359, A068780.

Select[Range[130], PrimeQ[NextPrime[#] - #] &] (* Amiram Eldar, Jan 01 2022 *)

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

from sympy import isprime, nextprime
def ok(n): return n > 0 and isprime(nextprime(n) - n)
print([k for k in range(130) 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

A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

Original entry on oeis.org

11, 14, 41, 49, 94, 111, 119, 122, 128, 133, 155, 166, 182, 188, 191, 199, 212, 218, 221, 229, 236, 263, 281, 289, 292, 298, 313, 326, 331, 362, 368, 386, 449, 494, 515, 551, 559, 595, 616, 623, 632, 638, 661, 683, 779, 797, 812, 818, 821, 829, 836, 863, 881

Offset: 1

Ryan Bresler, May 29 2021

If k is in the sequence then all anagrams of k are in the sequence. - David A. Corneth, May 29 2021

Trivially, this sequence has infinite elements. A031974 is an infinite sequence that is found in this sequence - Ryan Bresler, May 30 2021

			11 is a term because its digit sum is 2 (prime) and its digit product is 1 (perfect square > 0).

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A028834 and A050626.
Subsequence of A052382.
A031974 is a subsequence of this sequence.

q:= n-> (l-> not 0 in l and isprime(add(i, i=l)) and
         issqr(mul(i, i=l)))(convert(n, base, 10)):
select(q, [$0..999])[];  # Alois P. Heinz, May 29 2021

from math import prod
from sympy import isprime, integer_nthroot
def ok(n):
  d = list(map(int, str(n)))
  return 0 not in d and isprime(sum(d)) and integer_nthroot(prod(d), 2)[1]
print(list(filter(ok, range(1000)))) # Michael S. Branicky, May 29 2021

A341269 Number of non-extendable self-avoiding walks in an n X n grid starting at the top left corner.

Original entry on oeis.org

1, 2, 20, 548, 40440, 8442742, 5088482972, 8963926817126, 46591697187961736

Offset: 1

Ryan Bresler, Feb 07 2021

A self-avoiding walk is non-extendable if it ends on a square which has all its neighbors already visited. Not all paths are Hamiltonian. See examples.

All paths that start by moving one square to the right are symmetrical with all paths that start by moving one square down. This symmetry results in a(n) divisible by 2 for n > 1.

			Example of a self-avoiding walk on a 3 X 3 grid that visits every node (Hamiltonian path):
.
  1---2---3
          |
  6---5---4
  |
  7---8---9
.
Two examples of a self-avoiding walk on a 3 X 3 grid that do not visit every node:
.
  1---2   7
      |   |
  X   3   6
      |   |
  X   4---5
.
or
.
  1   8---7
  |       |
  2---3   6
      |   |
  X   4---5
.

Cf. A145157 (Hamiltonian case).

a(8)-a(9) from Andrew Howroyd, Feb 08 2021

a(6) corrected by Henry Bottomley, Oct 07 2021

cp's OEIS Frontend

User: Ryan Bresler

A352002 a(n) = prime(n)# + prime(n), where prime(n)# is the n-th primorial number A002110(n).

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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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A350460 Positive integers k such that if p is the next prime > k then p - k 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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Maple

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A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

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Maple

Python

A341269 Number of non-extendable self-avoiding walks in an n X n grid starting at the top left corner.

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