A354154 a(1) = 0; for n>1, a(n) = prime(n-1) - A090252(n).

0, 0, 0, 0, 3, 4, 4, 6, 6, 6, 21, 12, 14, 16, 22, 18, 22, 22, 20, 24, 24, 20, 63, 24, 28, 30, 30, 30, 52, 30, 86, 78, 48, 48, 42, 48, 48, 50, 54, 54, 46, 48, 44, 52, 44, 46, 173, 54, 60, 60, 56, 54, 58, 50, 58, 60, 64, 58, 186, 156, 58, 56, 236, 78, 150, 80, 78, 90, 86, 90, 86, 84, 88, 90, 92, 96, 90, 82, 86, 88, 92, 88, 84, 84, 84, 86, 84, 82, 84

Offset: 1

Author

N. J. A. Sloane, May 28 2022

Keywords

• nonn

Comments

Theorem: a(n) >= 0 for all n.

Examples

			A090252 begins
  1, 2, 3, 5, 4,  7,  9, 11, 13, ...
and we subtract these numbers from
  1, 2, 3, 5, 7, 11, 13, 17, 19, ...
to get
  0, 0, 0, 0, 3,  4,  4,  6,  6, ...
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A090252.

Programs

Python

from math import gcd, prod
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink, p = [1], {1}, 2, 1
    yield 0
    for n in count(2):
        k, s, p = mink, n - n//2, nextprime(p)
        prodall = prod(alst[n-n//2-1:n-1])
        while k in aset or gcd(prodall, k) != 1: k += 1
        alst.append(k); aset.add(k); yield p - k
        while mink in aset: mink += 1
print(list(islice(agen(), 89))) # Michael S. Branicky, May 28 2022