A360297 a(n) = minimal positive k such that the sum of the primes prime(n) + prime(n+1) + ... + prime(n+k) is divisible by prime(n+k+1), or -1 if no such k exists.

1, 3, 7, 11, 26, 20, 27, 52, 1650, 142, 53, 168234, 212, 7, 13

Offset

1,2

Comments

In the first 100 terms there are twenty values for which a(n) is currently unknown; for all of these values a(n) is at least 10^9. These unknown terms are for n = 16, 22, 24, 34, 41, 42, 45, 48, 50, 54, 55, 62, 68, 70, 72, 75, 80, 87, 88, 98. In this same range the largest known value is a(76) = 749597506, where prime(76) = 383 leads to a sum of primes of 6173644601523754801 that is divisible by 16865554301.

See A360311 for the sum of the k+1 primes. See A360312 for prime(n+k+1).

Links

Table of n, a(n) for n=1..15.

Scott R. Shannon, Terms for n = 1..100. The terms currently unknown are set to -1.

Example

a(1) = 1 as prime(1) + prime(2) = 2 + 3 = 5, which is divisible by prime(3) = 5.
a(4) = 11 as prime(4) + ... + prime(15) = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 = 318, which is divisible by prime(16) = 53.

Prog

(Python)
from sympy import prime, nextprime
def A360297(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p+q, 1
    while s%(q:=nextprime(q)):
        k += 1
        s += q
    return k # Chai Wah Wu, Feb 03 2023

Crossrefs

Cf. A360311, A360312, A000040, A007504, A332542, A332580.

Sequence in context: A099902 A336897 A316962 * A092284 A024459 A001645

Adjacent sequences: A360294 A360295 A360296 * A360298 A360299 A360300

Keyword

nonn,more

Author

Scott R. Shannon, Feb 02 2023

Status

approved