A341280 Numbers k such that A073837(k) is a multiple of k.

1, 4, 6, 8, 10, 12, 17, 20, 31, 34, 52, 85, 92, 555, 1723, 2870, 2904, 3943, 19325, 41708, 145474, 225476, 240632, 666862, 8911645, 10249751, 138543006, 209659550, 265831784, 540388470, 949428097, 2813155218, 12323589092, 407224380494, 1704233306223, 3361207818001

Offset: 1

Author

J. M. Bergot and Robert Israel, Feb 16 2021

Keywords

• nonn

Comments

Numbers k such that the sum of primes from k to 2*k is divisible by k.

Primes in the sequence include 17, 31, 1723, 3943.

Conjecture: For n > 1, a(n) is prime if and only if a(n) is odd and not a multiple of 5. - Chai Wah Wu, Feb 17 2021

The conjecture is false because a(35) = 1704233306223 is divisible by 3 and a(36) = 3361207818001 is divisible by 11. - Martin Ehrenstein, Feb 21 2021

Examples

			a(3) = 6 is a term because A073837(6) = 7+11 = 18 is divisible by 6.
		

Links

• Table of n, a(n) for n = 1..36

Crossrefs

Cf. A073837.

Programs

Maple

R:= 1: S:= [2,3]: s:= 5: q:= 5: count:= 1:
for n from 3 while count < 24 do
  if n = S[1]+1 then S:= S[2..-1]; s:= s-n+1 fi;
if q <= 2*n then S:= [op(S), q]; s:= s+q; q:= nextprime(q) fi;
if s mod n = 0 then count:= count+1; R:= R, n fi;
od:
R;

Python

from sympy import isprime
k, k2, d, A341280_list = 1, 3, 2, []
while k < 10**10:
    if d % k == 0:
        A341280_list.append(k)
    if isprime(k):
        d -= k
    if isprime(k2):
        d += k2
    k += 1
    k2 += 2 # Chai Wah Wu, Feb 16 2021

Extensions

a(26)-a(31) from Chai Wah Wu, Feb 16 2021

a(32) from Chai Wah Wu, Feb 17 2021

a(33)-a(36) from Martin Ehrenstein, Feb 21 2021