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

1,2

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

Links

Table of n, a(n) for n = 1..36

Example

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

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;

Prog

(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

Crossrefs

Cf. A073837.

Sequence in context: A194579 A194594 A163248 * A034288 A131984 A359330

Adjacent sequences: A341277 A341278 A341279 * A341281 A341282 A341283

Keyword

nonn

Author

J. M. Bergot and Robert Israel, 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

Status

approved