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Numbers k such that A073837(k) is a multiple of k.
2

%I #79 Jan 05 2023 10:16:46

%S 1,4,6,8,10,12,17,20,31,34,52,85,92,555,1723,2870,2904,3943,19325,

%T 41708,145474,225476,240632,666862,8911645,10249751,138543006,

%U 209659550,265831784,540388470,949428097,2813155218,12323589092,407224380494,1704233306223,3361207818001

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

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

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

%C 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

%C 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

%H <a href="/A341280/b341280.txt">Table of n, a(n) for n = 1..36</a>

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

%p R:= 1: S:= [2,3]: s:= 5: q:= 5: count:= 1:

%p for n from 3 while count < 24 do

%p if n = S[1]+1 then S:= S[2..-1]; s:= s-n+1 fi;

%p if q <= 2*n then S:= [op(S), q]; s:= s+q; q:= nextprime(q) fi;

%p if s mod n = 0 then count:= count+1; R:= R, n fi;

%p od:

%p R;

%o (Python)

%o from sympy import isprime

%o k, k2, d, A341280_list = 1, 3, 2, []

%o while k < 10**10:

%o if d % k == 0:

%o A341280_list.append(k)

%o if isprime(k):

%o d -= k

%o if isprime(k2):

%o d += k2

%o k += 1

%o k2 += 2 # _Chai Wah Wu_, Feb 16 2021

%Y Cf. A073837.

%K nonn

%O 1,2

%A _J. M. Bergot_ and _Robert Israel_, Feb 16 2021

%E a(26)-a(31) from _Chai Wah Wu_, Feb 16 2021

%E a(32) from _Chai Wah Wu_, Feb 17 2021

%E a(33)-a(36) from _Martin Ehrenstein_, Feb 21 2021