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

%I A341280 #79 Jan 05 2023 10:16:46
%S A341280 1,4,6,8,10,12,17,20,31,34,52,85,92,555,1723,2870,2904,3943,19325,
%T A341280 41708,145474,225476,240632,666862,8911645,10249751,138543006,
%U A341280 209659550,265831784,540388470,949428097,2813155218,12323589092,407224380494,1704233306223,3361207818001
%N A341280 Numbers k such that A073837(k) is a multiple of k.
%C A341280 Numbers k such that the sum of primes from k to 2*k is divisible by k.
%C A341280 Primes in the sequence include 17, 31, 1723, 3943.
%C A341280 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 A341280 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 A341280 <a href="/A341280/b341280.txt">Table of n, a(n) for n = 1..36</a>
%e A341280 a(3) = 6 is a term because A073837(6) = 7+11 = 18 is divisible by 6.
%p A341280 R:= 1: S:= [2,3]: s:= 5: q:= 5: count:= 1:
%p A341280 for n from 3 while count < 24 do
%p A341280   if n = S[1]+1 then S:= S[2..-1]; s:= s-n+1 fi;
%p A341280 if q <= 2*n then S:= [op(S), q]; s:= s+q; q:= nextprime(q) fi;
%p A341280 if s mod n = 0 then count:= count+1; R:= R, n fi;
%p A341280 od:
%p A341280 R;
%o A341280 (Python)
%o A341280 from sympy import isprime
%o A341280 k, k2, d, A341280_list = 1, 3, 2, []
%o A341280 while k < 10**10:
%o A341280     if d % k == 0:
%o A341280         A341280_list.append(k)
%o A341280     if isprime(k):
%o A341280         d -= k
%o A341280     if isprime(k2):
%o A341280         d += k2
%o A341280     k += 1
%o A341280     k2 += 2 # _Chai Wah Wu_, Feb 16 2021
%Y A341280 Cf. A073837.
%K A341280 nonn
%O A341280 1,2
%A A341280 _J. M. Bergot_ and _Robert Israel_, Feb 16 2021
%E A341280 a(26)-a(31) from _Chai Wah Wu_, Feb 16 2021
%E A341280 a(32) from _Chai Wah Wu_, Feb 17 2021
%E A341280 a(33)-a(36) from _Martin Ehrenstein_, Feb 21 2021