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Least prime p such that Sum_{q prime <= p} q is divisible by the first n primes.
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%I #21 Aug 23 2018 02:13:25

%S 2,269,269,3823,8539,729551,1416329,23592593,1478674861,20458458289,

%T 7558026467353,201008815538749

%N Least prime p such that Sum_{q prime <= p} q is divisible by the first n primes.

%C a(1)-a(9) are taken from De Koninck's book.

%C The sequence of indices of these primes is 1, 57, 57, 531, 1065, 58751, 108243, 1483151, 73716417, 901526695, 264119914199, 6301058125383.

%D Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 66.

%e 2 + 3 + ... + 269 = 2 * 3 * 1145

%e 2 + 3 + ... + 269 = 2 * 3 * 5 * 229

%e 2 + 3 + ... + 3823 = 2 * 3 * 5 * 7 * 4473

%e 2 + 3 + ... + 8539 = 2 * 3 * ... * 11 * 1826

%e 2 + 3 + ... + 729551 = 2 * 3 * ... * 13 * 682263

%e 2 + 3 + ... + 1416329 = 2 * 3 * ... * 17 * 143884

%e 2 + 3 + ... + 23592593 = 2 * 3 * ... * 19 * 1742804

%e 2 + 3 + ... + 1478674861 = 2 * 3 * ... * 23 * 237859969

%e 2 + 3 + ... + 20458458289 = 2 * 3 * ... * 29 * 1392427664

%e 2 + 3 + ... + 7558026467353 = 2 * 3 * ... * 31 * 4886311486119

%e 2 + 3 + ... + 201008815538749 = 2 * 3 * ... * 37 * 83956482342243

%t c=0; pr=2; p=2; s=2; q=2; While[c<6, While[!Divisible[s, pr], q = NextPrime[q]; s+=q]; Print[ q]; c++; p = NextPrime[p]; pr*=p]

%o (PARI) my(c=0, pr=2, p=2, s=2, q=2); while(c<6, while(s%pr!=0, q = nextprime(q+1); s+=q); print1(q,", "); c++; p = nextprime(p+1); pr*=p)

%Y Cf. A051838.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Aug 20 2018

%E a(11) from _Giovanni Resta_, Aug 20 2018

%E a(12) from _Giovanni Resta_, Aug 22 2018