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%I #37 May 12 2023 01:35:26
%S 2,3,5,7,17
%N Primes p such that the smallest positive integer k for which p# + k is square satisfies p# + k = k^2, where p# = 2*3*5*7*11*...*p is a primorial.
%C The corresponding values of k are 2, 3, 6, 15, 715 = A215659.
%C The equation p# + k = k^2 has an integer solution k if and only if 1 + 4*p# is a square.
%C Conjecture: Not the same sequence as A192579, which is finite.
%C When p is in this sequence, p# = k(k-1) is in A161620, the intersection of A002110 and A002378. - _Jeppe Stig Nielsen_, Mar 27 2018
%H C. Aebi and G. Cairns, <a href="http://www.parabola.unsw.edu.au/vol45_no1/vol45_no1_1.pdf">Partitions of primes</a>, Parabola 45, Issue 1 (2009); see p. 5.
%F A145781(n) = A216144(n) if and only if prime(n) is a member.
%F a(n)# = A215659(n)*(A215659(n)-1).
%e The smallest square > 17# = 510510 is 715^2 = 17# + 715, so 17 is a member.
%t t = {}; pm = 1; Do[pm = pm*p; s = Floor[Sqrt[pm]]; If[pm == s*(s+1), AppendTo[t, p]], {p, Prime[Range[100]]}]; t (* _T. D. Noe_, Sep 05 2012 *)
%o (PARI) for (n=1, 10, if (ceil(sqrt(prod(i=1, n, prime(i))))^2 - prod(i=1, n, prime(i)) - ceil(sqrt(prod(i=1, n, prime(i)))) == 0, print(prime(n)));); \\ _Michel Marcus_, Sep 05 2012
%o (Python)
%o from sympy import primorial, integer_nthroot, prime
%o A215658_list = [prime(i) for i in range(1,10**2) if integer_nthroot(4*primorial(i)+1,2)[1]] # _Chai Wah Wu_, Apr 01 2021
%Y Cf. A002110, A060797, A145781, A216144, A215659, A161620.
%K nonn,more
%O 1,1
%A _Jonathan Sondow_, Sep 02 2012
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