A215658 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.
2, 3, 5, 7, 17
Offset: 1
Examples
The smallest square > 17# = 510510 is 715^2 = 17# + 715, so 17 is a member.
Links
- C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009); see p. 5.
Programs
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Mathematica
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 *) -
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
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Python
from sympy import primorial, integer_nthroot, prime 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
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