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%I #7 May 26 2019 15:00:08
%S 2,5,307,277
%N Smallest prime with E-irregularity index n.
%C Smallest prime p such that A308243(i) = n, where i is the index of p in A000040.
%C E-regular primes have E-irregularity index 0, so a(0) = 2, since 2 is the smallest E-regular prime (A092217).
%C Does such a prime exist for every n?
%C a(4) > 2003 if it exists.
%H R. Ernvall and T. Metsänkylä, <a href="https://doi.org/10.1090/S0025-5718-1978-0482273-9">Cyclotomic invariants and E-irregular primes</a>, Mathematics of Computation 32 (1978), 617-629.
%o (PARI) a000364(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1) \\ after _Charles R Greathouse IV_ in A000364
%o a308243(n) = my(p=prime(n), e=2, i=0); while(e <= p-3, if(a000364(e)%p==0, i++); e=e+2); i
%o a(n) = for(x=1, oo, if(a308243(x)==n, return(prime(x))))
%Y Cf. A000040, A061576, A092217, A308243.
%K nonn,hard,more
%O 0,1
%A _Felix Fröhlich_, May 17 2019