A308243 Index of E-irregularity of prime(n).

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 3, 1, 0, 1, 2, 2, 1, 1, 0, 1, 0, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1

Offset: 1

Author

Felix Fröhlich, May 16 2019

Keywords

• nonn

Comments

A prime p >= 5 is an E-irregular prime if there is an even integer 2*k such that 2 <= 2*k <= p-3 and p divides E(2*k), where E(i) is the i-th Euler number (A000364). The pair (p, 2*k) is called an E-irregular pair. The number of such pairs for a given p is called the index of E-irregularity of p (cf. Ernvall, Metsänkylä, 1978, p. 618).

In other words, a prime p is E-irregular if its index of E-irregularity is > 0, which is the case if p is a term of A092218. Otherwise, p is E-regular and is a term of A092217.

Links

• R. Ernvall and T. Metsänkylä, Cyclotomic invariants and E-irregular primes, Mathematics of Computation 32 (1978), 617-629.

Crossrefs

Cf. A000364, A092217, A092218, A120337.

Programs

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
a(n) = my(p=prime(n), e=2, i=0); while(e <= p-3, if(a000364(e)%p==0, i++); e=e+2); i