A134670 First position k such that A046694(k) = A046694(k+1) =.. 0 are n consecutive zeros starting with A046694(k), where A046694 = Ramanujan tau numbers mod 691.

1381, 16581, 290217, 1409635, 1118176194, 107792931954, 673751534392, 2587409974788

Offset: 1

Author

Alexander Adamchuk, Nov 05 2007

Keywords

• hard
• nonn
• more

Comments

Most probably a(5) = 1118176194, because it is a starting point of a string of 5 zeros, but the fact that this is the least such number needs to be confirmed.

Note that zeros of A046694(n) have the indices equal to the terms of arithmetic progressions of the type k*p, where primes p belong to A134671. Thus: a(1) = 1381 = 2*691 - 1, a(2) = 16581 = 3*5527 = 3*(8*691 - 1), a(3) = 290217 = 3*96739 = 3*(140*691 - 1), a(4) = 1409635 = 5*281927 = 5*(408*691 - 1), a(5) = 1118176194 = 6*186362699 = 6*(269700*691 - 1).

Also, note that all listed terms have the form a(n) = k*p - 1, where prime p is a prime of the form p = 2m*691 - 1 that belong to A134671. a(1) = 2*691 - 1, a(2) = 2*8291 - 1, a(3) = 2*145109 - 1, a(4) = 4*352409 - 1, a(5) = 5*223635239 - 1.

Examples

			a(1) = 1381 because A046694(1381) = 0 is the first zero in A046694(n).
a(2) = 16581 because A046694(16581) = A046694(16582) = 0 are the first two consecutive zeros in A046694(n).
		

Links

• Eric Weisstein's World of Mathematics, Ramanujan's Tau Function

Crossrefs

Cf. A046694, A121733, A121734, A121742, A121743.

Programs

Maple

A134670 := proc(n)
    option remember;
    if n = 1 then
        1381 ;
    else
        for a from procname(n-1)+1 do
            wrks := true;
            for k from a to a+n-1 do
                if A046694(k) <> 0 then
                    wrks := false ;
                    break;
                end if;
            end do:
            if wrks then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Feb 01 2013

Extensions

a(5) confirmed by Jud McCranie Oct 17 2020

a(6) by Jud McCranie Oct 17 2020

a(7) by Jud McCranie Oct 19 2020

a(8) by Jud McCranie Oct 22 2020