A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 10, 2, 1, 1, 1, 6, 3, 2, 1, 9, 2, 3, 3, 2, 2, 6, 1, 13, 9, 1, 1, 10, 5, 1, 3, 2, 8, 3, 2, 2, 1, 1, 10, 3, 8, 7, 9, 2, 2, 3, 1, 2, 26, 1, 3, 9, 4, 2, 9, 4, 1, 6, 1, 18, 9, 1, 7, 3, 2, 1, 3, 2, 5, 10, 1, 10, 6, 38, 3, 3, 4, 1, 41, 2

Offset

1,4

Comments

a(n) is the number of integers k in range 1 .. A016754(n-1)-1 such that A000225(k) is an integral multiple of A016754(n-1). - Antti Karttunen, Nov 15 2014

Conjecture: the positions of 1's, a(k)=1, are exactly given by the 2k-1 which are elements of A167791. - Antti Karttunen, Nov 15 2014

From Charlie Neder, Oct 18 2018: (Start)

It would appear that, if 2k-1 is in A167791, then so is (2k-1)^2, and so a(k) = 1 would follow by definition.

Conjecture: Let B be the first value such that (2k-1)^2 divides 2^B - 1. Then either 2k-1 divides B, or 2k-1 is a Wieferich prime (A001220). (End)

Links

Kevin P. Thompson, Table of n, a(n) for n = 1..1560 (terms 1..128 from Antti Karttunen)

Formula

a(n) = floor( 4*n*(n-1) / A002326(2*n*(n-1)) ). - Max Alekseyev, Oct 11 2023

Example

a(2) = 1 because (2^6 - 1)/(2*2 - 1)^2 = 7 is integer and 6 < 9.
a(3) = 1 because (2^20 - 1)/(2*3 - 1)^2 = 41943 is integer and 20 < 25.
a(3) = 2 because (2^21 - 1)/(2*4 - 1)^2 = 42799 is integer and 21 < 49; and also (2^42 - 1)/(2*4 - 1)^2 = 89756051247 is integer and 42 < 49.

Maple

A246702 := proc(n)
    local a, klim, k ;
    a := 0 ;
    klim := (2*n-1)^2 ;
    for k from 1 to klim-1 do
        if modp(2^k-1, klim) = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A246702(n), n=1..80) ; # R. J. Mathar, Nov 15 2014

Mathematica

A246702[n_] := Module[{a, klim, k}, a = 0; klim = (2*n-1)^2; For[k = 1, k <= klim-1, k++, If[Mod[2^k-1, klim] == 0, a = a+1]]; a];
Table[A246702[n], {n, 1, 84}] (* Jean-François Alcover, Oct 04 2017, translated from R. J. Mathar's Maple code *)

Prog

(Scheme) (define (A246702 n) (let ((u (A016754 (- n 1)))) (let loop ((k (- u 1)) (s 0)) (cond ((zero? k) s) ((zero? (modulo (A000225 k) u)) (loop (- k 1) (+ s 1))) (else (loop (- k 1) s)))))) ;; Antti Karttunen, Nov 15 2014
(PARI) a(n)=my(t=(2*n-1)^2, m=Mod(1, t)); sum(k=1, t-1, m*=2; m==1) \\ Charles R Greathouse IV, Nov 16 2014
(PARI) a246702(n) = my(m=(2*n-1)^2); (m-1)\znorder(Mod(2, m)); \\ Max Alekseyev, Oct 11 2023

Crossrefs

A246703 gives the positions of records.

Cf. A000225, A002326, A016754, A049094, A237043.

Sequence in context: A262747 A016442 A076360 * A089398 A345272 A373381

Adjacent sequences: A246699 A246700 A246701 * A246703 A246704 A246705

Keyword

nonn

Author

Juri-Stepan Gerasimov, Nov 15 2014

Extensions

Corrected by R. J. Mathar, Nov 15 2014

Status

approved