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A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer. 6
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 (list; graph; refs; listen; history; text; internal format)
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

Antti Karttunen, Table of n, a(n) for n = 1..128

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,

(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-Fran├žois 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

CROSSREFS

A246703 gives the positions of records.

Cf. A000225, A016754, A049094, A237043.

Sequence in context: A262747 A016442 A076360 * A089398 A331183 A284082

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

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Last modified January 27 21:45 EST 2020. Contains 331297 sequences. (Running on oeis4.)