A064896 Numbers of the form (2^{mr}-1)/(2^r-1) for positive integers m, r.

1, 3, 5, 7, 9, 15, 17, 21, 31, 33, 63, 65, 73, 85, 127, 129, 255, 257, 273, 341, 511, 513, 585, 1023, 1025, 1057, 1365, 2047, 2049, 4095, 4097, 4161, 4369, 4681, 5461, 8191, 8193, 16383, 16385, 16513, 21845, 32767, 32769, 33825, 37449, 65535, 65537

Offset

1,2

Comments

Binary expansion of n consists of single 1's diluted by (possibly empty) equal-sized blocks of 0's.

A064894(a(n)) = A056538(n)

According to Stolarsky's Theorem 2.1, all numbers in this sequence are sturdy numbers; this sequence is a subsequence of A125121. - T. D. Noe, Jul 21 2008

Links

T. D. Noe, Table of n, a(n) for n=1..1000

T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arithmetica, 29 (1976), 227-250.

K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica, 38 (1980), 117-128.

Example

73 is included because it is 1001001 in binary, whose 1's are diluted by blocks of two 0's.

Maple

f := proc(p) local m, r, t1; t1 := {}; for m from 1 to 10 do for r from 1 to 10 do t1 := {op(t1), (p^(m*r)-1)/(p^r-1)}; od: od: sort(convert(t1, list)); end; f(2); # very crude!
# Alternative:
N:= 10^6: # to get all terms <= N
A:= sort(convert({1, seq(seq((2^(m*r)-1)/(2^r-1), m=2..1/r*ilog2(N*(2^r-1)+1)), r=1..ilog2(N-1))}, list)); # Robert Israel, Jun 12 2015

Prog

(PARI) lista(nn) = {v = [1]; x = (2^nn-1); for (m=2, nn, r = 1; while ((y = (2^(m*r)-1)/(2^r-1)) <=x, v = Set(concat(v, y)); r++); ); v; } \\ Michel Marcus, Jun 12 2015

Crossrefs

Cf. A064894, A056538.

Cf. A076270 (k=3), A076275 (k=4), A076284 (k=5), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).

Primes in this sequence: A245730.

Sequence in context: A163410 A329419 A235264 * A076188 A265852 A335132

Adjacent sequences: A064893 A064894 A064895 * A064897 A064898 A064899

Keyword

base,easy,nonn

Author

Marc LeBrun, Oct 11 2001

Status

approved