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%I #36 Feb 16 2024 14:45:17
%S 1,3,5,7,9,15,17,21,31,33,63,65,73,85,127,129,255,257,273,341,511,513,
%T 585,1023,1025,1057,1365,2047,2049,4095,4097,4161,4369,4681,5461,8191,
%U 8193,16383,16385,16513,21845,32767,32769,33825,37449,65535,65537
%N Numbers of the form (2^(m*r)-1)/(2^r-1) for positive integers m, r.
%C Binary expansion of n consists of single 1's diluted by (possibly empty) equal-sized blocks of 0's.
%C 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
%C These are the numbers k > 0 for which k + 2^m = k*2^n + 1 has a solution m,n > 0. For k > 1, these are numbers k such that (k - 2^x)*2^y + 1 = k has a solution in positive integers x,y. In other words, (k - 1)/(k - 2^x) = 2^y for some x,y > 0. If t = (2^m - 1)/(2^n - 1) is a term of this sequence (i.e. if and only if n|m), then t' = t + 2^m = t*2^n + 1 is also a term. Primes in this sequence (A245730) include: all Mersenne primes (A000668), all Fermat primes (A019434), and other primes (73, 262657, 4432676798593, ...). - _Thomas Ordowski_, Feb 14 2024
%H T. D. Noe, <a href="/A064896/b064896.txt">Table of n, a(n) for n=1..1000</a>
%H T. Chinburg and M. Henriksen, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa29/aa2932.pdf">Sums of k-th powers in the ring of polynomials with integer coefficients</a>, Acta Arithmetica, 29 (1976), 227-250.
%H K. B. Stolarsky, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3825.pdf">Integers whose multiples have anomalous digital frequencies</a>, Acta Arithmetica, 38 (1980), 117-128.
%F A064894(a(n)) = A056538(n).
%e 73 is included because it is 1001001 in binary, whose 1's are diluted by blocks of two 0's.
%p 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!
%p # Alternative:
%p N:= 10^6: # to get all terms <= N
%p 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
%o (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
%Y Cf. A064894, A056538.
%Y 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).
%Y Primes in this sequence: A245730.
%K base,easy,nonn
%O 1,2
%A _Marc LeBrun_, Oct 11 2001
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