A245730 -id:A245730 - cp's OEIS frontend

A064896 Numbers of the form (2^(m*r)-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

Marc LeBrun, Oct 11 2001

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

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

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.

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.

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

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

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

A297625 Primes of the form (2^(p^k) - 1)/(2^(p^(k - 1)) - 1), with p prime and k > 1.

Original entry on oeis.org

5, 17, 73, 257, 65537, 262657, 4432676798593

Offset: 1

Arkadiusz Wesolowski, Jan 04 2018

nonn

Primes of the form Phi(x, 2), where x is a proper prime power and Phi is the cyclotomic polynomial.
Together with 3, supersequence of A019434.
Also called Mersenne-Fermat primes.
a(8) has 1031 digits and is too large to include.

Fredrick Kennard, Unsolved Problems in Mathematics, Lulu Press, 2015, p. 160.

Wikipedia, Mersenne-Fermat primes

Cf. A019434, A051156, A140797, A187823, A245730, A246547.

lst:=[]; r:=7; pr:=PrimesUpTo(r); for k in [2..r] do for c in [1..#pr] do p:=pr[c]; if p^k le r^2 then MF:=Truncate((2^(p^k)-1)/(2^(p^(k-1))-1)); if IsPrime(MF) then Append(~lst, MF); end if; end if; end for; end for; Sort(lst);

cp's OEIS Frontend

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

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