A076287 -id:A076287 - 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).

A076270 Numbers of the form (3^{mr}-1)/(3^r-1) for positive integers m, r.

Original entry on oeis.org

1, 4, 10, 13, 28, 40, 82, 91, 121, 244, 364, 730, 757, 820, 1093, 2188, 3280, 6562, 6643, 7381, 9841, 19684, 20440, 29524, 59050, 59293, 66430, 88573, 177148, 265720, 531442, 532171, 538084, 551881, 597871, 797161, 1594324, 2391484, 4782970, 4785157, 5380840

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

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

Cf. A064896 (k=2), A076270 (k=3), A076275 (k=4), A076284 (k=5), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).

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(3); # very crude!

A076289 Numbers of the form (10^(m*r)-1)/(10^r-1) for positive integers m, r.

Original entry on oeis.org

1, 11, 101, 111, 1001, 1111, 10001, 10101, 11111, 100001, 111111, 1000001, 1001001, 1010101, 1111111, 10000001, 11111111, 100000001, 100010001, 101010101, 111111111, 1000000001, 1001001001, 1111111111, 10000000001, 10000100001, 10101010101, 11111111111

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..7046
T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arithmetica, 29 (1976), 227-250.

Cf. A064896 (k=2), A076270 (k=3), A076275 (k=4), A076284 (k=5), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).

N::= 20: # for terms <= 10^N
R:= {1}:
for n from 1 to 2*N do
  R:= R union select(`<=`, {seq((10^n-1)/(10^d-1), d = numtheory:-divisors(n))},10^N);
od:
sort(convert(R,list)); # Robert Israel, May 19 2024

A076286 Numbers of the form (7^{mr}-1)/(7^r-1) for positive integers m, r.

Original entry on oeis.org

1, 8, 50, 57, 344, 400, 2402, 2451, 2801, 16808, 19608, 117650, 117993, 120100, 137257, 823544, 960800, 5764802, 5767203, 5884901, 6725601, 40353608, 40471600, 47079208, 282475250, 282492057, 288360150, 329554457, 1977326744, 2306881200, 13841287202

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..8539
T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arithmetica, 29 (1976), 227-250.

Cf. A064896 (k=2), A076270 (k=3), A076275 (k=4), A076284 (k=5), A076285 (k=6), A076287 (k=8), A076288 (k=9), A076289 (k=10).

[1,seq(seq((7^(d+r)-1)/(7^r-1),r = sort(convert(numtheory:-divisors(d),list),`>`)),d=1..20)]; # Robert Israel, Aug 27 2018

A076275 Numbers of the form (4^{mr}-1)/(4^r-1) for positive integers m, r.

Original entry on oeis.org

1, 5, 17, 21, 65, 85, 257, 273, 341, 1025, 1365, 4097, 4161, 4369, 5461, 16385, 21845, 65537, 65793, 69905, 87381, 262145, 266305, 349525, 1048577, 1049601, 1118481, 1398101, 4194305, 5592405, 16777217, 16781313, 16843009, 17043521, 17895697, 22369621

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

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

Cf. A064896 (k=2), A076270 (k=3), A076284 (k=5), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).

A076284 Numbers of the form (5^{mr}-1)/(5^r-1) for positive integers m, r.

Original entry on oeis.org

1, 6, 26, 31, 126, 156, 626, 651, 781, 3126, 3906, 15626, 15751, 16276, 19531, 78126, 97656, 390626, 391251, 406901, 488281, 1953126, 1968876, 2441406, 9765626, 9768751, 10172526, 12207031, 48828126, 61035156, 244140626, 244156251, 244531876, 246109501

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

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

Cf. A064896 (k=2), A076270 (k=3), A076275 (k=4), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076288 (k=9), A076289 (k=10).

A076288 Numbers of the form (9^{mr}-1)/(9^r-1) for positive integers m, r.

Original entry on oeis.org

1, 10, 82, 91, 730, 820, 6562, 6643, 7381, 59050, 66430, 531442, 532171, 538084, 597871, 4782970, 5380840, 43046722, 43053283, 43584805, 48427561, 387420490, 387952660, 435848050, 3486784402, 3486843451, 3530369206, 3922632451, 31381059610, 35303692060

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

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

Cf. A064896 (k=2), A076270 (k=3), A076275 (k=4), A076284 (k=5), A076285 (k=6), A076286 (k=7), A076287 (k=8), A076289 (k=10).

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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A076270 Numbers of the form (3^{mr}-1)/(3^r-1) for positive integers m, r.

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A076289 Numbers of the form (10^(m*r)-1)/(10^r-1) for positive integers m, r.

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A076286 Numbers of the form (7^{mr}-1)/(7^r-1) for positive integers m, r.

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A076275 Numbers of the form (4^{mr}-1)/(4^r-1) for positive integers m, r.

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A076284 Numbers of the form (5^{mr}-1)/(5^r-1) for positive integers m, r.

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A076288 Numbers of the form (9^{mr}-1)/(9^r-1) for positive integers m, r.

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