A076289 -id:A076289 - 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!

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

A076287 Numbers of the form (8^{mr}-1)/(8^r-1) for positive integers m, r.

Original entry on oeis.org

1, 9, 65, 73, 513, 585, 4097, 4161, 4681, 32769, 37449, 262145, 262657, 266305, 299593, 2097153, 2396745, 16777217, 16781313, 17043521, 19173961, 134217729, 134480385, 153391689, 1073741825, 1073774593, 1090785345, 1227133513, 8589934593, 9817068105

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

Numbers which in base 8 contain only 0's and 1's, with equally spaced 1's. - Robert Israel, Jan 15 2019

Robert Israel, Table of n, a(n) for n = 1..7932
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), A076288 (k=9), A076289 (k=10).

1, seq(seq((8^(k+r)-1)/(8^r-1), r = ListTools:-Reverse(sort(convert(numtheory:-divisors(k),list)))), k=1..20); # Robert Israel, Jan 15 2019

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).

A265181 Prime numbers resulting from the concatenation of at least two copies of a cubic number followed by a trailing "1.".

Original entry on oeis.org

881, 27271, 7297291, 133113311, 337533751, 19683196831, 42875428751, 68921689211, 1038231038231, 1574641574641, 2053792053791, 2744274427441, 4218754218751, 6859685968591, 7290007290001, 7297297297291, 106120810612081, 224809122480911, 274400027440001, 280322128032211, 317652331765231, 500021150002111, 812060181206011, 1251251251251251, 1757617576175761, 1968319683196831, 5931959319593191

Offset: 1

Thomas S. Pedigo, Dec 03 2015

Subsequence of A030430 (primes of the form 10n+1). - Michel Marcus, Dec 04 2015
If m is a term then (m-1)/10 is divisible by a cube (A000578) and the resulting quotient, different from 1, is in A076289. - Michel Marcus, Dec 05 2015
Without the "repeated at least twice" constraint, A168147 would be a subsequence. - Michel Marcus, Dec 05 2015

			8 = 2^3; 881 is prime.
27 = 3^3; 27271 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A030430, A066592, A076289, A168147 , A232066.

N:= 20: # to get all terms with at most N digits
M:= floor((N-1)/2):
res:= {}:
for s from 1 to floor(10^(M/3)) do
   x:= s^3;
   m:= 1+ilog10(x);
   for k from 2 to floor((N-1)/m) do
     p:= x*add(10^(1+m*i),i=0..k-1)+1;
     if isprime(p) then res:= res union {p} fi;
   od
od:
sort(convert(res,list)); # Robert Israel, Jan 13 2016

Take[Sort@ Flatten[Select[#, PrimeQ] & /@ Table[FromDigits@ Append[Flatten@ IntegerDigits@ Table[n^3, {#}], 1] & /@ Range[2, 20], {n, 1, 300}] /. {} -> Nothing], 27] (* Michael De Vlieger, Jan 05 2016 *)

from itertools import count, islice
from sympy import isprime
def A265181_gen(): # generator of terms
    return filter(isprime,(int(str(k**3)*2)*10+1 for k in count(1)))
A265181_list = list(islice(A265181_gen(),20)) # Chai Wah Wu, Feb 20 2023

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

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A076287 Numbers of the form (8^{mr}-1)/(8^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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A265181 Prime numbers resulting from the concatenation of at least two copies of a cubic number followed by a trailing "1.".

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