A049096 -id:A049096 - cp's OEIS frontend

A069226 a(n) = gcd(n, 2^n + 1).

2, 1, 1, 3, 1, 1, 1, 1, 1, 9, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 27, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 25, 3, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 17, 3, 5, 1, 1, 1, 1, 3, 1, 1, 13, 1, 1, 81, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1

Offset: 0

Vladeta Jovovic, Apr 12 2002

nonn

First occurrence of n: a(1)=1, a(3)=3, a(10)=5, a(9)=9, a(55)=11, a(78)=13, a(68)=17, a(50)=25, a(27)=27, a(406)=29, a(165)=33, a(666)=37, a(301)=43, a(1378)=53, a(1711)=59, a(390)=65, a(81)=81, a(3403)=83, a(2328)=97, a(495)=99, ... - R. J. Mathar, Dec 14 2016

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A006521 (fixed points), A014491, A049095, A049096, A060444.

Table[GCD[n,2^n+1],{n,100}] (* Harvey P. Dale, Dec 12 2012 *)

A069226(n) = gcd(n, 1+(1<Antti Karttunen, Jan 15 2025

Term a(0) = 2 prepended by Antti Karttunen, Jan 15 2025

A272361 Numbers n such that (2^n + 1) / gcd(n, 2^n + 1) is not squarefree.

Original entry on oeis.org

182, 546, 910, 1274, 1638, 2002, 2366, 2730, 3094, 3458, 3822, 4186, 4550, 4914, 5278, 5642, 6006, 6370, 6734, 7098, 7462, 7826, 8190, 8554, 8918, 9282, 9646, 10010, 10374, 10738, 11102, 11466, 11830, 12194, 12558, 12922, 13286, 13650, 14014, 14378, 14742, 15106, 15470, 15834

Offset: 1

Thomas Ordowski, Apr 27 2016

nonn

Odd multiples of integer A002326((q-1)/2)/2, where q is a Wieferich prime A001220.

Cf. A049096, A272359.

isok(n) = my(m=2^n+1); !issquarefree(m/gcd(n, m)); \\ Michel Marcus, Apr 27 2016

A334213 Numbers m such that m^k + 1 is squarefree for all 0 <= k <= m.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 16, 30, 36, 46, 256

Offset: 1

Gionata Neri, Apr 18 2020

m = 2^i is a term iff k*i is not in A049096 with 0 < k < m + 1. Up to i = 128, there are no more terms of the form 2^i. a(12) > 10^7, if it exists. - Jinyuan Wang, May 01 2020

			4^0 + 1 = 2 is squarefree, 4^1 + 1 = 5 is squarefree, 4^2 + 1 = 17 is squarefree, 4^3 + 1 = 5*13 is squarefree and 4^4 + 1 = 257 is squarefree, so 4 is in the sequence.

Cf. A049096, A334212, A334214.

Do[L=Length[a];a=Select[a=m^Range[0,m-1]+1,SquareFreeQ[#]&];If[L==m-1,Print[m-1]],{m,0,1000}] (* Metin Sariyar, Apr 21 2020 *)

isOK(m) = k=0; until(k>m, if(!issquarefree(m^k+1), return(0)); k++); 1;
for(m=0, 99, if(isOK(m), print1(m, ", ")))

a(11) from Jinyuan Wang, May 01 2020

A349989 a(n) is the smallest k such that k^n + (k+1)^n is divisible by a square > 1.

Original entry on oeis.org

4, 3, 1, 113, 2, 3, 3, 19, 1, 1, 4, 113, 4, 3, 1, 765, 4, 3, 4, 87, 1, 3, 4, 19, 2, 2, 1, 28, 4, 1, 4, 151, 1, 3, 2, 113, 4, 3, 1, 19, 4, 3, 4, 113, 1, 3, 4, 335, 3, 1, 1, 113, 4, 3, 1, 19, 1, 3, 4, 87, 4, 3, 1, 379, 2, 3, 4, 1, 1, 1, 4, 19, 4, 3, 1, 113, 3, 1, 4

Offset: 1

Jon E. Schoenfield, Dec 07 2021

nonn

a(64) <= 379; a(76) <= 113. Terms a(65)..a(79): 2, 3, 4, 1, 1, 1, 4, 19, 4, 3, 1, a(76), 3, 1, 4.

At k=4, k^n + (k+1)^n = 4^n + 5^n is a multiple of 9 for all odd n, and at k=3, k^n + (k+1)^n = 3^n + 4^n is a multiple of 25 for all n == 2 (mod 4). Thus, a(n) <= 4 if n is not a multiple of 4.

Kevin P. Thompson, Factorizations to support a(n) for n = 1..79

Cf. A280302, A280547, A289629, A289985, A349988.

a(n) = my(k=1); while(issquarefree(k^n + (k+1)^n), k++); k; \\ Michel Marcus, Dec 09 2021

from sympy.ntheory.factor_ import core
def squarefree(n): return core(n, 2) == n
def a(n):
    k = 1
    while squarefree(k**n + (k+1)**n): k += 1
    return k
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Dec 09 2021

a(n) = A289629(n) if n is even.
a(k) = 1 for k in A049096.
a(n) <= 4 if 4 does not divide n; among terms where 4 divides n, certain terms appear repeatedly. E.g.,
  a(n) <= 113 for n ==  4 (mod 8):   for all such n,  17^2 divides 113^n + 114^n;
  a(n) <=  19 for n ==  8 (mod 16):  for all such n,  17^2 divides  19^n +  20^n;
  a(n) <= 765 for n == 16 (mod 32):  for all such n,  97^2 divides 765^n + 766^n;
  a(n) <=  87 for n == 20 (mod 40):  for all such n,  41^2 divides  87^n +  88^n;
  a(n) <=  28 for n == 68 (mod 136): for all such n,  17^2 divides  28^n +  29^n;
  a(n) <= 151 for n == 32 (mod 64):  for all such n, 257^2 divides 151^n + 152^n;
  a(n) <= 335 for n == 48 (mod 96):  for all such n, 769^2 divides 335^n + 336^n.

a(64)-a(79) from Kevin P. Thompson, Feb 23 2022

A187965 Numbers k such that 2^k - 1 and 2^k + 1 are not squarefree.

Original entry on oeis.org

21, 30, 63, 78, 90, 105, 110, 147, 150, 189, 204, 210, 231, 234, 270, 273, 310, 315, 330, 340, 357, 390, 399, 441, 450, 465, 483, 510, 525, 546, 550, 567, 570, 609, 612, 630, 651, 657, 666, 690, 693, 702, 735, 750, 759, 770, 777, 810, 819, 858, 861, 870, 903, 930, 945, 987, 990, 1014, 1020, 1029, 1050, 1071

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 18 2011

nonn

If k is in the sequence, then so is m*k for any odd m. - Thomas Ordowski, Nov 23 2015
Note that 110, 310, 340, 550, 770 are not divisible by 3.
Let b(p) be the multiplicative order of 2 modulo p^2. Then k is in this sequence if and only if there exists odd primes p, q such that b(p) | k and k == b(q)/2 (mod b(q)) with even b(q). For example, we have b(7) = 21, b(3) = 6 so b(7) | 21, 21 == b(3)/2 (mod b(3)), hence 21 is a term; likewise, b(3) = 6, b(5) = 20, so b(3) | 30, 30 == b(5)/2 (mod b(5)), hence 30 is a term. - Jianing Song, Jan 20 2021

			2^21 - 1 = 7^2 * 127 * 337, 2^21 + 1 = 3^2 * 43 * 5419.

Cf. A005117, A049094, A049096.

Cf. A243905 (multiplicative orders of 2 modulo p^2), A242777 (k+1 is prime).

[n: n in [1..250] | not IsSquarefree(2^n-1) and not IsSquarefree(2^n+1)]; // Vincenzo Librandi, Nov 23 2015

Select[ Range@ 500, !(SquareFreeQ[2^# - 1] || SquareFreeQ[2^# + 1]) &]
Select[Range[1100],NoneTrue[2^#+{1,-1},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2019 *)

is(n) = !issquarefree(2^n-1) && !issquarefree(2^n+1);
for(n=1, 1e3, if(is(n), print1(n, ", "))) \\ Altug Alkan, Nov 22 2015

More terms from Joerg Arndt, Nov 23 2015

A172522 Partial sums of A049094.

Original entry on oeis.org

6, 18, 36, 56, 77, 101, 131, 167, 207, 249, 297, 351, 411, 474, 540, 612, 690, 770, 854, 944, 1040, 1140, 1242, 1347, 1455, 1565, 1679, 1799, 1925, 2057, 2193, 2331, 2471, 2615, 2762, 2912, 3067, 3223, 3383, 3545, 3713, 3887, 4067, 4253, 4442, 4634, 4832

Offset: 1

Jonathan Vos Post, Feb 06 2010

nonn

The subsequence of primes in this sequence begins: 101, 131, 167, 3067.

			a(20) = 6 + 12 + 18 + 20 + 21 + 24 + 30 + 36 + 40 + 42 + 48 + 54 + 60 + 63 + 66 + 72 + 78 + 80 + 84 + 90.

Cf. A000040, A014491, A049093, A049096, A049095, A001220.

a(n) = SUM[i=1..n] {i such that 2^i - 1 is divisible by a square}.

More terms from R. J. Mathar, Feb 16 2010

cp's OEIS Frontend

A069226 a(n) = gcd(n, 2^n + 1).

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A272361 Numbers n such that (2^n + 1) / gcd(n, 2^n + 1) is not squarefree.

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A334213 Numbers m such that m^k + 1 is squarefree for all 0 <= k <= m.

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A349989 a(n) is the smallest k such that k^n + (k+1)^n is divisible by a square > 1.

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A187965 Numbers k such that 2^k - 1 and 2^k + 1 are not squarefree.

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Magma

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A172522 Partial sums of A049094.

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