A237043 -id:A237043 - cp's OEIS frontend

A282631 The smallest square referenced in A237043 (Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.).

9, 25, 49, 121, 289, 961, 529, 1194649, 1849, 5329, 841, 16129, 7921, 2209

Offset: 1

Robert Price, Feb 19 2017

			A237043(1)=6, 2^6-1=63, 63=3*3*7, 63 is not squarefree and the square is 3*3=9.

Cf. A237043, A282632.

A282632 Square root of the smallest square referenced in A237043 (Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.).

Original entry on oeis.org

3, 5, 7, 11, 17, 31, 23, 1093, 43, 73, 29, 127, 89, 47

Offset: 1

Robert Price, Feb 19 2017

nonn

			A237043(1)=6, 2^6-1=63, 63=3*3*7, 63 is not squarefree and the square is 3*3=9, the square root being 3.

Cf. A237043, A282631.

A049094 Numbers m such that 2^m - 1 is divisible by a square > 1.

Original entry on oeis.org

6, 12, 18, 20, 21, 24, 30, 36, 40, 42, 48, 54, 60, 63, 66, 72, 78, 80, 84, 90, 96, 100, 102, 105, 108, 110, 114, 120, 126, 132, 136, 138, 140, 144, 147, 150, 155, 156, 160, 162, 168, 174, 180, 186, 189, 192, 198, 200, 204, 210, 216, 220, 222, 228, 231, 234, 240

Offset: 1

Labos Elemer

nonn

Conjecture: 2^n-1 is squarefree iff gcd(n,2^n-1)=1. If true, the conjecture would imply that Mersenne numbers (A001348) are squarefree. - Vladeta Jovovic, Apr 12 2002. But the conjecture is not true: counterexamples are n = 364 and n = 1755, i.e., gcd(364,2^364-1) = 1 and (2^364-1) mod 1093^2 = 0 and gcd(1755,2^1755-1) = 1 and (2^1755-1) mod 3511^2 = 0, cf. A001220. - Vladeta Jovovic, Nov 01 2005. The conjecture is true with assumption that n is not a multiple of A002326((q-1)/2), where q is a Wieferich prime A001220. - Thomas Ordowski, Nov 17 2015
If d|n and 2^d-1 is not squarefree, then 2^n-1 cannot be squarefree. For example, if 6 is in the sequence then 6*d is also. - Enrique Pérez Herrero, Feb 28 2009
If p(p-1)|n then p^2|2^n-1, where p is an odd prime. - Thomas Ordowski, Jan 22 2014
The primitive elements of this sequence are A237043. - Charles R Greathouse IV, Feb 05 2014
Dilcher & Ericksen prove that this sequence is exactly the set of numbers divisible by either t(p)p for a Wieferich prime p>2 or t(p) for a non-Wieferich prime p, where t(p) is the order of 2 modulo p (see Proposition 3.1). - Kellen Myers, Jun 09 2015
If d^2 divides 2^n-1 then d divides n, where n is not a multiple of 364, 1755, ...; i.e., A002326((q-1)/2) for Wieferich primes q, A001220. - Thomas Ordowski, Nov 15 2015
(1, 2^n-1, 2^n) is an abc triple iff 2^n-1 is not squarefree. - William Hu, Jul 04 2024

			a(2)=12 because 2^12 - 1 = 4095 = 5*(3^2)*7*13 is divisible by a square.

R. K. Guy, Unsolved Problems in Number Theory, A3.

Max Alekseyev, Table of n, a(n) for n = 1..296
Karl Dilcher and Larry Ericksen, The Polynomials of Mahler and Roots of Unity, The American Mathematical Monthly, Vol. 122, No. 04 (April 2015), pp. 338-353.
Enrique Pérez Herrero, Mersenne Numbers Treasure Map, Psych Geom blogspot, 02/17/09
Andy Steward, Factorizations of Generalized Repunits [Dead link]

Cf. A001220, A001348, A002326, A014491, A049093, A049096, A049095, A237043, A282242, A282243.

[n: n in [1..250] | not IsSquarefree(2^n-1)]; // Vincenzo Librandi, Jul 14 2015

N:= 250:
B:= Vector(N):
for n from 1 to N do
  if B[n] <> 1 then
    F:= ifactors(2^n-1,easy)[2];
    if max(seq(t[2],t=F)) > 1 or (hastype(F,symbol)
            and not numtheory:-issqrfree(2^n-1)) then
       B[[seq(n*k,k=1..floor(N/n))]]:= 1;
    fi
  fi;
od:
select(t -> B[t]=1, [$1..N]); # Robert Israel, Nov 17 2015

Select[Range[240], !SquareFreeQ[2^#-1]&] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)

default(factor_add_primes,1);
is(n)=my(f=factor(n>>valuation(n,2))[,1],N,o); for(i=1,#f,if(n%(f[i]-1) == 0, return(1))); N=2^n-1; fordiv(n,d,f=factor(2^d-1)[,1]; for(i=1,#f, if(d==n,return(!issquarefree(N))); o=valuation(N,f[i]); if(o>1, return(1)); N/=f[i]^o)) \\ Charles R Greathouse IV, Feb 02 2014

is(n)=!issquarefree(2^n-1) \\ Charles R Greathouse IV, Feb 04 2014

More terms from Vladeta Jovovic, Apr 12 2002

Definition corrected by Jonathan Sondow, Jun 29 2010

A086251 Number of primitive prime factors of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

A280302 Smallest k such that (n+1)^k - n^k is divisible by a square > 1.

Original entry on oeis.org

6, 10, 4, 2, 21, 20, 3, 20, 33, 6, 20, 2, 2, 5, 21, 6, 10, 6, 6, 4, 4, 2, 7, 2, 6, 3, 10, 4, 18, 6, 2, 10, 20, 6, 57, 17, 2, 14, 42, 2, 10, 10, 6, 39, 14, 4, 10, 20, 2, 21, 20, 6, 4, 21, 6, 20, 10, 2, 5, 2, 5, 2, 20, 6, 42, 14, 2, 6, 55, 6, 3, 7, 2, 42, 3, 2

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2016

nonn

a(209) > 70.

a(n) <= p^2 - p, where p = A053670(n). - Jinyuan Wang, May 15 2020

			a(1) = 6 is because (1+1)^6 - 1^6 = 63 is divisible by 9 = 3^2.

Lars Blomberg, Table of n, a(n) for n = 1..208

Cf. A049094, A053670, A237043, A280203, A280208, A280209, A280296, A282176, A282177, A280307.

Cf. A289629, A280547, A289985.

a(n) = {my(k = 1); while (issquarefree((n+1)^k - n^k), k++); k;} \\ Michel Marcus, Jan 14 2017

More terms from Lars Blomberg, Jan 10 2017

A280203 Numbers n such that 3^n - 2^n is not squarefree, but 3^d - 2^d is squarefree for all proper divisors d of n.

Original entry on oeis.org

10, 11, 42, 52, 57, 203, 272, 497

Offset: 1

Juri-Stepan Gerasimov and Charles R Greathouse IV, Dec 28 2016

Primitive members of A280149: members of A280149 which are not multiples of any earlier term.

547 <= a(9) <= 689. 689, 732, 776, 903, 1055, 1081, 1332, 2525, 2628 are terms. - Chai Wah Wu, Jul 20 2020

			10 is in this sequence because all 3^1 - 2^1 = 1, 3^2 - 2^2 = 5, 3^5 - 2^5 = 211 are squarefrees and 3^10 - 2^10 = 58025 = 5^2*2321 is not squarefree.

Cf. A001047, A280149.

Cf. Numbers n such that (k+1)^n - k^n is not squarefree, but (k+1)^d - k^d is squarefree for every proper divisor d of n: A237043 (k = 1), this sequence (k = 2), A280208 (k = 3), A280209 (k = 4).

Function[s, DeleteCases[#, 0] &@ MapIndexed[#1 Boole[Total@ Boole@ Map[Function[k, Divisible[#1, k]], Take[s, First@ #2 - 1]] == 0] &, s]]@ Select[Range@ 60, ! SquareFreeQ[3^# - 2^#] &]  (* Michael De Vlieger, Dec 30 2016 *)

is(n)=fordiv(n,d, if(!issquarefree(3^d-2^d), return(d==n))); 0 \\ Charles R Greathouse IV, Mar 01 2018

A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 10, 2, 1, 1, 1, 6, 3, 2, 1, 9, 2, 3, 3, 2, 2, 6, 1, 13, 9, 1, 1, 10, 5, 1, 3, 2, 8, 3, 2, 2, 1, 1, 10, 3, 8, 7, 9, 2, 2, 3, 1, 2, 26, 1, 3, 9, 4, 2, 9, 4, 1, 6, 1, 18, 9, 1, 7, 3, 2, 1, 3, 2, 5, 10, 1, 10, 6, 38, 3, 3, 4, 1, 41, 2

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

a(n) is the number of integers k in range 1 .. A016754(n-1)-1 such that A000225(k) is an integral multiple of A016754(n-1). - Antti Karttunen, Nov 15 2014
Conjecture: the positions of 1's, a(k)=1, are exactly given by the 2k-1 which are elements of A167791. - Antti Karttunen, Nov 15 2014
From Charlie Neder, Oct 18 2018: (Start)
It would appear that, if 2k-1 is in A167791, then so is (2k-1)^2, and so a(k) = 1 would follow by definition.
Conjecture: Let B be the first value such that (2k-1)^2 divides 2^B - 1. Then either 2k-1 divides B, or 2k-1 is a Wieferich prime (A001220). (End)

			a(2) = 1 because (2^6 - 1)/(2*2 - 1)^2 = 7 is an integer and 6 < 9.
a(3) = 1 because (2^20 - 1)/(2*3 - 1)^2 = 41943 is an integer and 20 < 25.
a(3) = 2 because (2^21 - 1)/(2*4 - 1)^2 = 42799 is an integer and 21 < 49; and also (2^42 - 1)/(2*4 - 1)^2 = 89756051247 is an integer and 42 < 49.

Kevin P. Thompson, Table of n, a(n) for n = 1..1560 (terms 1..128 from Antti Karttunen)

A246703 gives the positions of records.

Cf. A000225, A002326, A016754, A049094, A237043.

A246702 := proc(n)
    local a,klim,k ;
    a := 0 ;
    klim := (2*n-1)^2 ;
    for k from 1 to klim-1 do
        if modp(2^k-1,klim) = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A246702(n),n=1..80) ; # R. J. Mathar, Nov 15 2014

A246702[n_] := Module[{a, klim, k}, a = 0; klim = (2*n-1)^2; For[k = 1, k <= klim-1, k++, If[Mod[2^k-1, klim] == 0, a = a+1]]; a];
Table[A246702[n], {n, 1, 84}] (* Jean-François Alcover, Oct 04 2017, translated from R. J. Mathar's Maple code *)

a(n)=my(t=(2*n-1)^2,m=Mod(1,t)); sum(k=1,t-1,m*=2;m==1) \\ Charles R Greathouse IV, Nov 16 2014

a246702(n) = my(m=(2*n-1)^2); (m-1)\znorder(Mod(2,m)); \\ Max Alekseyev, Oct 11 2023

(define (A246702 n) (let ((u (A016754 (- n 1)))) (let loop ((k (- u 1)) (s 0)) (cond ((zero? k) s) ((zero? (modulo (A000225 k) u)) (loop (- k 1) (+ s 1))) (else (loop (- k 1) s)))))) ;; Antti Karttunen, Nov 15 2014

a(n) = floor( 4*n*(n-1) / A002326(2*n*(n-1)) ). - Max Alekseyev, Oct 11 2023

Corrected by R. J. Mathar, Nov 15 2014

A280208 Numbers m such that 4^m - 3^m is not squarefree, but 4^d - 3^d is squarefree for every proper divisor d of m.

Original entry on oeis.org

4, 14, 55, 78, 111, 253, 342, 355

Offset: 1

Juri-Stepan Gerasimov, Dec 28 2016

Where numbers m such that 4^m - 3^m is not squarefree: numbers of the form i*a(j) for i >= 1.
The smallest squares of 4^m - 3^m as defined above are 25, 49, 121, 169, 1369, 529, 361, 5041. - Robert Price, Mar 07 2017
431 <= a(9) <= 1081. 1081, 3403 are terms. - Chai Wah Wu, Jul 20 2020

			4 is in this sequence because all 4^1 - 3^1 = 1, 4^2 - 3^2 = 7 are squarefrees where 1, 2 are proper divisors of 4 and 4^4 - 3^4 = 175 = 7*5^2 is not squarefree;
14 is in this sequence because all 4^1 = 3^2 = 1, 4^2 - 3^2 = 7, 4^7 - 3^7 = 14197 are squarefrees where 1, 2, 7 are proper divisors of 14 and 4^14 - 3^14 = 263652487 = 7^2*3591*14197 is not squarefree.

Cf. A005061.

Cf. Numbers m such that (k+1)^m - k^m is not squarefree, but (k+1)^d - k^d is squarefree for every proper divisor d of m: A237043 (k = 1), A280203 (k = 2), this sequence (k = 3), A280209 (k = 4), A280307 (k = 6).

Function[s, DeleteCases[#, 0] &@ MapIndexed[#1 Boole[! AnyTrue[Take[s, First@ #2 - 1], Function[k, Divisible[#1, k]]]] &, s]]@ Select[Range@ 80, ! SquareFreeQ[4^# - 3^#] &] (* Michael De Vlieger, Dec 30 2016 *)

a(6)-a(8) from Jinyuan Wang, May 15 2020

A280209 Numbers m such that 5^m - 4^m is not squarefree, but 5^d - 4^d is squarefree for every proper divisor d of m.

Original entry on oeis.org

2, 55, 171, 183, 203

Offset: 1

Juri-Stepan Gerasimov, Dec 28 2016

Where numbers m such that 5^m - 4^m is not squarefree: numbers of the form i*a(j) for i >= 1.
Numbers m such that (k+1)^m - k^m is not squarefree, but (k+1)^d - k^d is squarefree for every proper divisor d of m:
A237043 (k = 1): 6, 20, 21, 110, 136, 155, 253, 364, 602, 657, 812, 889, 979, 1081, ... a(15) >= 1207 - Max Alekseyev, Sep 28 2015;
A280203 (k = 2): 10, 11, 42, 52, 57, 203, 272, 497, ... a(9) > 497 - Charles R Greathouse IV, Dec 27 2016;
A280208 (k = 3): 4, 14, 55, 78, 111, 253, 342, 355, ... a(9) >= 431;
this sequence (k = 4): 2, 55, 171, 183, 203, ... a(6) >= 367;
A... (k = 5): 21, 22, 39, 136, 186, 203, 244, 333, ... a(9) >= 337;
A280307 (k = 6): 20, 26, 55, 68, 171, 258, 310, ... a(8) >= 323;
A... (k = 7): 3, 10, 55, 272, ... a(5) >= 289;
A... (k = 8): 20, 21, 22, 34, 93, 116, 138, 156, 166, 205, 253, ... a(12) >= 277;
A... (k = 9): 33, 38, 42, 78, 110, 155, ... a(7) >= 263;
A... (k = 10): 6, 14, 52, 68, 253, ... a(6) >= 263;
A... (k = 11): 20, 42, 46, 53, 114, 136, 156, 205, ... a(9) >= 251.
The smallest square of 5^m - 4^m as defined above are 9, 121, 361, 3721, 841. - Robert Price, Mar 07 2017
a(6) <= 465. 465, 955, 1027 are terms. - Chai Wah Wu, Jul 20 2020

			2 is in this sequence because 5^1 - 4^1 = 1 is squarefree where 1 is proper divisor of 2 and 5^2 - 4^2 = 9 = 3^2 is not squarefree.

Cf. A005060, A237043, A280203, A280208.

a(3)-a(5) from Jinyuan Wang, May 15 2020

A280296 Squarefree numbers k such that 2^k - 1 is divisible by a square > 1.

Original entry on oeis.org

6, 21, 30, 42, 66, 78, 102, 105, 110, 114, 138, 155, 174, 186, 210, 222, 231, 246, 253, 258, 273, 282, 310, 318, 330, 354, 357, 366, 390, 399, 402, 426, 438, 462, 465, 474, 483, 498, 506, 510, 534, 546, 570, 582, 602, 606, 609, 618, 642, 651, 654, 678, 690, 714, 759, 762, 770, 777, 786, 798, 822

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2016

nonn

Intersection of A049094 and A005117. - Michel Marcus, Dec 31 2016

			6 is in this sequence because 2^6 - 1 = 63 is divisible by 9 = 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..89

Cf. A005117, A049094, A237043, A280203, A280208, A280209.

[n: n in [1..200] | IsSquarefree(n) and not IsSquarefree(2^n-1)];

a(38)=498 inserted by Amiram Eldar, Oct 23 2019

cp's OEIS Frontend

A282631 The smallest square referenced in A237043 (Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.).

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A282632 Square root of the smallest square referenced in A237043 (Numbers n such that 2^n - 1 is not squarefree, but 2^d - 1 is squarefree for every proper divisor d of n.).

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A049094 Numbers m such that 2^m - 1 is divisible by a square > 1.

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A086251 Number of primitive prime factors of 2^n - 1.

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A280302 Smallest k such that (n+1)^k - n^k is divisible by a square > 1.

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A280203 Numbers n such that 3^n - 2^n is not squarefree, but 3^d - 2^d is squarefree for all proper divisors d of n.

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A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

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