A280307 -id:A280307 - cp's OEIS frontend

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

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

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

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

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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.

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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.

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