A280203 -id:A280203 - cp's OEIS frontend

A282874 The smallest square referenced in A280203 (Numbers n such that 3^n - 2^n is not squarefree not divisible by any smaller number of the same form).

25, 529, 49, 169, 361, 841, 289, 5041

Offset: 1

Robert Price, Feb 23 2017

nonn

			A280203(1)=10, 3^10-2^10=58025, 58025=5*5*11*211, so 58025 is not squarefree the square being 5*5=25.

Cf. A280149, A280203, A282875.

A282875 Square root of the smallest square referenced in A280203 (Numbers n such that 3^n - 2^n is not squarefree not divisible by any smaller number of the same form).

Original entry on oeis.org

5, 23, 7, 13, 19, 29, 17, 71

Offset: 1

Robert Price, Feb 23 2017

nonn

			A280203(1)=10, 3^10-2^10=58025, 58025=5*5*11*211, so 58025 is not squarefree the square being 5*5=25, the square root being 5.

Cf. A280149, A280203, A282874.

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

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

A280307 Numbers m such that 7^m - 6^m is not squarefree, but 7^d - 6^d is squarefree for every proper divisor d of m.

Original entry on oeis.org

20, 26, 55, 68, 171, 258, 310

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2016

Numbers m such that 7^m - 6^m is not squarefree not divisible by any smaller number of the same form.
7^m - 6^m is nonsquarefree if and only if m is divisible by a term of this sequence. - Jon E. Schoenfield, Jan 01 2017
The smallest squares of 7^m - 6^m as defined above are 25, 169, 121, 289, 361, 1849, 961. - Robert Price, Mar 07 2017
a(8) >= 323. - Jinyuan Wang, May 15 2020
a(8) <= 381. 381, 406, 506, 610, 689, 979, 1027, 1081, 1332 are terms. - Chai Wah Wu, Jul 20 2020

			20 is in this sequence because 7^20 - 6^20 = 43242508113549025 is not squarefree but 7^d - 6^d is squarefree for every proper divisor d of 20 (i.e., for d = 1, 2, 4, 5, and 10): 7^1 - 6^1 = 1, 7^2 - 6^2 = 13, 7^4 - 6^4 = 1105, 7^5 - 6^5 = 13682, 7^10 - 6^10 = 222009013 are all squarefree.

Cf. A016169, A237043, A280203, A280208, A280209, A280302.

a(5)-a(7) from Jinyuan Wang, May 15 2020

cp's OEIS Frontend

A282874 The smallest square referenced in A280203 (Numbers n such that 3^n - 2^n is not squarefree not divisible by any smaller number of the same form).

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A282875 Square root of the smallest square referenced in A280203 (Numbers n such that 3^n - 2^n is not squarefree not divisible by any smaller number of the same form).

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

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A280307 Numbers m such that 7^m - 6^m is not squarefree, but 7^d - 6^d is squarefree for every proper divisor d of m.

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