cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Juri-Stepan Gerasimov, Dec 31 2016

Keywords

Comments

a(209) > 70.
a(n) <= p^2 - p, where p = A053670(n). - Jinyuan Wang, May 15 2020

Examples

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

Links

Crossrefs

Cf. A049094, A053670, A237043, A280203, A280208, A280209, A280296, A282176, A282177, A280307.
Cf. A289629, A280547, A289985.

Programs

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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

Crossrefs

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).

Programs

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

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

Views

Author

Juri-Stepan Gerasimov, Dec 28 2016

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A005060, A237043, A280203, A280208.

Extensions

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

Views

Author

Juri-Stepan Gerasimov, Dec 31 2016

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Juri-Stepan Gerasimov, Dec 31 2016

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Extensions

a(5)-a(7) from Jinyuan Wang, May 15 2020
Showing 1-5 of 5 results.

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