A280149 -id:A280149 - cp's OEIS frontend

A282688 Square root of the smallest square referenced in A280149 (Numbers n such that 3^n - 2^n is not squarefree).

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

Offset: 1

Robert Price, Feb 20 2017

nonn

			A280149(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.

Robert Price, Table of n, a(n) for n = 1..127

Cf. A280149, A282689.

A282689 The smallest square referenced in A280149 (Numbers n such that 3^n - 2^n is not squarefree).

Original entry on oeis.org

25, 529, 25, 529, 25, 529, 25, 49, 529, 25, 169, 529, 361, 25, 529, 25, 529, 25, 49, 529, 25, 529, 25, 169, 25, 361, 25, 529, 49, 25, 529, 25, 529, 25, 529, 169, 25, 529, 49, 25, 361, 529, 25, 529, 25, 529, 25, 841, 169, 529, 25, 25, 361, 25, 529, 25, 529

Offset: 1

Robert Price, Feb 20 2017

nonn

			A280149(1)=10, 3^10-2^10=58025, 58025=5*5*11*211, so 58025 is not squarefree the square being 5*5=25.

Robert Price, Table of n, a(n) for n = 1..127

Cf. A280149, A282688.

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

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).

Original entry on oeis.org

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.

A280170 Primes p such that both 2^(p-1) - 1 and 2^(p+1) - 1 are not squarefree.

Original entry on oeis.org

19, 41, 79, 101, 109, 137, 139, 199, 271, 281, 311, 379, 401, 439, 461, 499, 521, 601, 619, 641, 701, 727, 739, 761, 769, 811, 821, 859, 881, 919, 941, 953, 1013, 1039, 1061, 1087, 1181, 1279, 1301, 1361, 1399, 1429, 1459, 1481, 1549, 1579, 1601, 1699, 1721, 1759, 1777, 1871, 1879, 1901

Offset: 1

Juri-Stepan Gerasimov, Dec 27 2016

nonn

			19 is in this sequence because 2^(19-1) - 1 = 262143 = 3^3*7*19*73 and 2^(19+1) - 1 = 1048575 = 3*5^2*11*31*41.

Cf. A013929, A049094, A280149.

[p: p in PrimesUpTo(200) | not IsSquarefree(2^(p-1)-1) and not
IsSquarefree(2^(p+1)-1)];

Select[Prime[Range[200]], ! SquareFreeQ[ 2^(#-1) - 1 ] && ! SquareFreeQ[ 2^(#+1) - 1 ] &] (* Robert Price, Feb 26 2017 *)
Select[Prime[Range[300]],NoneTrue[{2^(#-1)-1,2^(#+1)-1},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2020 *)

is(n)=isprime(n) && !issquarefree(2^(n-1)-1) && !issquarefree(2^(n+1)-1) \\ Charles R Greathouse IV, Aug 26 2017

Inserted terms 727 and 739 by Robert Price, Feb 26 2017

Added terms a(38)-a(54) by Robert Price, Feb 26 2017

cp's OEIS Frontend

A282688 Square root of the smallest square referenced in A280149 (Numbers n such that 3^n - 2^n is not squarefree).

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A282689 The smallest square referenced in A280149 (Numbers n such that 3^n - 2^n is not squarefree).

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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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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).

Views

Author

Keywords

Examples

Crossrefs

A280170 Primes p such that both 2^(p-1) - 1 and 2^(p+1) - 1 are not squarefree.

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Author

Keywords

Examples

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Programs

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Mathematica

PARI

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