cp's OEIS Frontend

A280547 Smallest number k such that (k+1)^n - k^n is divisible by a square greater than 1.

%I A280547 #56 Dec 07 2021 11:10:15
%S A280547 4,7,3,14,1,23,3,7,2,2,1,75,3,7,3,36,1,2476,1,1,2,165,1,14,4,7,3,149,
%T A280547 1,2972,3,2,4,14,1,977,4,5,1,34,1,135,2,7,4,136,1,23,2,7,2,11,1,2,3,2,
%U A280547 4
%N A280547 Smallest number k such that (k+1)^n - k^n is divisible by a square greater than 1.
%C A280547 a(31) > 2882.
%C A280547 a(31) <= 2972 (since (2972+1)^31 - 2972^31 is divisible by 1489^2). - _Jon E. Schoenfield_, Jan 20 2017
%C A280547 From _Jon E. Schoenfield_, Jan 22 2017: (Start)
%C A280547 Observation: let f(n,k) = (k+1)^n - k^n; then, for each n <= 100, there exists a value of k <= 3735 and a prime p <= 1489 such that p^2 | f(n,k), and the smallest value of k (for a given n) at which any prime-squared divisor p^2 of f(n,k) is found establishes an upper bound on a(n). For n <= 100, there is no prime p in [1490..10^6] whose square divides f(n,k) at a smaller value of k than the upper bound described above; values of that upper bound, for n = 31..100, are 2972, 3, 2, 4, 14, 1, 977, 4, 5, 1, 34, 1, 135, 2, 7, 4, 136, 1, 23, 2, 7, 2, 11, 1, 2, 3, 2, 4, 1155, 1, 3735, 4, 1, 3, 14, 1, 1068, 3, 7, 2, 715, 1, 415, 4, 7, 3, 2, 1, 533, 1, 7, 4, 509, 1, 14, 4, 7, 2, 23, 1, 23, 3, 7, 4, 14, 1, 1550, 3, 2, 1.
%C A280547 Conjecture: for n <= 100, there is no square greater than (10^6)^2 = 10^12 that divides (k+1)^n - k^n at any value of k lower than the upper bound described above; i.e., the upper bound described above is equal to a(n) for each n <= 100. (End)
%C A280547 Confirmed Jon E. Schoenfield's conjecture through a(58). - _Robert Price_, Feb 04 2017
%C A280547 If p is a prime that divides (k+1)^n - k^n for some k but does not divide n, then by Hensel's lemma there is some k for which p^2 divides (k+1)^n - k^n.  In particular, all terms exist. - _Robert Israel_, Feb 08 2017
%e A280547 a(2) = 4 because (4+1)^2 - 4^2 = 9 is a square.
%t A280547 A280547 = {};
%t A280547 For[n = 2, n < 11, n++,
%t A280547     k = 0;
%t A280547     While[SquareFreeQ[(k + 1)^n - k^n], k++];
%t A280547     AppendTo[A280547, k]];
%t A280547 A280547 (* _Robert Price_, Feb 04 2017 *)
%o A280547 (PARI) a(n) = {my(k = 1); while (issquarefree((k+1)^n - k^n), k++); k;} \\ _Michel Marcus_, Jan 14 2017
%Y A280547 Cf. A280302, A281996, A282174.
%Y A280547 Cf. A289629, A289985.
%K A280547 nonn,more
%O A280547 2,1
%A A280547 _Juri-Stepan Gerasimov_, Jan 06 2017
%E A280547 a(19)-a(30) from _Lars Blomberg_, Jan 14 2017
%E A280547 a(31)-a(58) from _Robert Price_, Feb 04 2017