A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).

3, 39, 225, 249, 321, 447, 471, 519, 681, 831, 921, 993, 1119, 1191, 1473, 1641, 1671, 1857, 1929, 1983, 2361, 2391, 2463, 2625, 2631, 2913, 3321, 3369, 3561, 3591, 3777, 3807, 3831, 3903, 4119, 4281, 4287, 4359, 4545, 4569, 4791, 5001, 5025, 5079, 5241, 5481

Offset: 1

Author

Jianing Song, Apr 01 2021

Keywords

• nonn
• easy

Comments

Numbers m such that m^2-1 is divisible by d(m^2-1) and m^2 is divisible by d(m^2), d = A000005.

Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. Such k must be of the form m^2-1 for some odd m.

The smallest term not divisible by 3 is a(66) = 9025.

For the first terms we have d(a(n)^2-1) > d(a(n)^2). But this is not always the case. The smallest counterexample is a(30) = 3591, where d(3591^2-1) = 40 and d(3591^2) = 63. The terms m such that d(m^2-1) < d(m^2) are listed in A342970. [Note that d(m^2-1) = d(m^2) is impossible since d(m^2-1) is even and d(m^2) is odd. - Jianing Song, Nov 21 2021]

Examples

			39 is a term since 39^2-1 = 1520 is divisible by d(1520) = 20 and 39^2 = 1521 is divisible by d(1521) = 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3110 from Jianing Song)
• Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Crossrefs

Cf. A000005, A033950, A036896, A036898, A114617, A342970.

Programs

Mathematica

refQ[n_] := Divisible[n, DivisorSigma[0, n]]; Select[Range[6000], And @@ refQ /@ (#^2 - {1, 0}) &] (* Amiram Eldar, Feb 03 2025 *)

PARI

isrefac(n) = ! (n % numdiv(n));
isA342969(n) = (n>1) && isrefac(n^2-1) && isrefac(n^2)

Formula

A036898(2*n+1) = A114617(n+1) = a(n)^2 - 1; A036898(2*n+2) = A114617(n+1) + 1 = a(n)^2.