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

1,1

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]

Links

Jianing Song, Table of n, a(n) for n = 1..3110 (all terms <= 10^6).

Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Formula

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

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A209366 A292542 A212664 * A050392 A292294 A366995

Adjacent sequences: A342966 A342967 A342968 * A342970 A342971 A342972

Keyword

nonn,easy

Author

Jianing Song, Apr 01 2021

Status

approved