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A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950). 2
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 (list; graph; refs; listen; history; text; internal format)
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. Those terms are listed in A342970.

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) = a(n)^2 - 1; A036898(2*n+2) = 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, A036898, A114617, A342970.

Sequence in context: A209366 A292542 A212664 * A050392 A292294 A191468

Adjacent sequences:  A342966 A342967 A342968 * A342970 A342971 A342972

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Apr 01 2021

STATUS

approved

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Last modified September 25 09:40 EDT 2021. Contains 347654 sequences. (Running on oeis4.)