A036898 List of pairs of consecutive refactorable numbers.

1, 2, 8, 9, 1520, 1521, 50624, 50625, 62000, 62001, 103040, 103041, 199808, 199809, 221840, 221841, 269360, 269361, 463760, 463761, 690560, 690561, 848240, 848241, 986048, 986049, 1252160, 1252161, 1418480, 1418481, 2169728, 2169729, 2692880

Offset

1,2

Comments

Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. As a result, a(n) == n-1 (mod 2) for n >= 3. See also A114617. - Jianing Song, Apr 01 2021

Links

Donovan Johnson, Table of n, a(n) for n = 1..2000

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

S. Colton, HR - Automatic Theory Formation in Pure Mathematics

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

Example

8 is refactorable because tau(8)=4 and 4 divides 8.
9 is refactorable because tau(9)=3 and 3 divides 9.

Mathematica

SequencePosition[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 27*10^5}], {1, 1}]//Flatten (* Harvey P. Dale, Dec 07 2021 *)

Prog

(PARI) isrefac(n) = ! (n % numdiv(n));
lista(nn) = {for (n = 1, nn, if (isrefac(n) && isrefac(n+1), print1(n, ", ", n+1, ", ")); ); } \\ Michel Marcus, Aug 31 2013

Crossrefs

Cf. A033950, A114617.

Sequence in context: A381317 A109351 A111134 * A053372 A088155 A005876

Adjacent sequences: A036895 A036896 A036897 * A036899 A036900 A036901

Keyword

nonn

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Status

approved