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

Author

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

Keywords

• nonn

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

Examples

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

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.

Crossrefs

Cf. A033950, A114617.

Programs

Mathematica

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

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