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a(1) = 10; for n>1, a(n) = (a(n-1)-1) * (smallest odd prime factor of a(n-1)) + 1.
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%I #51 Feb 28 2022 13:43:03

%S 10,46,1036,7246,26248636,11628145306,461742021916246,

%T 7849614372576166,44750651538056716666,17139499539075722482696,

%U 188534494929832947309646,69192159639248691662639716,2144956948816709441541831166,13721289601580490297543093962506

%N a(1) = 10; for n>1, a(n) = (a(n-1)-1) * (smallest odd prime factor of a(n-1)) + 1.

%C Note: this sequence will terminate if a power of 2 occurs.

%C Conjecture: this sequence is infinite (so it contains no powers of 2). The similarly defined sequence starting with a(1) = 6 terminates after 2 terms: 6 and 16.

%C Conjecture is true if it turns out that one number in this sequence is one more than a multiple of 41. Any number of the form 2^n-1 that is divisible by 41 is also divisible by 25, which is the square of a prime number greater than 3. Because numbers one less than this sequence's terms are always 9 times a squarefree number, this proves that if a number one less than this sequence is divisible by 41, then this sequence is infinite. - _J. Lowell_, Jul 17 2017

%C Conjecture is also true if it turns out that one number in this sequence is a multiple of either 19 or 73. Any number of the form 2^n-1 that is divisible by 19 is divisible by 27, and any number of the form 2^n-1 that is divisible by both 73 and 9 is divisible by 27. Numbers one less than this sequence are always divisible by 9 but not by 27. - _J. Lowell_, Feb 27 2022

%e a(1) = 10; 10-1 = 9; 9*5 (smallest odd prime factor of 10) is 45; 45+1=46, so a(2) = 46.

%t NestList[DeleteCases[FactorInteger[#], w_ /; First@ w == 2][[1, 1]] (# - 1) + 1 &, 10, 13] (* _Michael De Vlieger_, Jul 18 2017 *)

%o (PARI) opf(n) = n = n>>valuation(n,2); if (n==1, 1, factor(n)[1,1]);

%o lista(nn) = {a = 10; for (n=2, nn, olda = a; print1(a, ", "); a = 1 + (olda-1)*opf(olda););} \\ _Michel Marcus_, May 17 2014

%K nonn

%O 1,1

%A _J. Lowell_, May 15 2014

%E a(13)-a(14) and name change from _Michel Marcus_, May 17 2014