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A116486 Numbers k such that both k and k + 1 are logarithmically smooth.

%I A116486 #26 Dec 05 2019 05:39:19
%S A116486 8,24,80,125,224,2400,3024,4224,4374,6655,9800,10647,123200,194480,
%T A116486 336140,601425,633555,709631,5142500,5909760,11859210,1611308699
%N A116486 Numbers k such that both k and k + 1 are logarithmically smooth.
%C A116486 N is logarithmically smooth if its largest prime factor p <= ceiling(log_2(n)).
%C A116486 Is the sequence finite?
%C A116486 No more terms with largest prime factor <= 47. - _Joerg Arndt_, Jul 02 2012
%H A116486 Discussion titled <a href="http://www.mersenneforum.org/showthread.php?t=5630">Special Smooth numbers</a>, (postings in mersenneforum.org), starting March 20 2006.
%e A116486 125 is in the sequence because 125 = 5 * 5 * 5, 126 = 2 * 3 * 3 * 7; no prime factor is greater than ceiling(log_2(125)) = 7.
%t A116486 logCeilSmoothQ[n_, b_:E] := FactorInteger[n][[-1, 1]] <= Ceiling[Log[b, n]]; Select[Range[10000], logCeilSmoothQ[#, 2] && logCeilSmoothQ[# + 1, 2] &] (* _Alonso del Arte_, Nov 27 2019 *)
%o A116486 (PARI)
%o A116486 fm=97;  /* max factor for factorizing, 2^97 >= searchlimit */
%o A116486 lpf(n)={ vecmax(factor(n, fm)[, 1]) } /* largest prime factor */
%o A116486 lsm(n)=if ( lpf(n)<=#binary(n-1), 1, 0 );  /* whether log-smooth, for n>=2 */
%o A116486 n0=3;  /* lower search limit */
%o A116486 l1=lsm(n0-1);
%o A116486 { for (n=n0, 10^10,
%o A116486     l0 = lsm(n);
%o A116486     if ( l0 && l1, print1(n-1,", ") );
%o A116486     l1 = l0;
%o A116486 ); }
%o A116486 /* _Joerg Arndt_, Jul 02 2012 */
%K A116486 nonn,hard,more
%O A116486 1,1
%A A116486 _Harsh R. Aggarwal_, Mar 20 2006
%E A116486 Edited by _Don Reble_, Apr 07 2006