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

8, 24, 80, 125, 224, 2400, 3024, 4224, 4374, 6655, 9800, 10647, 123200, 194480, 336140, 601425, 633555, 709631, 5142500, 5909760, 11859210, 1611308699

Offset

1,1

Comments

N is logarithmically smooth if its largest prime factor p <= ceiling(log_2(n)).

Is the sequence finite?

No more terms with largest prime factor <= 47. - Joerg Arndt, Jul 02 2012

Links

Table of n, a(n) for n=1..22.

Discussion titled Special Smooth numbers, (postings in mersenneforum.org), starting March 20 2006.

Example

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.

Mathematica

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 *)

Prog

(PARI)
fm=97;  /* max factor for factorizing, 2^97 >= searchlimit */
lpf(n)={ vecmax(factor(n, fm)[, 1]) } /* largest prime factor */
lsm(n)=if ( lpf(n)<=#binary(n-1), 1, 0 );  /* whether log-smooth, for n>=2 */
n0=3;  /* lower search limit */
l1=lsm(n0-1);
{ for (n=n0, 10^10,
    l0 = lsm(n);
    if ( l0 && l1, print1(n-1, ", ") );
    l1 = l0;
); }
/* Joerg Arndt, Jul 02 2012 */

Crossrefs

Sequence in context: A303402 A078158 A221906 * A179609 A241690 A141317

Adjacent sequences: A116483 A116484 A116485 * A116487 A116488 A116489

Keyword

nonn,hard,more

Author

Harsh R. Aggarwal, Mar 20 2006

Extensions

Edited by Don Reble, Apr 07 2006

Status

approved