A073491 Numbers having no prime gaps in their factorization.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 59, 60, 61, 64, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 90, 96, 97, 101, 103, 105, 107, 108, 109, 113, 120, 121, 125, 127, 128, 131, 135

Offset

1,2

Comments

A073490(a(n)) = 0; subsequences are: A000040, A000961, A006094, A002110, A000142, A073485.

A137721(n) = number of terms not greater than n; A137794(a(n))=1; complement of A073492. - Reinhard Zumkeller, Feb 11 2008

Essentially the same as A066311. - R. J. Mathar, Sep 23 2008

The Heinz numbers of the partitions that have no gaps. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: (i) 18 (= 2*3*3) is in the sequence because it is the Heinz number of the partition [1,2,2]; (ii) 10 (= 2*5) is not in the sequence because it is the Heinz number of the partition [1,3]. - Emeric Deutsch, Oct 02 2015

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Example

360 is a term, as 360 = 2*2*2*3*3*5 with consecutive prime factors.

Mathematica

ok[n_] := (p = FactorInteger[n][[All, 1]]; PrimePi[Last@p] - PrimePi[First@p] == Length[p] - 1); Select[Range[135], ok] (* Jean-François Alcover, Apr 29 2011 *)
npgQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]}, f==Prime[Range[ PrimePi[ f[[1]]], PrimePi[f[[-1]]]]]]; Join[{1}, Select[Range[2, 200], npgQ]] (* Harvey P. Dale, Apr 12 2013 *)

Prog

(Haskell)
a073491 n = a073491_list !! (n-1)
a073491_list = filter ((== 0) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013
(PARI) is(n)=my(f=factor(n)[, 1]); for(i=2, #f, if(precprime(f[i]-1)>f[i-1], return(0))); 1 \\ Charles R Greathouse IV, Apr 28 2015

Crossrefs

Cf. A137791, A137792, A137793, A137895.

Sequence in context: A251726 A362981 A193671 * A066311 A069899 A081306

Adjacent sequences: A073488 A073489 A073490 * A073492 A073493 A073494

Keyword

nonn,nice

Author

Reinhard Zumkeller, Aug 03 2002

Status

approved