A097319 Numbers with more than one prime factor and, in the ordered factorization, the exponents are strictly increasing.

18, 50, 54, 75, 98, 108, 147, 162, 242, 245, 250, 324, 338, 363, 375, 486, 500, 507, 578, 605, 648, 686, 722, 845, 847, 867, 972, 1029, 1058, 1083, 1125, 1183, 1250, 1372, 1445, 1458, 1587, 1682, 1715, 1805, 1859, 1875, 1922, 1944, 2023, 2250

Offset

1,1

Comments

If n = Product[k=1..m, p(k)^e(k)], then m>1 and e(1) < e(2) <...< e(m).

Links

T. D. Noe, Table of n, a(n) for n = 1..1180

Example

507 is 3^1*13^2, A001221(507)=2 and 1<2, so 507 is in sequence.
150 is 2^1*3^1*5^2 is not in the sequence because 1,1,2 is not strictly increasing (although it is nondecreasing).

Mathematica

fQ[n_] := Module[{d, f = FactorInteger[n]}, If[Length[f] == 1, False, d = Differences[Transpose[f][[2]]]; And @@ ((# > 0) & /@ d)]]; Select[Range[2250], fQ] (* T. D. Noe, Apr 09 2013 *)

Prog

(PARI) for(n=1, 3000, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]>=F[k+1, 2], t=1; break)); if(!t, print1(n", "))))
(PARI) is(n) = my(f = factor(n)[, 2]); #f > 1 && vecsort(f, , 8) == f \\ Rick L. Shepherd, Jan 17 2018

Crossrefs

Subset of A126706. Cf. A097318, A097320.

Sequence in context: A317258 A071365 A360252 * A258211 A354929 A093617

Adjacent sequences: A097316 A097317 A097318 * A097320 A097321 A097322

Keyword

nonn

Author

Ralf Stephan, Aug 04 2004

Status

approved