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A097319
Numbers with more than one prime factor and, in the ordered factorization, the exponents are strictly increasing.
6
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).
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
KEYWORD
nonn
AUTHOR
Ralf Stephan, Aug 04 2004
STATUS
approved