A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114

Offset

1,1

Comments

If n = Product_{k=1..m} p(k)^e(k), then m > 1, e(1) >= e(2) >= ... >= e(m).

These are numbers whose ordered prime signature is weakly decreasing. Weakly increasing is A304678. Ordered prime signature is A124010. - Gus Wiseman, Nov 10 2019

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

S. Ramanujan, Asymptotic formulas for the distribution of integers of various types, Proc. London Math. Soc. 2, 16 (1917), 112-132.

Example

60 is 2^2*3^1*5^1, A001221(60)=3 and 2>=1>=1, so 60 is in sequence.

Maple

q:= n-> (l-> (t-> t>1 and andmap(i-> l[i, 2]>=l[i+1, 2],
        [$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
select(q, [$1..120])[];  # Alois P. Heinz, Nov 11 2019

Mathematica

fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] <= 0]; Select[Range[2, 200], fQ] (* T. D. Noe, Nov 04 2013 *)

Prog

(PARI) for(n=1, 130, 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", "))))

Crossrefs

Subset of A024619. Cf. A097319, A097320, A230766.

Cf. A001222, A025487, A118914, A124010, A304678, A329138, A329142.

Sequence in context: A201463 A119313 A362980 * A080363 A289558 A082300

Adjacent sequences: A097315 A097316 A097317 * A097319 A097320 A097321

Keyword

nonn

Author

Ralf Stephan, Aug 04 2004

Status

approved