A325608 Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.

147, 245, 294, 357, 490, 511, 539, 588, 595, 637, 681, 714, 735, 845, 847, 853, 867, 903, 980, 1022, 1029, 1043, 1078, 1083, 1135, 1176, 1183, 1190, 1239, 1241, 1267, 1274, 1309, 1362, 1421, 1428, 1445, 1470, 1505, 1519, 1547, 1553, 1563, 1617, 1631, 1690

Offset: 1

Author

Gus Wiseman, May 12 2019

Keywords

• nonn

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example, 147 = q(1)^5 q(2) q(4)^2 has multiplicities (5,1,2), which are not weakly decreasing, so 147 belongs to the sequence.

Crossrefs

Cf. A001222, A056239, A112798, A118914.

Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.

q-factorization: A324922, A324923, A324924, A324931, A325613, A325614, A325615, A325660, A325662.

Programs

Mathematica

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[1000],!GreaterEqual@@Length/@Split[difac[#]]&]