A357188 Numbers with (WLOG adjacent) prime indices x <= y such that the greatest prime factor of x is greater than the least prime factor of y.

35, 65, 70, 95, 105, 130, 140, 143, 145, 169, 175, 185, 190, 195, 209, 210, 215, 245, 247, 253, 260, 265, 280, 285, 286, 290, 305, 315, 319, 323, 325, 338, 350, 355, 370, 377, 380, 385, 390, 391, 395, 407, 418, 420, 429, 430, 435, 445, 455, 473, 475, 481, 490

Offset: 1

Author

Gus Wiseman, Sep 30 2022

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The terms and corresponding multisets of multisets:
   35: {{2},{1,1}}
   65: {{2},{1,2}}
   70: {{},{2},{1,1}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  130: {{},{2},{1,2}}
  140: {{},{},{2},{1,1}}
  143: {{3},{1,2}}
  145: {{2},{1,3}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  185: {{2},{1,1,2}}
		

Crossrefs

These are the positions of non-weakly increasing rows in A357139.

A000961 lists prime powers.

A003963 multiples prime indices.

A056239 adds up prime indices.

Cf. A000720, A001221, A001222, A007716, A275024, A302242, A302243, A302505, A324926, A325032, A325034.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MatchQ[primeMS[#],{_,x_,y_,_}/;Max@@primeMS[x]>Min@@primeMS[y]]&]
Select[Range[100],!LessEqual@@Join@@primeMS/@primeMS[#]&]