A302243 -id:A302243 - cp's OEIS frontend

A302602 Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 25, 27, 31, 32, 41, 59, 64, 67, 81, 83, 109, 121, 125, 127, 128, 157, 179, 191, 211, 241, 243, 256, 277, 283, 289, 331, 353, 367, 401, 431, 461, 509, 512, 547, 563, 587, 599, 617, 625, 709, 729, 739, 773, 797, 859, 877, 919

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			25 is in the sequence because 25 = prime(3)^2 and 3 is a prime number.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
08: {{},{},{}}
09: {{1},{1}}
11: {{3}}
16: {{},{},{},{}}
17: {{4}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
59: {{7}}
64: {{},{},{},{},{},{}}
67: {{8}}
81: {{1},{1},{1},{1}}
83: {{9}}

Cf. A000961, A001222, A003963, A005117, A007425, A007716, A056239, A275024, A281113, A295931, A301764, A302242, A302243.

Select[Range[1000],#===1||MatchQ[FactorInteger[#],{{?(#===2||PrimeQ[PrimePi[#]]&),}}]&]

isok(n) = (n==1) || ((isprimepower(n, &p)) && ((p==2) || isprime(primepi(p)))); \\ Michel Marcus, Apr 10 2018

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.

Original entry on oeis.org

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

Gus Wiseman, Sep 30 2022

nonn

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}}.

			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}}

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.

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[#]&]

cp's OEIS Frontend

A302602 Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.

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