A322027 -id:A322027 - cp's OEIS frontend

A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 5, 2, 1, 1, 6, 1, 7, 4, 1, 3, 2, 5, 1, 2, 9, 1, 8, 1, 1, 6, 11, 1, 10, 7, 3, 4, 1, 1, 2, 3, 13, 2, 1, 5, 12, 1, 1, 2, 1, 9, 14, 1, 1, 8, 15, 1, 2, 1, 17, 6, 1, 11, 4, 1, 3, 10, 19, 7, 2, 3, 1, 4, 1, 1, 18, 1, 5, 2, 1, 3, 16, 13, 23, 2, 21, 1, 2, 5, 1, 12, 1, 1, 22, 1, 3, 2, 1, 1, 20, 9, 1, 14, 1, 1, 6

Offset: 1

Antti Karttunen, Nov 26 2017

The number of applications to reach 1 is A322027(n). Positions of first appearances are A076610. - Gus Wiseman, Jan 17 2020

			For n = 360 = 2^3 * 3^2 * 5 = prime(1)^3 * prime(2)^2 * prime(3), 1 is not a prime, but 2 and 3 are, thus a(360) = 2^2 * 3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A010051, A295666.
Cf. also A003963, A257538.
Positions of 1's are A320628.
Positions of terms > 1 are A331386.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The number of prime prime indices is A257994.
The number of nonprime prime indices is A330944.
Numbers whose prime indices are not all prime are A330945.
Cf. A001222, A007097, A018252, A056239, A112798, A320633, A322027, A330947, A330948.

Table[Times@@Cases[FactorInteger[n],{p_?(PrimeQ[PrimePi[#]]&),k_}:>PrimePi[p]^k],{n,40}] (* Gus Wiseman, Jan 17 2020 *)

(definec (A295665 n) (if (= 1 n) 1 (let ((k (A055396 n))) (* (if (zero? (A010051 k)) 1 k) (A295665 (A032742 n))))))

Multiplicative with a(p^e) = A000720(p)^(e*A010051(A000720(p))).
a(1) = 1; for n > 1, if A055396(n) is a prime, then a(n) = A055396(n) * a(A032742(n)), otherwise a(n) = a(A032742(n)).
Other identities. For all n >= 1:
a(A006450(n)) = A000040(n).
a(A007097(n)) = A007097(n-1).
a(A294876(n)) = A295666(n).

A322028 Number of distinct orders of primeness among the prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322027, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> nops(map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Length[Union[Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]]]]],{n,100}]

A322030 Numbers whose prime factors all have the same order of primeness.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 26, 27, 28, 29, 31, 32, 37, 38, 41, 43, 46, 47, 49, 51, 52, 53, 56, 58, 59, 61, 64, 67, 71, 73, 74, 76, 79, 81, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 121, 122, 123

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			182 is in the sequence because its prime factors 2, 7, 13 all have order of primeness 1.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A322027, A322028.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while nops(map(p, factorset(k)))>1 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018

ordpri[n_]:=If[!PrimeQ[n],0,Length[NestWhileList[PrimePi,PrimePi[n],PrimeQ]]];
Select[Range[200],SameQ@@ordpri/@FactorInteger[#][[All,1]]&]

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A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

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A322028 Number of distinct orders of primeness among the prime factors of n.

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A322030 Numbers whose prime factors all have the same order of primeness.

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