A331593 - OEIS

%I #21 Feb 02 2020 13:56:38

%S 1,2,3,4,5,7,9,11,12,13,16,17,18,19,20,23,24,25,28,29,31,37,40,41,43,

%T 44,45,47,48,49,50,52,53,54,56,59,61,63,67,68,71,72,73,75,76,79,80,81,

%U 83,88,89,92,96,97,98,99,101,103,104,107,108,109,112,113,116,117,121,124,127,131,135,136,137,139,144,147,148,149

%N Numbers k that have the same number of distinct prime factors as A225546(k).

%C Numbers k for which A001221(k) = A331591(k).

%C Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.

%C If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.

%C k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

%F {a(n)} = {k : A001221(k) = A000120(A267116(k))}.

%e There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.

%e There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

%t Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* _Michael De Vlieger_, Jan 26 2020 *)

%o (PARI)

%o A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));

%o k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

%Y Cf. A000120, A046523, A225546, A267116, A329332.

%Y Sequences with related definitions: A001221, A331591, A331592.

%Y Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.

%Y Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

%K nonn

%O 1,2

%A _Antti Karttunen_ and _Peter Munn_, Jan 21 2020