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A067340
Numbers k such that (number of distinct prime factors of k) divides (number of prime factors of k).
39
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91
OFFSET
1,1
COMMENTS
From Peter Luschny, Jul 19 2023: (Start)
If the name means 'Numbers k such that (number of prime factors of k) is divisible by the (number of distinct prime factors of k)', then 1 has to be prepended to the data since A001221(1) = A001222(1) = 0 and 0 is divisible by 0.
Note that the expression 'A001222(k)/A001221(k)' is read as 'the quotient of A001222(k) and A001221(k)' and is not defined in the case k = 1 because A001221(1) = 0. On the other hand, the expression 'A001221(k) | A001222(k)' is read as 'A001221(k) divides A001222(k)' and is well defined also if k = 1 and has the value 'True'. (End)
LINKS
FORMULA
A001222(k)/A001221(k) is an integer.
EXAMPLE
Primes and prime powers are included in this sequence. Another example: 24, since A001222(24)/A001222(24) = 4/2 = 2.
MATHEMATICA
ff[x_] := Flatten[FactorInteger[x]]; f1[x_] := Length[FactorInteger[x]]; f2[x_] := Apply[Plus, Table[Part[ff[x], 2*w], {w, 1, f1[x]}]]; Do[s=f2[n]/f1[n]; If[IntegerQ[s], Print[n]], {n, 2, 256}]
Select[Range[2, 91], Divisible[PrimeOmega[#], PrimeNu[#]]&] (* Ivan N. Ianakiev, Dec 07 2015 *)
PROG
(PARI) v=[]; for(n=2, 100, if(denominator(bigomega(n)/omega(n)) == 1, v=concat(v, n))); v
(PARI) is(n)=my(f=factor(n)[, 2]); #f && vecsum(f)%#f==0 \\ Charles R Greathouse IV, Oct 15 2015
(SageMath)
def dpf(n): return sloane.A001221(n)
def tpf(n): return sloane.A001222(n)
a = [k for k in range(1, 92) if ZZ(dpf(k)).divides(tpf(k))]
print(a) # Peter Luschny, Jul 19 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jan 16 2002
STATUS
approved