A067340 Numbers k such that (number of distinct prime factors of k) divides (number of prime factors of k).

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

Stanislav Sykora, Table of n, a(n) for n = 1..20000

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

Cf. A001221, A001222, A245080.

Sequence in context: A285100 A368007 A360553 * A263837 A283550 A271113

Adjacent sequences: A067337 A067338 A067339 * A067341 A067342 A067343

Keyword

nonn

Author

Labos Elemer, Jan 16 2002

Status

approved