A396766 - OEIS

%I #19 Jul 14 2026 21:31:33

%S 19,20,30,37,38,56,64,65,66,73,74,84,92,102,109,119,120,127,138,146,

%T 155,156,163,164,165,174,181,199,200,209,218,228,236,245,246,254,255,

%U 264,271,272,273,282,289,299,300,307,318,326,335,336,343,344,345,354,361,362,371,372,379,397

%N Numbers whose number of distinct prime factors equals their digital root.

%C Since the digital root of n lies in {1..9}, every term has between 1 and 9 distinct prime factors. The sequence is infinite: by Dirichlet's theorem there are infinitely many primes p == 1 (mod 9), and each such prime has exactly one distinct prime factor and digital root 1, hence belongs to the sequence.

%C For each d from 1 to 9, there are infinitely many terms k with A001221(k) = A010888(k) = d. - _Robert Israel_, Jul 08 2026

%F { k : A001221(k) = A010888(k) }.

%e 56 = 2^3 * 7 has 2 distinct prime factors {2, 7}; digital root of 56 is 5+6 = 11, then 1+1 = 2; since 2 = 2, 56 is a term.

%e 120 = 2^3 * 3 * 5 has 3 distinct prime factors; digital root is 1+2+0 = 3; since 3 = 3, 120 is a term.

%p q:= x-> is(((x-1) mod 9)+1=nops(ifactors(x)[2])):

%p select(q, [$2..397])[];  # _Alois P. Heinz_, Jul 07 2026

%t Select[Range[400], PrimeNu[#] == (Mod[# - 1, 9] + 1) &]

%o (PARI) select(n->omega(n)==(n-1)%9+1, [1..400])

%o (Python) from sympy import primefactors

%o print([n for n in range(1, 400) if len(primefactors(n)) == 1+(n-1)%9])

%Y Cf. A001221, A010888, A061237 (subsequence).

%K nonn,base,easy,new

%O 1,1

%A _Alexander Pokorny_, Jul 07 2026