cp's OEIS Frontend

A385932 Composite numbers m such that the sum of digits of m divides the sum of digits of prime factors of m (counted with multiplicity).

%I A385932 #16 Jul 17 2025 00:49:24
%S A385932 4,10,22,27,32,42,58,60,70,85,94,100,104,121,152,166,200,202,231,265,
%T A385932 274,315,316,319,322,330,342,346,355,361,378,382,391,402,406,430,438,
%U A385932 450,454,483,510,517,526,535,540,562,576,588,602,610,612,627,632,634,636,645,648
%N A385932 Composite numbers m such that the sum of digits of m divides the sum of digits of prime factors of m (counted with multiplicity).
%C A385932 Equivalently, numbers m such that A007953(m) | A118503(m).
%C A385932 Union of the k-Smith numbers for all the positive integers k.
%D A385932 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.1.14 and 3.1.16 on pages 84-85.
%H A385932 Wayne L. McDaniel, <a href="https://www.fq.math.ca/Scanned/25-1/mcdaniel.pdf">The Existence of Infinitely Many k-Smith Numbers</a>, The Fibonacci Quarterly, 25(1), 76-80, (1987).
%e A385932 10 = 2*5 is a term since it is a 7-Smith number: 1 + 0 = 1 | 7 = 2 + 5;
%e A385932 60 = 2^2*3*5 is term since it is a 2-Smith number: 6 + 0 = 6 | 12 = 2 + 2 + 3 + 5;
%e A385932 382 = 2*191 is a term since it is a Smith number (k=1): 3 + 8 + 2 = 13 | 13 = 2 + 1 + 9 + 1;
%e A385932 635 = 5*127 is not a term since 6 + 3 + 5 = 14 does not divide 15 = 5 + 1 + 2 + 7.
%t A385932 fQ[n_]:=!PrimeQ[n] && n>1 && Divisible[Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]] & /@ FactorInteger[n]]], Total[IntegerDigits[n]]]; Select[ Range@ 650, fQ]
%Y A385932 Supersequence of A006753, A103125, A103126, A104390, A104391.
%Y A385932 Cf. A002808, A007953, A118503.
%K A385932 nonn,base,easy
%O A385932 1,1
%A A385932 _Stefano Spezia_, Jul 12 2025