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).

4, 10, 22, 27, 32, 42, 58, 60, 70, 85, 94, 100, 104, 121, 152, 166, 200, 202, 231, 265, 274, 315, 316, 319, 322, 330, 342, 346, 355, 361, 378, 382, 391, 402, 406, 430, 438, 450, 454, 483, 510, 517, 526, 535, 540, 562, 576, 588, 602, 610, 612, 627, 632, 634, 636, 645, 648

Offset: 1

Author

Stefano Spezia, Jul 12 2025

Keywords

• nonn
• base
• easy

Comments

Equivalently, numbers m such that A007953(m) | A118503(m).

Union of the k-Smith numbers for all the positive integers k.

Examples

			10 = 2*5 is a term since it is a 7-Smith number: 1 + 0 = 1 | 7 = 2 + 5;
60 = 2^2*3*5 is term since it is a 2-Smith number: 6 + 0 = 6 | 12 = 2 + 2 + 3 + 5;
382 = 2*191 is a term since it is a Smith number (k=1): 3 + 8 + 2 = 13 | 13 = 2 + 1 + 9 + 1;
635 = 5*127 is not a term since 6 + 3 + 5 = 14 does not divide 15 = 5 + 1 + 2 + 7.
		

References

• 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.

Links

• Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).

Crossrefs

Supersequence of A006753, A103125, A103126, A104390, A104391.

Cf. A002808, A007953, A118503.

Programs

Mathematica

fQ[n_]:=!PrimeQ[n] && n>1 && Divisible[Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]] & /@ FactorInteger[n]]], Total[IntegerDigits[n]]]; Select[ Range@ 650, fQ]