A102217 3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

9, 24, 27, 42, 60, 72, 78, 81, 105, 114, 126, 132, 150, 180, 186, 192, 195, 204, 216, 222, 231, 234, 243, 258, 276, 285, 315, 330, 336, 342, 348, 357, 366, 375, 378, 396, 402, 429, 438, 450, 465, 474, 480, 483, 492, 510, 540, 555, 558, 564, 576, 582, 585

Offset: 1

Author

Eric W. Weisstein, Dec 30 2004

Keywords

• nonn

Comments

Composite numbers k such that the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) are both divisible by 3. - Amiram Eldar, Apr 23 2021

The new secondary definition is equal to the original because taking the decimal digit sum preserves congruence modulo 3. This is a multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jun 08 2024

Examples

			From _Antti Karttunen_, Jun 08 2024: (Start)
42 = 2*3*7 is a term as it is a multiple of 3, and also 2+3+7 = 12 is a multiple of 3.
60 = 2*2*3*5 is a term is it is a multiple of 3, and also 2+2+3+5 = 12 is a multiple of 3.
(End)
		

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
• Eric Weisstein's World of Mathematics, Suzanne Set.

Crossrefs

Subsequence of A177927.

Intersection of A008585 and A289142 without the initial 3.

Positions of multiples of 3 in A082299, after A082299(3).

Cf. A001414, A007953, A018252, A102216, A118503, A373371.

Programs

Mathematica

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; suz3Q[n_] := CompositeQ[n] && And @@ Divisible[{s[n], sp[n]}, 3]; Select[Range[600], suz3Q] (* Amiram Eldar, Apr 23 2021 *)

PARI

isA102217(n) = if(n<=3 || (n%3), 0, my(f=factor(n)); 0==(sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1]))%3)); \\ Antti Karttunen, Jun 08 2024

PARI

isA102217(n) = (n>3 && !(n%3) && A373371(n)); \\ Antti Karttunen, Jun 08 2024

Formula

a(n) = 3*A289142(1+n). - Antti Karttunen, Jun 08 2024

Extensions

Alternative definition added and keyword:base removed by Antti Karttunen, Jun 08 2024