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

%I A102217 #52 Feb 16 2025 08:32:55
%S A102217 9,24,27,42,60,72,78,81,105,114,126,132,150,180,186,192,195,204,216,
%T A102217 222,231,234,243,258,276,285,315,330,336,342,348,357,366,375,378,396,
%U A102217 402,429,438,450,465,474,480,483,492,510,540,555,558,564,576,582,585
%N A102217 3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.
%C A102217 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
%C A102217 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
%D A102217 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
%D A102217 James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.
%H A102217 Amiram Eldar, <a href="/A102217/b102217.txt">Table of n, a(n) for n = 1..10000</a>
%H A102217 Michael Smith, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/34-2/smith.pdf">Cousins of Smith Numbers: Monica and Suzanne Sets</a>, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
%H A102217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SuzanneSet.html">Suzanne Set</a>.
%F A102217 a(n) = 3*A289142(1+n). - _Antti Karttunen_, Jun 08 2024
%e A102217 From _Antti Karttunen_, Jun 08 2024: (Start)
%e A102217 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.
%e A102217 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.
%e A102217 (End)
%t A102217 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 *)
%o A102217 (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
%o A102217 (PARI) isA102217(n) = (n>3 && !(n%3) && A373371(n)); \\ _Antti Karttunen_, Jun 08 2024
%Y A102217 Subsequence of A177927.
%Y A102217 Intersection of A008585 and A289142 without the initial 3.
%Y A102217 Positions of multiples of 3 in A082299, after A082299(3).
%Y A102217 Cf. A001414, A007953, A018252, A102216, A118503, A373371.
%K A102217 nonn
%O A102217 1,1
%A A102217 _Eric W. Weisstein_, Dec 30 2004
%E A102217 Alternative definition added and keyword:base removed by _Antti Karttunen_, Jun 08 2024