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

1,1

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

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.

Formula

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

Example

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)

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 *)

Prog

(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

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.

Sequence in context: A223379 A223297 A317493 * A124065 A121453 A003343

Adjacent sequences: A102214 A102215 A102216 * A102218 A102219 A102220

Keyword

nonn

Author

Eric W. Weisstein, Dec 30 2004

Extensions

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

Status

approved