OFFSET
1,1
COMMENTS
3-Monica numbers are composite positive integers k for which 3 divides S(k)-Sp(k), where S(k) denotes the sum of the digits of k and Sp(k) denotes the sum of the digits in an extended prime factorization of k.
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, Cambridge University Press, 2005, page 93.
E. W. Weisstein, The CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pages 1192-1193.
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, Monica Set.
EXAMPLE
S(10)=1+0=1, 10=2*5, Sp(10)=2+5=7, S(10)-Sp(10)=-6 which is divisible by 3.
MATHEMATICA
s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; mon3Q[n_] := CompositeQ[n] && Divisible[s[n] - sp[n], 3]; Select[Range[300], mon3Q] (* Amiram Eldar, Apr 23 2021 *)
CROSSREFS
Cf. A006753 (Smith numbers are a subset of every n-Monica sequence).
Cf. A102217 (n-Suzanne numbers are a subset of n-Monica numbers).
Cf. A102219 (This list of '3-Monica' numbers is incorrect. It does not contain all the Smith numbers and appears to be based on S(n)+Sp(n) ==0 (mod 3), instead of S(n)-Sp(n) == 0 (mod 3)).
KEYWORD
nonn,base
AUTHOR
Chris Fry, Dec 26 2010
STATUS
approved