A177927 3-Monica numbers.

4, 9, 10, 22, 24, 25, 27, 34, 42, 46, 55, 58, 60, 72, 78, 81, 82, 85, 94, 105, 106, 114, 115, 118, 121, 126, 128, 132, 142, 145, 150, 166, 178, 180, 186, 187, 192, 195, 202, 204, 205, 214, 216, 222, 224, 226, 231, 234, 235, 243, 253, 256, 258, 262, 265, 274, 276, 285, 289, 295

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)).

Cf. A007953, A118503.

Sequence in context: A109412 A061766 A109448 * A062398 A030754 A244863

Adjacent sequences: A177924 A177925 A177926 * A177928 A177929 A177930

Keyword

nonn,base

Author

Chris Fry, Dec 26 2010

Status

approved