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Monica Set


The nth Monica set M_n is defined as the set of composite numbers x for which n|[S(x)-S_p(x)], where

x=a_0+a_1(10^1)+...+a_d(10^d)
(1)
=p_1p_2...p_m,
(2)

and

S(x)=sum_(j=0)^(d)a_j
(3)
S_p(x)=sum_(i=1)^(m)S(p_i).
(4)

Every Monica set has an infinite number of elements.

The Monica set M_n is a superset of the Suzanne set S_n.

The following table gives the first few Monica numbers in S_n for small n.

nOEISM_n
1A0182521, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ...
2A1022184, 8, 10, 12, 14, 15, 22, 26, 27, 35, 42, 44, ...
3A1022199, 16, 24, 27, 28, 32, 40, 42, 49, 52, 56, 60, ...

If x is a Smith number, then it is a member of the Monica set M_n for all n in N. For any integer k>1, if x is a k-Smith number, then x in M_(k-1).


See also

Smith Number, Suzanne Set

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequences A018252, A102218, and A102219 in "The On-Line Encyclopedia of Integer Sequences."Smith, M. "Cousins of Smith Numbers: Monica and Suzanne Sets." Fib. Quart. 34, 102-104, 1996.

Referenced on Wolfram|Alpha

Monica Set

Cite this as:

Weisstein, Eric W. "Monica Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MonicaSet.html

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