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%I #29 Apr 23 2021 05:24:03
%S 4,9,10,22,24,25,27,34,42,46,55,58,60,72,78,81,82,85,94,105,106,114,
%T 115,118,121,126,128,132,142,145,150,166,178,180,186,187,192,195,202,
%U 204,205,214,216,222,224,226,231,234,235,243,253,256,258,262,265,274,276,285,289,295
%N 3-Monica numbers.
%C 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.
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
%D James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 93.
%D E. W. Weisstein, The CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pages 1192-1193.
%H Amiram Eldar, <a href="/A177927/b177927.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Smith, <a href="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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MonicaSet.html">Monica Set</a>.
%e S(10)=1+0=1, 10=2*5, Sp(10)=2+5=7, S(10)-Sp(10)=-6 which is divisible by 3.
%t 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 *)
%Y Cf. A006753 (Smith numbers are a subset of every n-Monica sequence).
%Y Cf. A102217 (n-Suzanne numbers are a subset of n-Monica numbers).
%Y 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)).
%Y Cf. A007953, A118503.
%K nonn,base
%O 1,1
%A _Chris Fry_, Dec 26 2010
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