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%I #46 Jul 16 2022 22:52:25
%S 0,1,135,144
%N Numbers k such that k = (product of digits of k) * (sum of digits of k).
%C The list is complete. Proof: One shows that the number of digits is at most 84 and then it is only necessary to consider numbers of the forms 2^i*3^j*7^k and 3^i*5^j*7^k. - _David W. Wilson_, May 16 2003
%H Alan Beardon, <a href="https://doi.org/10.2307/3618679">S.P numbers</a>, The Mathematical Gazette, 83(496), 25-32 (1999).
%H Alan Beardon, <a href="https://nrich.maths.org/1336">Sums and Products of Digits and SP Numbers</a>, NRICH, University of Cambridge, 1998.
%H Alan Beardon, <a href="https://nrich.maths.org/1339">Recent Developments on S.P. Numbers</a>, NRICH, University of Cambridge, 1998-2011.
%H E. Bussmann, <a href="https://doi.org/10.2307/3622010">S.P numbers in bases other than 10</a>, The Mathematical Gazette, 85(503), 245-248 (2001).
%H K. McLean, <a href="https://doi.org/10.2307/3618680">There are only three S.P numbers!</a>, The Mathematical Gazette, 83(496), 32-38 (1999).
%H S. Parameswaran, <a href="https://doi.org/10.2307/3619204">Numbers and their digits - a structural pattern</a>, Note 81.24, The Mathematical Gazette, 81(491), 263-263 (1997).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sum-ProductNumber.html">Sum-Product Number.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Digit.html">Digit.</a>
%F a(n) = A007953(a(n)) * A007954(a(n)).
%e 144 belongs to the sequence because 1*4*4=16, 1+4+4=9 -> 16*9=144
%t pdsdQ[n_]:=Module[{idn=IntegerDigits[n]},(Total[idn]Times@@idn)==n]; Select[Range[0,150],pdsdQ] (* _Harvey P. Dale_, Apr 23 2011 *)
%o (PARI) is(n)=my(d=digits(n)); factorback(d)*vecsum(d)==n \\ _Charles R Greathouse IV_, Feb 06 2017
%Y Cf. A007953, A007954, A066308.
%Y Cf. A038364, A062237, A066282.
%K nice,nonn,fini,base,full
%O 1,3
%A _Felice Russo_
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