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A128290 If p(x) is the product of the digits of the number x and s(x) the sum of the digits then the sequence lists all the numbers k for which p(s(k)) = s(p(k)), with k >= 1.

%I A128290 #28 Feb 17 2025 01:55:37
%S A128290 1,2,3,4,5,6,7,8,9,22,36,63,109,123,132,158,185,190,199,208,213,231,
%T A128290 280,289,298,307,312,321,333,370,406,458,460,469,485,496,505,518,548,
%U A128290 550,556,559,565,581,584,595,604,640,649,655,667,676,694,703,730,766,802
%N A128290 If p(x) is the product of the digits of the number x and s(x) the sum of the digits then the sequence lists all the numbers k for which p(s(k)) = s(p(k)), with k >= 1.
%H A128290 Michael S. Branicky, <a href="/A128290/b128290.txt">Table of n, a(n) for n = 1..10000</a>
%e A128290 496 is a term: s(496) = 4+9+6 = 19, p(s(496)) = 1*9 = 9, p(496) = 4*9*6 = 216, s(p(496)) = 2+1+6 = 9.
%e A128290 845 is a term: s(845) = 8+4+5 = 17, p(s(845)) = 1*7 = 7, p(845) = 8*4*5 = 160, s(p(845)) = 1+6+0 = 7.
%e A128290 From _Jon E. Schoenfield_, Jun 15 2024: (Start)
%e A128290 Expressed more visually:
%e A128290 .
%e A128290           Sum                         Sum
%e A128290    496 --------> 19            845 --------> 17
%e A128290     |    4+9+6    |             |    8+4+5    |
%e A128290   P | 4         P |           P | 8         P |
%e A128290   r | *         r | 1         r | *         r | 1
%e A128290   o | 9         o | *         o | 4         o | *
%e A128290   d | *         d | 9         d | *         d | 7
%e A128290     | 6           |             | 5           |
%e A128290     v     Sum     v             v     Sum     v
%e A128290    216 -------->  9            160 -------->  7
%e A128290          2+1+6                       1+6+0
%e A128290 (End)
%p A128290 P:=proc(q) local a,b,c; a:=convert(q,base,10): b:=convert(a,`+`): c:=convert(a,`*`):
%p A128290 if convert(convert(b,base,10),`*`)=convert(convert(c,base,10),`+`) then q; fi; end: seq(P(i),i=1..10^3); # _Paolo P. Lava_, Jun 15 2024
%t A128290 p[n_] := Times @@ IntegerDigits[n]; Select[Range[1000], p[DigitSum[#]] == DigitSum[p[#]] &] (* _Paolo Xausa_, Jun 17 2024 *)
%o A128290 (Python)
%o A128290 from math import prod
%o A128290 def ok(n):
%o A128290     d = list(map(int, str(n)))
%o A128290     p, s = prod(d), sum(d)
%o A128290     return sum(map(int, str(p))) == prod(map(int, str(s)))
%o A128290 print([k for k in range(1, 803) if ok(k)]) # _Michael S. Branicky_, Jun 15 2024
%Y A128290 Cf. A007953, A007954, A128212, A128244.
%K A128290 easy,nonn,base
%O A128290 1,2
%A A128290 _Paolo P. Lava_ and _Giorgio Balzarotti_, May 04 2007