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A321990 Positive numbers for which the product of digits is equal to the power tower of digits (right-associative).

%I A321990 #70 Jul 29 2019 09:58:15
%S A321990 1,2,3,4,5,6,7,8,9,11,21,22,31,41,51,61,71,81,91,111,211,221,311,411,
%T A321990 511,611,711,811,911,1111,2111,2211,2412,3111,3313,4111,4212,5111,
%U A321990 6111,6213,7111,8111,8214,9111,11111,21111,22111,22212,24112,24121,28128,28144
%N A321990 Positive numbers for which the product of digits is equal to the power tower of digits (right-associative).
%C A321990 Positive numbers k such that A007954(k) = A256229(k).
%C A321990 All numbers of the form xx1x...x with x x's are terms, as are numbers of the form xxx1x...x with x^x x's, and so on.
%C A321990 If the first two digits of a number are x,y, respectively, and if (x^(y-1))/y is a positive integer, then the number of the form xy1(...), where (...) is a sequence of digits whose product is (x^(y-1))/y, is a term. - _Michal Gren_, Nov 29 2018
%H A321990 Michal Gren, <a href="/A321990/b321990.txt">Table of n, a(n) for n = 1..10000</a>
%e A321990 6213 is a term since 6^2^1^3 = 6*2*1*3 = 36.
%e A321990 8^4 = 4096. 8*4 = 32. So 841 followed by any sequence of digits whose product is 4096/32 = 128 is in the sequence. - _David A. Corneth_, Nov 28 2018
%t A321990 aQ[n_] := Module[{digits = IntegerDigits[n]}, If[MemberQ[digits, 0], False, Power@@digits == Times@@digits]]; Select[Range[1000], aQ] (* for small terms, or: *) aQ[n_] := Module[{d=IntegerDigits[n]}, If[MemberQ[d, 0], Return[False]]; p = Times@@d; If[MemberQ[d, 1], If[d[[1]]==1, Return[p==1]]; d = d[[1 ;; FirstPosition[d, 1][[1]]-1]]]; Do[p = Log[d[[i]], p], {i,1,Length[d]}]; p==1]; Select[Range[1000], aQ] (* _Amiram Eldar_, Nov 24 2018 *)
%o A321990 (PARI) a007954(n) = my(d=digits(n)); vecprod(d);
%o A321990 f256229(n, pd)= my(p=1); until(!n\=10, p=(n%10)^p; if (p>pd, return (-p))); p;
%o A321990 isok(k) = my(pd = a007954(k)); pd == f256229(k, pd); \\ _Michel Marcus_, Nov 25 2018
%Y A321990 Cf. A001055, A007954, A256229.
%K A321990 nonn,base
%O A321990 1,2
%A A321990 _Michal Gren_, Nov 23 2018