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%I #18 Jun 22 2017 16:59:26
%S 0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,3,1,2,1,2,1,2,0,1,1,1,1,
%T 2,1,3,1,1,0,1,0,1,0,0,0,1,0,0,0,0,1,2,1,1,1,2,1,2,0,1,0,0,0,0,0,1,0,
%U 0,0,1,1,2,1,1,1,1,1,2,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0
%N Number of ways in which one can insert * and ^ into the decimal digits of n to create a valid (see comments) base-ten factorization statement.
%C A base-ten factorization statement is valid when it is the product of base-ten powers of (left to right) strictly increasing base-ten primes. A single prime (with or without an exponent) is acceptable. No prime and no exponent may begin with a zero. No exponent may be equal to one.
%C Excepting 1, a(n) is the number of occurrences of n in A080670.
%H Hans Havermann, <a href="/A288818/b288818.txt">Table of n, a(n) for n = 1..10000</a>
%H Hans Havermann, <a href="http://gladhoboexpress.blogspot.ca/2017/06/asterisks-and-circumflexes.html">Asterisks and circumflexes</a>
%H StackExchange, <a href="https://mathematica.stackexchange.com/questions/148155/generating-prime-factorizations-from-an-integers-digits-by-inserting-and-in">Generating prime factorizations from an integer's digits by inserting * and ^ into the digit sequence</a>
%e a(12) = 0 because there are no valid solutions.
%e a(1111) = 1 because 11^11 is the only valid statement.
%e a(7013) = 2 because 7013 and 701^3 are the only solutions.
%e a(2353797) = 75 because there are 75 valid solutions.
%e a(13^532*3853*96179) = 1593300019. There are 1593300019 ways of creating valid factorization statements using this 602-digit integer.
%t See the StackExchange link. (* or *)
%t ric[d_, lp_] := Block[{p, e, i, j, n = Length@d}, If[n == 0, cnt++, If[d[[1]] > 0, Do[p = FromDigits@ Take[d, i]; If[p > lp && PrimeQ@p, ric[Take[d, i - n], p]; Do[e = Take[d, {i + 1, j}]; If[e[[1]] > 0 && e != {1}, ric[Take[d, j - n], p]], {j, i+1, n}]], {i, n}]]]]; a[n_] := (cnt = 0; ric[ IntegerDigits@ n, 1]; cnt); Array[a, 100] (* _Giovanni Resta_, Jun 19 2017 *)
%Y Cf. A288819 (records), A288820 (records' position), A080670.
%K nonn,base
%O 1,23
%A _Hans Havermann_, Jun 17 2017
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