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A288818
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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.
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5
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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, 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, 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
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OFFSET
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1,23
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COMMENTS
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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.
Excepting 1, a(n) is the number of occurrences of n in A080670.
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LINKS
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EXAMPLE
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a(12) = 0 because there are no valid solutions.
a(1111) = 1 because 11^11 is the only valid statement.
a(7013) = 2 because 7013 and 701^3 are the only solutions.
a(2353797) = 75 because there are 75 valid solutions.
a(13^532*3853*96179) = 1593300019. There are 1593300019 ways of creating valid factorization statements using this 602-digit integer.
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MATHEMATICA
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See the StackExchange link. (* or *)
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 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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