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A288818 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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,23

COMMENTS

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.

LINKS

Hans Havermann, Table of n, a(n) for n = 1..10000

Hans Havermann, Asterisks and circumflexes

StackExchange, Generating prime factorizations from an integer's digits by inserting * and ^ into the digit sequence

EXAMPLE

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A288819 (records), A288820 (records' position), A080670.

Sequence in context: A205555 A165913 A256253 * A276988 A084642 A271418

Adjacent sequences:  A288814 A288816 A288817 * A288819 A288820 A288821

KEYWORD

nonn,base

AUTHOR

Hans Havermann, Jun 17 2017

STATUS

approved

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Last modified July 27 04:53 EDT 2017. Contains 289841 sequences.