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A035232
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Number of substrings of n which are primes (counted with multiplicity).
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62
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0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 2, 0, 1, 0, 2, 0, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 0, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 2, 0, 0, 1, 2, 2, 3, 1, 2, 1, 2, 1, 2, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 1, 1, 2, 0, 1
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OFFSET
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1,13
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COMMENTS
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No leading 0's allowed in substrings.
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LINKS
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FORMULA
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Trivial upper bound: a(n) <= binomial(floor(log(n)/log(10)+2), 2) ~ k*log^2 n with k = 0.09430584850580... = 1/log(10)^2/2. - Charles R Greathouse IV, Nov 15 2022
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EXAMPLE
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The primes occurring as substrings of 37 are 3, 7, 37, so a(37) = 3.
a(22) = 2, since the prime 2 occurs twice as a substring.
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MAPLE
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a:= n-> (s-> nops(select(t -> t[1]<>"0" and isprime(parse(t)),
[seq(seq(s[i..j], i=1..j), j=1..length(s))])))(""||n):
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MATHEMATICA
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a[n_] := Block[{s = IntegerDigits[n], c = 0, d = {}}, l = Length[s]; t = Flatten[ Table[ Take[s, {i, j}], {i, 1, l}, {j, i, l}], 1]; k = l(l + 1)/2; While[k > 0, If[ t[[k]][[1]] != 0, d = Append[d, FromDigits[ t[[k]] ]]]; k-- ]; Count[ PrimeQ[d], True]]; Table[ a[n], {n, 1, 105}]
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PROG
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(Python)
from sympy import isprime
def a(n):
s = str(n)
ss = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1))
return sum(1 for w in ss if w[0] != "0" and isprime(int(w)))
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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