A039997 Number of distinct primes which occur as substrings of the digits of n.

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 2, 0, 1, 0, 2, 0, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 0, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 2, 0, 0, 1, 2, 2, 3, 1, 2, 1, 1, 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

Offset

1,13

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(A062115(n)) = 0; a(A093301(n)) = n and a(m) <> n for m < A093301(n). - Reinhard Zumkeller, Jul 16 2007

a(A163753(n)) > 0; a(A205667(n)) = 1. [Reinhard Zumkeller, Jan 31 2012]

Example

a(22) = 1 because 22 has two substrings which are prime but they are identical. a(103) = 2, since the primes 3 and 103 occur as substrings.

Maple

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):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 09 2022

Mathematica

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[ Union[d]], True]]; Table[ a[n], {n, 1, 105}]

Prog

(Haskell)
import Data.List (isInfixOf)
a039997 n = length [p | p <- takeWhile (<= n) a000040_list,
                        show p `isInfixOf` show n]
a039997_list = map a039997 [1..]
-- Reinhard Zumkeller, Jan 31 2012
(PARI) dp(n)=if(n<12, return(if(isprime(n), [n], []))); my(v=vecsort(select(isprime, eval(Vec(Str(n)))), , 8), t); while(n>9, if(gcd(n%10, 10)>1, n\=10; next); t=10; while((t*=10)<n*10, if(isprime(n%t), v=concat(v, n%t))); v=vecsort(v, , 8); n\=10); v
a(n)=#dp(n) \\ Charles R Greathouse IV, Jul 10 2012
(Python)
from sympy import isprime
def a(n):
    s = str(n)
    ss = (int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))
    return len(set(k for k in ss if isprime(k)))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 07 2022

Crossrefs

Different from A039995 after the 100th term. Cf. A035232.

Sequence in context: A361414 A193403 A354272 * A039995 A035232 A359269

Adjacent sequences: A039994 A039995 A039996 * A039998 A039999 A040000

Keyword

nonn,base

Author

David W. Wilson

Extensions

Edited by Robert G. Wilson v, Feb 24 2003

Status

approved