A039996 Number of distinct primes embedded in prime(n) as substrings.
1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 2, 3, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 1, 4, 3, 4, 5, 3, 1, 2, 3, 2, 3, 5, 4, 1, 2, 3, 4, 2, 3, 4, 3, 3, 4, 4, 3, 3, 4, 3, 2, 4, 3, 2, 4, 4, 3, 4, 4, 5, 3, 4, 4, 2, 4, 4, 4, 5, 5, 3, 3, 4, 1, 1, 3, 2, 4, 3, 3, 3, 1, 3, 2, 2, 3, 4, 2, 1, 1, 3, 2, 3, 5, 3, 4, 3, 3, 2, 4
Offset: 1
Examples
a(26) = 1 since the only prime substring of "101" is 101. a(48) = 4 since the only distinct prime substrings of "223" are 2, 3, 23, 223. - _David A. Corneth_, Jul 06 2020
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000
Programs
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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))})))(""||(ithprime(n))): seq(a(n), n=1..105); # Alois P. Heinz, Jul 29 2025 -
Mathematica
f[n_] := Block[{id = IntegerDigits@ Prime@n, len = Floor[ Log[10, Prime@n] + 1]}, Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[id, k, 1], {k, len}], 1]], True]]; Array[f, 105] (* Robert G. Wilson v, Jun 28 2010 *) -
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)
Formula
a(n) = A039997(prime(n)).
a(n) <= A039994(n). - Charles R Greathouse IV, Apr 22 2015
a(n) = A079066(n) + 1. - Alois P. Heinz, Jul 29 2025
Extensions
Name corrected by David A. Corneth, Jul 06 2020