A039994 -id:A039994 - cp's OEIS frontend

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

David W. Wilson

			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

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A039994, A079066, A178597.

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

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 *)

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)Charles R Greathouse IV, Apr 22 2015

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

Name corrected by David A. Corneth, Jul 06 2020

A372600 Least prime whose decimal representation has n distinct substrings which are also prime.

Original entry on oeis.org

2, 13, 23, 113, 131, 137, 1013, 1031, 1279, 1237, 1973, 6173, 10139, 10193, 10391, 10739, 10937, 12739, 12379, 36137, 100937, 101693, 102931, 102397, 103913, 106319, 106937, 109379, 123479, 123973, 125791, 123719, 361973, 852937, 1006937, 1019731, 1023173, 1021973, 1024379, 1027319

Offset: 1

Robert G. Wilson v, May 06 2024

First occurrence n of pi(a(n)) in A039994: 1, 6, 9, 30, 32, 33, 170, 173, 207, 203, 298, 804, ..., .

			A039994(1) = 1, is the first occurrence of 1 which is the index of the prime 2;
A039994(6) = 2, is the first occurrence of 2 and 6 is the index of the prime 13;
A039994(9) = 3, is the first occurrence of 3 and 9 is the index of the prime 23;
A039994(30) = 4, is the first occurrence of 4 and 30 is the index of the prime 113; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..688

Cf. A039994.

f[p_] := Count[ PrimeQ[ Union[ FromDigits@# & /@ Rest@ Subsets[ IntegerDigits@ p]]], True]; t[_] := 0; p = 2; While[p < 10000001, a = f@ p; If[ t[a] == 0, t[a] = p]; p = NextPrime@ p]; t /@ Range@ 56

cp's OEIS Frontend

A039996 Number of distinct primes embedded in prime(n) as substrings.

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