A093301 -id:A093301 - cp's OEIS frontend

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

David W. Wilson

			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.

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

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

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

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

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}]

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, Jul 10 2012

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

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]

Edited by Robert G. Wilson v, Feb 24 2003

A094535 a(n) is the smallest integer m such that A039995(m)=n.

Original entry on oeis.org

1, 2, 13, 23, 113, 131, 137, 1013, 1031, 1273, 1237, 1379, 6173, 10139, 10193, 10379, 10397, 10937, 12397, 12379, 36137, 36173, 101397, 102371, 101937, 102973, 103917, 106937, 109371, 109739, 123797, 123917, 123719, 346137, 193719, 346173

Offset: 0

Farideh Firoozbakht, May 08 2004

			a(6) = 137 because 137 is the smallest number m such that A039995(m) = 6; the six numbers 3, 7, 13, 17, 37 & 137 are primes.
See also A205956 for a(100) = 39467139.

Giovanni Resta, Table of n, a(n) for n = 0..500 (first 101 terms from Reinhard Zumkeller)
Carlos Rivera, Puzzle 265. Primes embedded, The Prime Puzzles & Problems Connection.
Reinhard Zumkeller, Illustration of initial terms

Cf. A039995, A093301, A039997.

Cf. A205956.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a094535 n = a094535_list !! n
a094535_list = map ((+ 1) . fromJust . (`elemIndex` a039995_list)) [0..]
-- Reinhard Zumkeller, Feb 01 2012

cnt[n_] := Count[ PrimeQ@ Union[ FromDigits /@ Subsets[ IntegerDigits[n]]], True]; a[n_] := Block[{k = 1}, While[cnt[k] != n, k++]; k]; Array[a, 21, 0] (* Giovanni Resta, Jun 16 2017 *)

from sympy import isprime
from itertools import chain, combinations as combs, count, islice
def powerset(s): # nonempty subsets of s
    return chain.from_iterable(combs(s, r) for r in range(1, len(s)+1))
def A039995(n):
    ss = set(int("".join(s)) for s in powerset(str(n)))
    return sum(1 for k in ss if isprime(k))
def agen():
    adict, n = dict(), 0
    for k in count(1):
        v = A039995(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 36))) # Michael S. Branicky, Aug 07 2022

A039995(a(n)) = n and A039995(m) != n for m < a(n). - Reinhard Zumkeller, Feb 01 2012

A168169 Primes with d digits (d>0) which have more than 2d distinct primes as substrings.

Original entry on oeis.org

23719, 31379, 52379, 113171, 113173, 113797, 123719, 153137, 179719, 199739, 211373, 213173, 229373, 231197, 231379, 233113, 233713, 236779, 237331, 237619, 237971, 241973, 259397, 291373, 313739, 317971, 327193, 337397, 343373, 353173

Offset: 1

M. F. Hasler, Nov 28 2009

"Substrings" includes the whole number in itself.
This is a subsequence of A168167.
The least palindrome in this sequence is 9179719.

			The least number with d digits to have over 2d distinct prime substrings is the prime a(1)=23719, with 5 digits and #{2, 3, 7, 19, 23, 37, 71, 719, 2371, 3719, 23719} = 11.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A093301, A069490, A131648, A012884, A168167, A168168.

filter:= proc(n) local i,j,count,d,S,x,y;
  if not isprime(n) then return false fi;
  d:= ilog10(n)+1;
  count:= 0; S:= {};
  for i from 0 to d-1 do
    x:= floor(n/10^i);
    for j from i to d-1 do
      y:= x mod 10^(j-i+1);
      if not member(y,S) and isprime(y) then count:= count+1; S:= S union {y}; if count > 2*d then return true fi fi
  od od;
  false
end proc:
select(filter, [seq(i,i=1..10^6,2)]); # Robert Israel, Nov 11 2020

{forprime( p=1, default(primelimit), #prime_substrings(p) > #Str(p)*2 & print1(p", "))} /* see A168168 for prime_substrings() */

cp's OEIS Frontend

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

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A094535 a(n) is the smallest integer m such that A039995(m)=n.

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Haskell

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A168169 Primes with d digits (d>0) which have more than 2d distinct primes as substrings.

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Programs

Maple

PARI