A039995 -id:A039995 - cp's OEIS frontend

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

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

A035232 Number of substrings of n which are primes (counted with multiplicity).

Original entry on oeis.org

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

Offset: 1

Erich Friedman

No leading 0's allowed in substrings.

			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.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A039997, A039995.

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

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)))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 07 2022

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

Edited by Robert G. Wilson v, Feb 24 2003

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

Original entry on oeis.org

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

A205956 Distinct primes occurring as not necessarily consecutive subsequences in decimal representation of 39467139.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 97, 139, 313, 347, 349, 367, 373, 379, 397, 419, 439, 461, 463, 467, 479, 613, 619, 673, 719, 739, 919, 941, 947, 967, 971, 3413, 3461, 3463, 3467, 3469, 3613, 3671, 3673, 3719, 3739, 3919, 3943, 3947

Offset: 1

Reinhard Zumkeller, Feb 02 2012

39467139 is the smallest number containing exactly 100 primes: A094535(100) = 39467139, A039995(39467139) = 100; 39467139 itself is not prime: 39467139 = 3*53*89*2789.

			a(10) = 61        <- ___6_1__;
a(20) = 367       <- 3__67___;
a(30) = 613       <- ___6_13_;
a(40) = 3413      <- 3_4__13_;
a(50) = 3919      <- 39___1_9;
a(60) = 9419      <- _94__1_9;
a(70) = 9719      <- _9__71_9;
a(80) = 39439     <- 394___39;
a(90) = 346139    <- 3_46_139;
a(100) = 3946139  <- 3946_139.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100 (full sequence)

Cf. A010051.

import Data.List (subsequences, nub, sort)
a205956 n = a205956_list !! (n-1)
a205956_list = sort $ filter ((== 1) . a010051) $
                      nub $ map read (tail $ subsequences "39467139")

A039994 Number of primes embedded in decimal expansion of prime(n) as substrings.

Original entry on oeis.org

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, 2, 3, 3, 2, 4, 4, 5, 6, 4, 2, 3, 4, 3, 4, 6, 5, 2, 3, 4, 5, 2, 3, 4, 3, 3, 4, 5, 3, 3, 4, 4, 3, 4, 3, 2, 5, 5, 4, 4, 4, 6, 3, 4, 5, 2, 4, 4, 5, 5, 5, 3, 3, 5, 2, 1, 3, 3, 5, 3, 3, 3, 1, 4, 3, 3, 4, 4, 3, 2, 1, 4, 3, 3, 6, 3, 4, 3, 4, 3, 4

Offset: 1

David W. Wilson

			a(26) = 2 since prime(26) = 101 and "101" has two prime subsequences, 101 and 11.

Cf. A039995.

Different from A039996.

a(n) = A039995(prime(n)).

Definition and example clarified by N. J. A. Sloane, Jun 03 2024

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

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A035232 Number of substrings of n which are primes (counted with multiplicity).

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

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A205956 Distinct primes occurring as not necessarily consecutive subsequences in decimal representation of 39467139.

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Haskell

A039994 Number of primes embedded in decimal expansion of prime(n) as substrings.

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Author

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Extensions