A205956 -id:A205956 - cp's OEIS frontend

A039995 Number of distinct primes which occur as subsequences of the sequence of 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, 2, 1, 3, 0, 1

Offset: 1

David W. Wilson

a(n) counts subsequences of digits of n which denote primes.

			a(103) = 3; the 3 primes are 3, 13 and 103.

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

A039997 counts only the primes which occur as substrings, i.e. contiguous subsequences. Cf. A035232.

Cf. A010051.

import Data.List (subsequences, nub)
a039995 n = sum $
   map a010051 $ nub $ map read (tail $ subsequences $ show n)
-- Reinhard Zumkeller, Jan 31 2012

cnt[n_] := Module[{d = IntegerDigits[n]}, Length[Union[Select[FromDigits /@ Subsets[d], PrimeQ]]]]; Table[cnt[n], {n, 105}] (* T. D. Noe, Jan 31 2012 *)

from sympy import isprime
from itertools import chain, combinations as combs
def powerset(s): # nonempty subsets of s
    return chain.from_iterable(combs(s, r) for r in range(1, len(s)+1))
def a(n):
    ss = set(int("".join(s)) for s in powerset(str(n)))
    return sum(1 for k in ss if isprime(k))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 07 2022

a(A094535(n)) = n and a(m) < n for m < A094535(n); A039995(39467139) = 100, cf. A205956. - Reinhard Zumkeller, Feb 01 2012

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

cp's OEIS Frontend

A039995 Number of distinct primes which occur as subsequences of the sequence of digits of n.

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