A154764 -id:A154764 - cp's OEIS frontend

A068690 Primes such that all digits (except perhaps the least significant digit) are even.

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 223, 227, 229, 241, 263, 269, 281, 283, 401, 409, 421, 443, 449, 461, 463, 467, 487, 601, 607, 641, 643, 647, 661, 683, 809, 821, 823, 827, 829, 863, 881, 883, 887, 2003, 2027, 2029, 2063, 2069, 2081, 2083, 2087

Offset: 1

Amarnath Murthy and Joseph L. Pe, Mar 03 2002

Essentially the same as A154764.

			2 is in the sequence even though the least significant digit of 2 is even instead of odd. - _David A. Corneth_, Sep 18 2019

Zak Seidov, Table of n, a(n) for n = 1..10775 (all terms < 10^8.)

Cf. A030096, A154764.

a068690 n = a068690_list !! (n-1)
a068690_list = filter (all (`elem` "02468") . init . show) a000040_list
-- Reinhard Zumkeller, Apr 28 2014

(*returns true if all but the last digit of n is even, false o.w.*) f[n_] := Module[{a, l, i, r = True}, a = IntegerDigits[n]; l = Length[a]; For[i = 1, i < l, i++, If[Mod[a[[i]], 2] == 1, r = False; Break[ ]]]; r]; Select[Range[1, 4*10^3], PrimeQ[ # ] && f[ # ] &]
m = 7; Prepend[Reap[Do[If[PrimeQ[fd = FromDigits[{a[1], a[2], a[3], a[4], a[5], a[6], a[m]}]], Sow[fd]], {a[1], 0, 8, 2}, {a[2], 0, 8, 2}, {a[3], 0, 8, 2}, {a[4], 0, 8, 2}, {a[5], 0, 8, 2}, {a[6], 0, 8, 2}, {a[m], 1, 9, 2}]][[2, 1]], 2] (* all terms < 10^8. Zak Seidov, Jan 29 2013 *)
Select[ Prime@ Range[10000], ContainsAll[{0, 2, 4, 6, 8}, Most@ IntegerDigits[#]] &] (* From version 10. Mikk Heidemaa, Feb 08 2016 *)
Select[Prime[Range[400]],AllTrue[Most[IntegerDigits[#]],EvenQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 18 2019 *)

Definition rephrased by N. J. A. Sloane, Dec 11 2007

A104638 Number of odd digits in n-th prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Zak Seidov, Mar 18 2005

The only zero is the first term. Sequence is unbounded. - Zak Seidov, Jan 12 2016
From Robert Israel, Jan 12 2016: (Start)
For any N, the asymptotic density of terms >= N is 1.
On the other hand, a(n) = 2 if prime(n) is in A159352, which is conjectured to be infinite.
Record values: a(2) = 1, a(5) = 2, a(30) = 3, a(187) = 4, a(1346) = 5, a(10545) = 6, a(86538) = 7, a(733410) = 8.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A104637, A159352.

Cf. A154764 (1 odd digit), A155071 (2 odd digits), A030096 (all digits odd).

seq(nops(select(type, convert(ithprime(i),base,10),odd)),i=1..100); # Robert Israel, Jan 12 2016
# alternative
A104638 := proc(n)
    local a,dgs,d ;
    ithprime(n) ;
    dgs := convert(%,base,10) ;
    a := 0 ;
    for d in dgs do
        a := a+modp(d,2) ;
    end do:
    a ;
end proc:
seq(A104638(n),n=1..40) ; # R. J. Mathar, Jul 13 2025

Table[Count[IntegerDigits[Prime[n]],?OddQ],{n,100}] (* _Harvey P. Dale, Jan 22 2012 *)
Table[Total[Mod[IntegerDigits[Prime[n]], 2]], {n, 100}] (* Vincenzo Librandi, Jan 13 2016 *)

a(n)=vecsum(digits(prime(n)%2)) \\ Zak Seidov, Jan 12 2016

a(n) = A196564(A000040(n)). - Michel Marcus, Oct 05 2013

A320256 k-digit primes with the same even digit repeated k-1 times and a single odd digit.

Original entry on oeis.org

3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 223, 227, 229, 443, 449, 661, 881, 883, 887, 2221, 4441, 4447, 6661, 8887, 22229, 44449, 88883, 444443, 444449, 666667, 888887, 22222223, 66666667, 88888883, 222222227, 444444443, 666666667, 888888883, 888888887

Offset: 1

Enrique Navarrete, Oct 08 2018

For the resulting number to be prime, the rightmost digit must be the odd one. - Michel Marcus, Oct 11 2018

			3, 5, 7 are in the sequence for k = 1.
229 is in the sequence because it is a 3-digit prime with the first 3-1 digits repeating even (2) and the last digit odd (9). - _David A. Corneth_, Oct 10 2018

Alois P. Heinz, Table of n, a(n) for n = 1..132

Cf. A055558, A068690, A105975, A141311, A154764.

Join[{3, 5, 7}, Select[Flatten@ Table[{1, 3, 7, 9} + 10 FromDigits@ ConstantArray[k, n], {n, 9}, {k, Range[2, 8, 2]}], PrimeQ]] (* Michael De Vlieger, Oct 31 2018 *)

first(n) = {n = max(n, 3); my(t = 3, res = List([3, 5, 7])); print1("3, 5, 7, "); for(i=1, oo, k=(10^i - 1) / 9; forstep(f = 2, 8, 2, forstep(d=1, 9, 2, c = 10 * f * k + d; if(isprime(c), print1(c", "); listput(res, c); t++; if(t>=n, return(res))))))} \\ David A. Corneth, Oct 10 2018

More terms from Michel Marcus, Oct 10 2018

A359813 Number of primes < 10^n with exactly one odd decimal digit.

Original entry on oeis.org

3, 12, 45, 171, 619, 2560, 10774, 46708, 202635, 904603, 4073767, 18604618, 85445767, 395944114, 1837763447, 8600149593

Offset: 1

Zhining Yang, Jan 14 2023

a(n) =~ Pi(10^n)/2^(n-1). Robert G. Wilson v, Feb 04 2023

			a(2)=12 as there are 12 primes less than 100 with exactly one odd decimal digit: 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89.

Cf. A030096, A068690, A154764, A358685, A358690.

c=1; k=0; lst={}; f[n_] := Block[{e = 10 FromDigits[2 IntegerDigits[n, 5]]}, Length@ Select[e + {1, 3, 7, 9}, PrimeQ]];
Do[ While[k< 5^n, c+=f@k; k++]; Print[c], {n, 0, 16}] (* Robert G. Wilson v, Feb 04 2023 *)

from sympy import isprime
from itertools import  product
def a(n):
    c=3
    if n==1:return(c)
    x=[[1,7],[1,3,7,9],[3,9],'2468','02468']
    for k in range(2,n+1):
        for f in x[3]:
            for m in product(x[4], repeat=k-2):
                s = int(f+"".join(m))*10
                t=s%3
                for last in x[t]:
                    if isprime(s+last):
                        c+= 1
    return(c)
print([a(n) for n in range(1,7)])

from sympy import primerange
def a(n):
    p=list(primerange(3,10**n))
    return(sum(1 for k in p if sum(str(k).count(d) for d in '13579')==1))
print([a(n) for n in range(1,7)])

a(16) from Robert G. Wilson v, Feb 04 2023

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A068690 Primes such that all digits (except perhaps the least significant digit) are even.

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A104638 Number of odd digits in n-th prime.

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A320256 k-digit primes with the same even digit repeated k-1 times and a single odd digit.

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A359813 Number of primes < 10^n with exactly one odd decimal digit.

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