A107689 -id:A107689 - cp's OEIS frontend

A090537 Least n-digit prime using digit 3 once and rest all 1, or 0 if no such prime exists.

3, 13, 113, 0, 11113, 113111, 0, 11111131, 111111113, 0, 11111111113, 111113111111, 0, 31111111111111, 0, 0, 11111111113111111, 111111111111111131, 0, 11111111111111111131, 131111111111111111111, 0, 11111111111111111311111

Offset: 1

Amarnath Murthy, Dec 08 2003

Cf. A090535, A090536, A090538.

Without zeros, subsequence of A107689.

from sympy import isprime
def agen():
  digits = 0
  while True:
    for i in range(digits+1):
      t = int("1"*(digits-i) + "3" + "1"*i)
      if isprime(t): yield t; break
    else: yield 0
    digits += 1
g = agen()
print([next(g) for i in range(23)]) # Michael S. Branicky, Mar 13 2021

a(3n+1) = 0.

More terms from David Wasserman, Jan 03 2006

A179302 Indices of primes occurring in A034050.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 11, 12, 18, 19, 20, 29, 30, 31, 32, 36, 37, 39, 43, 55, 56, 61, 62, 65, 72, 77, 104, 142, 154, 169, 191, 197, 200, 229, 258, 267, 276, 375, 407, 408, 411, 412, 437, 440, 456, 539, 597, 607, 621, 629, 653, 660, 663, 724, 742, 743, 759, 760, 768

Offset: 0

Dmitry Kamenetsky, Jul 10 2010

nonn

Dmitry Kamenetsky, Table of n, a(n) for n = 0..1000

Cf. A034050, A107689, A179303.

Flatten[Position[Sort[Flatten[Table[FromDigits/@Permutations[Join[{3},PadRight[{},n,1]]],{n,0,50}]]],?PrimeQ]]-1 (* _Harvey P. Dale, Jul 16 2012 *)

A135048 Indices of primes with digit product = 3.

Original entry on oeis.org

2, 6, 11, 30, 32, 64, 1346, 1349, 1367, 10715, 12253, 26886, 733412, 733420, 733533, 734596, 6363182, 6363183, 6363289, 7437658, 503193257, 503193259, 503279746, 504057767, 1346029458, 4555878955, 12238593622, 1035905650780, 309353882444942, 2907021742443975

Offset: 1

Zak Seidov, Feb 11 2008

The corresponding primes are in A107689.

Amiram Eldar, Table of n, a(n) for n = 1..37 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A107611, A107612, A107689.

a(n) = primepi(A107689(n)).

a(26)-a(30) from Amiram Eldar, Sep 06 2024

A199982 Composite numbers with digital product = 3.

Original entry on oeis.org

1113, 1131, 1311, 3111, 13111, 31111, 111113, 111131, 111311, 1111113, 1111131, 1111311, 1113111, 1131111, 1311111, 3111111, 11111113, 11311111, 13111111, 31111111, 111111311, 111131111, 111311111, 113111111, 311111111, 1111111113, 1111111131, 1111111311

Offset: 1

Jaroslav Krizek, Nov 13 2011

Also composite numbers whose multiplicative digital root is 3. Complement of A107689 with respect to A034050.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A107689 (primes with digital product = 3).

Cf. A034050 (numbers with digital product = 3).

Table[Select[FromDigits[Permutations[PadRight[{3},n,1]]],CompositeQ],{n,4,10}]//Flatten//Sort (* Harvey P. Dale, May 31 2025 *)

Incorrect 111113111 removed by Sean A. Irvine, Jan 06 2025

A342221 Integers k, not congruent to 1 mod 3, such that there is no prime of length k using digit 3 once and rest all 1.

Original entry on oeis.org

15, 26, 32, 41, 68, 93, 99, 113, 116, 119, 134, 150, 158, 161, 170, 173, 176, 177, 179, 204, 213, 252, 257, 266, 284, 299, 305, 312, 320, 333, 353, 357, 374, 392, 402, 404, 419, 434, 443, 450, 491, 495, 506, 509, 513, 518, 527, 548, 551, 554, 570, 582, 584, 593

Offset: 1

Michel Marcus, Mar 14 2021

The "not congruent to 1 mod 3" condition is due to the fact that A090537(3*k+1) = 0.

Cf. A090537, A107689.

Subsequence of A007494.

from sympy import isprime
A342221_list = []
for k in range(1,10**3):
    if k % 3 != 1:
        m, l = (10**k-1)//9, 2
        for i in range(k):
            if isprime(m+l):
                break
            l *= 10
        else:
            A342221_list.append(k) # Chai Wah Wu, Mar 15 2021

More terms from Chai Wah Wu, Mar 15 2021

A346275 Number of n-digit primes with digital product = 3.

Original entry on oeis.org

1, 2, 3, 0, 3, 3, 0, 4, 4, 0, 5, 2, 0, 1, 0, 0, 1, 2, 0, 3, 1, 0, 2, 1, 0, 0, 1, 0, 4, 3, 0, 0, 1, 0, 4, 3, 0, 1, 6, 0, 0, 4, 0, 2, 1, 0, 2, 6, 0, 2, 2, 0, 5, 3, 0, 4, 3, 0, 3, 1, 0, 1, 3, 0, 1, 4, 0, 0, 2, 0, 3, 4, 0, 4, 1, 0, 2, 4, 0, 3, 2, 0, 1, 6, 0, 2, 3, 0, 3, 4, 0

Offset: 1

Bernard Schott, Jul 13 2021

Equivalently: Number of n-digit terms of A107689 that are primes.
This sequence is similar to A346274 where digital product = 7.
a(n) = n if n = 1 or n = 2 or n = 3.
Conjecture: for n >= 4, there is at least one composite number among the n-digit numbers with digital product = 3 <==> a(n) < n for n >= 4 (guess proposed by Derek Holton in reference).
Also a(n) = 0 for n = 15, 26, 32, 68, ...

			3 is prime, hence a(1) = 1.
13 and 31 are primes, hence a(2) = 2.
11113, 11131, 11311 are primes, but 13111 =  7*1873 and 31111 = 53*587, hence a(5) = 3.

Derek Holton, A Second Step to Mathematical Olympiad Problems, Vol. 7, Mathematical Olympiad Series, World Scientific, 2011, Section 8.14 - 69, page 287.

Cf A107689, A346274.

a[n_] := Count[(10^n - 1)/9 + 2*10^Range[0, n - 1], ?PrimeQ]; Array[a, 100] (* _Amiram Eldar, Jul 13 2021 *)

from sympy import isprime
def a(n): return sum(isprime(int('1'*(n-1-i)+'3'+'1'*i)) for i in range(n))
print([a(n) for n in range(1, 92)]) # Michael S. Branicky, Jul 13 2021

a(3*k+1) = 0 for k > 0.

cp's OEIS Frontend

A090537 Least n-digit prime using digit 3 once and rest all 1, or 0 if no such prime exists.

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A179302 Indices of primes occurring in A034050.

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A135048 Indices of primes with digit product = 3.

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A199982 Composite numbers with digital product = 3.

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A342221 Integers k, not congruent to 1 mod 3, such that there is no prime of length k using digit 3 once and rest all 1.

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A346275 Number of n-digit primes with digital product = 3.

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