A107693 -id:A107693 - cp's OEIS frontend

A346274 Number of n-digit primes with digital product = 7.

1, 2, 0, 2, 2, 0, 3, 3, 0, 2, 1, 0, 1, 0, 0, 0, 2, 0, 3, 1, 0, 3, 1, 0, 2, 1, 0, 2, 3, 0, 2, 1, 0, 2, 2, 0, 3, 0, 0, 3, 0, 0, 2, 1, 0, 3, 3, 0, 4, 4, 0, 1, 2, 0, 4, 2, 0, 1, 2, 0, 1, 2, 0, 3, 3, 0, 2, 1, 0, 2, 2, 0, 1, 3, 0, 0, 3, 0, 1, 3, 0, 2, 8, 0, 1, 3, 0

Offset: 1

Bernard Schott, Jul 12 2021

Equivalently: Number of n-digit terms of A107693 that are primes.
This sequence is inspired by the 1st problem, submitted by USSR during the 31st International Mathematical Olympiad in 1990 at Beijing, but not used for the competition.
The problem was: Consider the n-digit numbers consisting of one '7' and n-1 '1'. For what values of n are all these numbers prime?
a(n) = n iff n = 1 or n = 2 (this is the answer to the Olympiad problem).
a(n) < n for n >= 3 <=> for n >= 3, there is always at least one composite number among the n-digit numbers with digital product = 7.
Steps of the proof by disjunction elimination (proof in Derek Holton in reference):
-> If n = 3*k, k>0, a(n) = 0 because the n-digit numbers with digital product = 7 are then divisible by 3.
-> If n = 4, 1117 and 1171 are primes, but 1711 = 29 * 59 and 7111 = 13 * 547; hence a(4) = 2 < 4.
-> If n > 4 and n <> 3k, there is always at least one n-digit number that is divisible by 7, these composites are in A346276.
Also, a(n) = 0 for n = 14, 16, 38, 41, 76, 104, 107, 110, 128, 134, 146, 152, 155, 164, 166, 178, 185, ... (comes from b-file in A107693).

			7 is prime, hence a(1) = 1.
17 and 71 are primes, hence a(2) = 2.
1117 and 1171 are primes, but 1711 = 29 * 59 and 7111 = 13 * 547; hence a(4) = 2.

Derek Holton, A Second Step to Mathematical Olympiad Problems, Vol. 7, Mathematical Olympiad Series, World Scientific, 2011, Section 8.2. USS 1 p. 260 and Section 8.14 Solutions pp 284-287.

Michael S. Branicky, Table of n, a(n) for n = 1..1000
Index to sequences related to Olympiads.

Cf. A107693, A346276.

a[n_] := Count[(10^n - 1)/9 + 6*10^Range[0, n - 1], ?PrimeQ]; Array[a, 100] (* _Amiram Eldar, Jul 12 2021 *)

a(n) = {my(s = 10^n\9); sum(i = 0, n-1, isprime(s + 6*10^i))} \\ David A. Corneth, Jul 12 2021

from sympy import isprime
def a(n):
    return sum(isprime(int('1'*(n-1-i) + '7' + '1'*i)) for i in range(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 12 2021

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

A179303 Indices of primes occurring in A034054.

Original entry on oeis.org

0, 1, 2, 6, 7, 10, 11, 23, 24, 25, 28, 29, 35, 51, 53, 64, 85, 137, 143, 172, 184, 185, 193, 233, 244, 245, 253, 305, 323, 331, 385, 397, 406, 417, 423, 472, 492, 499, 563, 567, 599, 601, 673, 692, 694, 780, 786, 792, 922, 930, 975, 1049, 1051, 1053, 1096, 1120

Offset: 0

Dmitry Kamenetsky, Jul 10 2010

nonn

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

Cf. A034054, A107693, A179302.

A346276 Composite m-digit numbers with m <> 3k and digital product = 7 that are divisible by 7.

Original entry on oeis.org

11711, 1111117, 7111111, 11171111, 1111711111, 11111111711, 11711111111, 1111111111117, 1111117111111, 7111111111111, 11111111171111, 11171111111111, 1111111111711111, 1111711111111111, 11111111111111711, 11111111711111111, 11711111111111111, 1111111111111111117, 1111111111117111111

Offset: 1

Bernard Schott, Jul 22 2021

Proposition: when m > 4 and m <> 3k, there is always at least one m-digit number that is divisible by 7 (proof in Derek Holton in reference), these composites are in this sequence (see A346274).
Every term has m = 5+6r, resp. (7+6r, 8+6r, 10+6r (r>=0)) digits and comes from the concatenation of a(1) = 11711, resp. (7, a(4) = 11171111, a(5) = 1111711111) with one or more strings of R_6 = 111111 = 7*15873 placed before and/or after a(1), resp. (7, a(4), a(5)).
The corresponding smallest term b(m) with m = 5+6r, resp. (7+6r, 8+6r, 10+6r (r>=0)) digits is b(m) = R_m + 6*10^k with k = 2, resp. (0, 4, 5) with R_m = A002275(m) is the repunit with a string of m 1's.
Alternatively, numbers viewed as strings over {1, 7} divisible by 7 are accepted by an automaton with 12 states, {0..6, 0'..6'} - {3, 3'}, with k (resp. k') representing remainder mod 7 where 7 has not (resp. has) been seen; transition function delta(k, 1) = 10*k+1 mod 7, delta(k, 7) = (10*k+7 mod 7)', delta(k', 1) = 10*k'+1 mod 7; start state 0; and accept state 0'. Accepted strings satisfy the regular expression (111111)*(7 + 11711 + 11171111 + 1111711111 + 111117111)(111111)*.  This sequence includes all terms matching (111111)*(7 + 11711 + 11171111 + 1111711111)(111111)* except the number 7 alone. - Michael S. Branicky, Jul 22 2021

			a(1) = 11711 because 11711 has 5 digits, a digital product = 7 and 11711 = 7^2 * 239.

Derek Holton, A Second Step to Mathematical Olympiad Problems, Vol. 7, Mathematical Olympiad Series, World Scientific, 2011, Section 8.2. USS 1 p. 260 and Section 8.14 Solutions pp. 284-287.

Cf. A107693, A346274.

seq[digmax_] := Module[{dig = Select[Range[5, digmax], Mod[#, 3] > 0 &], s = {}}, Do[s = Join[s, Select[(10^d - 1)/9 + 6*10^Range[0, d - 1], Divisible[#, 7] &]], {d, dig}]; s]; seq[19] (* Amiram Eldar, Jul 22 2021 *)

a(n) = { my(q = floor((-3+sqrt(1+8*n))/4), s = [1, 2, 1, 1] + [1, 1, 1, 1] * q, c, b = [2, 0, 4, 5]); n -= (2*q^2 + 3*q); for(i = 1, 4, if(s[i] < n, n-=s[i] , c = i; break; ) ); qd = ceil(6*q + 3*c/2+7/2); 10^qd\9 + 6*10^(6*(n-1)+b[c]) } \\ David A. Corneth, Jul 23 2021

from sympy import isprime
def auptod(digs):
  return [t for t in (int('1'*(m-1-i) + '7' + '1'*i) for m in range(5, digs+1) if m%3 != 0 for i in range(m)) if t%7 == 0]
print(auptod(19)) # Michael S. Branicky, Jul 22 2021

A201021 Composite numbers whose multiplicative digital root is 7.

Original entry on oeis.org

117, 171, 711, 1711, 7111, 11711, 17111, 71111, 111117, 111171, 111711, 117111, 171111, 711111, 1111117, 1111171, 1711111, 7111111, 11111711, 11117111, 11171111, 11711111, 17111111, 111111117, 111111171, 111111711, 111117111, 111171111, 111711111, 117111111, 171111111

Offset: 1

Jaroslav Krizek, Nov 25 2011

Also composite numbers whose product of digits is 7.

Complement of A107693 with respect to A034054. ~

Cf. A107693 (primes whose multiplicative digital root is 7), A034054 (numbers whose multiplicative digital root is 7).

Flatten[Table[Select[FromDigits/@Permutations[PadRight[{7},n,1]],CompositeQ],{n,9}]]//Sort (* Harvey P. Dale, Mar 29 2023 *)

Number 171 is in sequence because 1*7*1=7.

cp's OEIS Frontend

A346274 Number of n-digit primes with digital product = 7.

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A179303 Indices of primes occurring in A034054.

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A346276 Composite m-digit numbers with m <> 3k and digital product = 7 that are divisible by 7.

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PARI

Python

A201021 Composite numbers whose multiplicative digital root is 7.

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Author

Keywords

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Programs

Mathematica

Formula