A002371 -id:A002371 - cp's OEIS frontend

A347225 Lesser of twin primes (A001359) being both half-period primes (A097443).

197, 599, 881, 1277, 1997, 2081, 2237, 2801, 2999, 3359, 4721, 5279, 5879, 6197, 6959, 7877, 8837, 9239, 9719, 12161, 12239, 13721, 17921, 17957, 18521, 21839, 22637, 24917, 28277, 30557, 31319, 31721, 32117, 32441, 32717, 34757, 35081, 35279, 35837, 38921, 39239, 39839

Offset: 1

Lamine Ngom, Aug 24 2021

A proper subset of both A001359 and A097443.
Number of terms < 10^k: 0, 0, 3, 19, 86, 516, 3686, 27834, 216758, 1739358, …
A243096 provides lesser of twin primes being both full reptend primes (A001913, A006883): in other words, lesser of twin primes whose periods difference is 2.
This sequence lists lesser of twin primes whose periods difference is 1. Equivalently, these twin primes are both half-period primes (A097443).
The twin primes conjecture being true should imply that these two sequences are infinite.
Surprisingly, apart from 1 and 2, for any other value of k integer, it appears that the sequence "lesser of twin primes whose periods difference is k" is empty or contains at most two terms (no counterexample found for twin primes below 10^9).

			The decimal expansion 1/p for the prime p = 1277 has a periodic part length equal to (p-1)/2. 1277 is thus a half-period prime. The same applies for p + 2 = 1279 (prime). Hence 1277 is in sequence.

Cf. A002371, A001359, A097443, A243096, A001913, A006883

select(t -> isprime(t) and isprime(t + 2) and numtheory:-order(10, t) = (t - 1)/2 and numtheory:-order(10, t + 2) = (t + 1)/2, [seq(t, t = 3 .. 40000, 2)]);

a(n) is congruent to {11, 17, 29} mod 30.

A347226 Safe primes (A005385) that are half-period primes (A097443).

Original entry on oeis.org

83, 107, 227, 347, 359, 467, 479, 563, 587, 719, 839, 1187, 1283, 1307, 1319, 1439, 1523, 1907, 2027, 2039, 2879, 2963, 2999, 3119, 3203, 3467, 3803, 3947, 4079, 4283, 4547, 4679, 4787, 4799, 4919, 5387, 5399, 5483, 5507, 5639, 5879, 6599, 6719, 6827, 7079, 7187, 7523

Offset: 1

Lamine Ngom, Aug 24 2021

Apart from 5 and 11, a safe prime p is necessarily either a full reptend prime (A001913) or a half-period prime (A097443) since (p-1) is semiprime: 2*A005384(n) (Sophie Germain primes).
Safe primes being full reptend primes are listed in A000353.
a(n) is of the form 100*k + 10*{0, 2, 4, 6, 8} + {3, 7} or 100*k + 10*{1, 3, 5, 7, 9} + 9.
Number of terms < 10^k: 0, 1, 11, 56, 343, 2138, 15267, 114847, 886907, 7079602, ...

			(107-1)/2 = 53 is a prime, and the periodic part of the decimal expansion of 1/107 is of length 53.
Hence the safe prime 107 is in the sequence.

Cf. A005385, A097443, A005384, A000353, A001913, A002371, A006883.

select(t -> isprime(t) and isprime((t - 1)/2) and numtheory:-order(10, t) = (t - 1)/2, [seq(t, t = 3 .. 10000, 2)]);

Select[Prime@Range@1000,PrimeQ[(#-1)/2]&&Length[First@@RealDigits[1/#]]==(#-1)/2&] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

A005385 INTERSECTION A097443.
a(n) == {17, 23, 29} mod 30.
a(n) == 11 (mod 12). - Hugo Pfoertner, Aug 24 2021

A381578 For n > 0, for k > n, a(n) is the least k such that the pre-period and first period of the decimal expansion of n/k contains every digit of n at least as many times it is contained in n.

Original entry on oeis.org

6, 7, 8, 7, 7, 9, 8, 9, 10, 17, 17, 14, 17, 17, 17, 17, 19, 19, 21, 21, 23, 23, 29, 28, 27, 27, 28, 29, 32, 31, 34, 34, 38, 38, 38, 38, 38, 39, 46, 43, 42, 43, 46, 46, 46, 47, 49, 49, 51, 51, 53, 53, 57, 56, 57, 57, 58, 59, 61, 61, 62, 63, 65, 65, 68, 67, 68, 69

Offset: 1

Ctibor O. Zizka, Feb 28 2025

What is the density of a(n) = prime in this sequence ?

			n = 1:
  1/2 = 0.500...
  1/3 = 0.33...
  1/4 = 0.2500...
  1/5 = 0.200...
  1/6 = 0.166... contains the digit 1, thus a(1) = 6.
n = 2:
  2/3 = 0.66...
  2/4 = 0.500...
  2/5 = 0.400...
  2/6 = 0.33...
  2/7 = 0.285714285714...contains the digit 2, thus a(2) = 7.

Cf. A002371, A007732.

a[n_] := Module[{k = n+1, r = Range[0, 9]}, While[! AllTrue[Count[Flatten[RealDigits[n/k][[1]]], #] & /@ r - DigitCount[n, 10, r], # >= 0 &], k++]; k]; Array[a, 100] (* Amiram Eldar, Feb 28 2025 *)

A386519 Index of the smallest prime p such that the number of digits L in the repeating decimal period of 1/p equals the n-th prime.

Original entry on oeis.org

5, 12, 13, 52, 2431, 16, 153888, 27417323062119920, 223378173194137397198, 452, 406, 150886, 23, 40, 2153717, 28, 92971458509, 130, 40998

Offset: 1

Jean-Marc Rebert, Jul 24 2025

In general, for (q,2*5)=1, the length of the period of 1/q is equal to the multiplicative order of 10 modulo q, which is the smallest k such that 10^k == 1 (mod q). It follows that a(n) must be a prime divisor of 10^prime(n)-1. Hence, apart from a(2), we have prime(a(n)) = A147555(n) and a(20) is the index of the prime 241573142393627673576957439049. - Giovanni Resta, Jul 24 2025

			a(1) = 5, since the 5th prime, p = 11, has a repeating decimal period of length L = 2, and 2 = prime(1). There is no smaller prime for which the period length equals the 1st prime.
 n      a(n)         p  L
 1         5        11  2
 2        12        37  3
 3        13        41  5
 4        52       239  7
 5      2431     21649 11
 6        16        53 13
 7    153888   2071723 17

Cf. A072859, A002371, A249330, A147555.

a(8)-a(19) from Giovanni Resta, Jul 24 2025

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A347225 Lesser of twin primes (A001359) being both half-period primes (A097443).

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A347226 Safe primes (A005385) that are half-period primes (A097443).

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A381578 For n > 0, for k > n, a(n) is the least k such that the pre-period and first period of the decimal expansion of n/k contains every digit of n at least as many times it is contained in n.

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A386519 Index of the smallest prime p such that the number of digits L in the repeating decimal period of 1/p equals the n-th prime.

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