A341702 -id:A341702 - cp's OEIS frontend

A341701 a(n) = largest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

-1, -1, 2, 3, 3, 5, -1, 7, -1, -1, 9, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 21, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 39, 47, 41, -1, -1, 47, 37, 53, -1, 43, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

A variant of A341717. a(82) = 1. Are there other n such that a(n) = 1?

Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then a(n) is odd, n-a(n) !== 2 (mod 3) and n+a(n) !== 0 (mod 3).

			a(4) = 3 since 43 is prime, a(25) = 13 since 25242322212019181716151413 is prime.

Cf. A052088, A052089, A341702, A341715, A341716, A341717.

from sympy import isprime
def A341701(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return m+1 if isprime(k) else -1

a(p) = p if and only if p is prime.

A341931 a(n) = smallest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 2, 3, 3, 5, -1, 3, -1, -1, 7, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 19, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 3, 47, 17, -1, -1, 47, 37, 41, -1, 27, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(n) <= A341701(n). a(82) = 1, are there any other n such that a(n) = 1?
Primes p such that a(p) < p: 7, 53, 73, 79, 89, 103, ...
n such that a(n) < A341701(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then a(n) is odd, n-a(n) !== 2 (mod 3) and n+a(n) !== 0 (mod 3).

			a(7) = 3 since 76543 is prime and 765432, 7654321 are not. a(10) = 7 since 10987 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341932.

from sympy import isprime
def A341931(n):
    k, m, r = n, n-1, n if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = m
        m -= 1
    return r

A341932 a(n) = largest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, -1, 4, -1, -1, 3, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 3, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 43, 0, 31, -1, -1, 4, 15, 12, -1, 28, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(82) = 81, are there any other n such that a(n) = n-1?
Primes p such that a(p) > 0: 7, 53, 73, 79, 89, 103, ...
n such that a(n) > A341702(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

			a(22) = 3 since 22212019 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341931.

from sympy import isprime
def A341932(n):
    k, m, r = n, n-1, 0 if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = n-m
        m -= 1
    return r

a(n) = n-A341931(n) >= A341702(n).

cp's OEIS Frontend

A341701 a(n) = largest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

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A341931 a(n) = smallest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

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A341932 a(n) = largest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

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