A052088 -id:A052088 - cp's OEIS frontend

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.

-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).

A085720 Start of a run of 7 successive numbers which when concatenated form a prime.

Original entry on oeis.org

7, 37, 157, 185, 187, 271, 301, 355, 475, 485, 523, 533, 577, 611, 653, 661, 667, 731, 733, 755, 761, 791, 853, 911, 913, 937, 983, 1085, 1111, 1187, 1205, 1253, 1397, 1417, 1585, 1631, 1655, 1685, 1697, 1711, 1723, 1841, 1907, 1975, 2035, 2077, 2105, 2185

Offset: 1

Zak Seidov, Jun 27 2003

Concatenation of three and six successive numbers are always composite.

Primes as concatenation of two, four and five successive numbers are in A030458, A030471, A052087, A052088, A052089.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030458, A030471, A052087, A052088, A052089.

f[n_] := FromDigits[ Flatten[ Table[ IntegerDigits[i], {i, n, n + 6}]]]; Select[ Range[2190], PrimeQ[ f[ # ]] & ]
Select[Range[2500],PrimeQ[FromDigits[Flatten[IntegerDigits/@Range[#,#+6]]]]&] (* Harvey P. Dale, Aug 15 2021 *)

Edited by Robert G. Wilson v, Jun 28 2003

Edited by Charles R Greathouse IV, Apr 24 2010

A317744 Prime numbers which result as a concatenation of a decimal number and its binary representation.

Original entry on oeis.org

11, 311, 5101, 131101, 2511001, 37100101, 51110011, 59111011, 731001001, 931011101, 971100001, 1191110111, 12910000001, 13110000011, 13710001001, 15310011001, 17310101101, 19311000001, 21311010101, 21511010111, 24711110111, 25511111111, 319100111111

Offset: 1

Philip Mizzi, Aug 05 2018

The decimal number plus its Hamming weight A000120 must not be divisible by 3. - M. F. Hasler, Apr 05 2024

			11 is in the sequence because the binary representation of 1 is 1 and the concatenation of 1 and 1 gives 11, which is prime.
931011101 is in the sequence because it is the concatenation of 93 and 1011101 (the binary representation of 93) and is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000040, A030458, A052087, A052088, A052089, A127421, A236551, A287300, A287310.

Select[Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[n,2]]],{n,400}],PrimeQ] (* Harvey P. Dale, Jul 15 2020 *)

nb(n)=fromdigits(concat(n,binary(n)))
A317744_upto(N=666)=[p|n<-[1..N], is/*pseudo*/prime(p=nb(n))] \\ M. F. Hasler, Apr 05 2024

from sympy import isprime
def nbinn(n): return int(str(n)+bin(n)[2:])
def ok(n): return isprime(nbinn(n))
def aprefixupto(p): return [nbinn(k) for k in range(1, p+1, 2) if ok(k)]
print(aprefixupto(319)) # Michael S. Branicky, Dec 27 2020

cp's OEIS Frontend

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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A085720 Start of a run of 7 successive numbers which when concatenated form a prime.

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A317744 Prime numbers which result as a concatenation of a decimal number and its binary representation.

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Crossrefs

Programs

Mathematica

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Python