cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A362117 Concatenation of first n numbers in base 5.

Original entry on oeis.org

1, 12, 123, 1234, 123410, 12341011, 1234101112, 123410111213, 12341011121314, 1234101112131420, 123410111213142021, 12341011121314202122, 1234101112131420212223, 123410111213142021222324, 12341011121314202122232430, 1234101112131420212223243031
Offset: 1

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Comments

When regarded as decimal numbers, the first prime in this sequence is a(6) = 12341011.
a(8) is also prime and for n <= 2000, a(n) is not prime except for n = 6 or 8. - Chai Wah Wu, Apr 19 2023

Crossrefs

Programs

  • Mathematica
    A362117[n_]:=FromDigits[Flatten[IntegerDigits[Range[n],5]]];Array[A362117,20] (* Paolo Xausa, Nov 27 2023 *)
  • Python
    from gmpy2 import digits
    def A362117(n): return int(''.join(digits(n,5) for n in range(1,n+1))) # Chai Wah Wu, Apr 19 2023

A362119 Concatenate the base-6 strings for 1,2,...,n.

Original entry on oeis.org

1, 12, 123, 1234, 12345, 1234510, 123451011, 12345101112, 1234510111213, 123451011121314, 12345101112131415, 1234510111213141520, 123451011121314152021, 12345101112131415202122, 1234510111213141520212223, 123451011121314152021222324, 12345101112131415202122232425
Offset: 1

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The smallest prime occurs at n = 12891. - Michael S. Branicky, Apr 20 2023
The b-file has only 313 terms, since a(314) has 1001 digits.

Crossrefs

Programs

  • Mathematica
    A362119[n_]:=FromDigits[Flatten[IntegerDigits[Range[n],6]]];Array[A362119,20] (* Paolo Xausa, Nov 27 2023 *)
  • Python
    from sympy.ntheory import digits
    from itertools import count, islice
    def agen(s="", base=6): yield from (int(s:=s+"".join(map(str, digits(n, base)[1:]))) for n in count(1))
    print(list(islice(agen(), 20)))

A362429 Smallest k such that the concatenation of the numbers 123...k in base n is prime when interpreted as a decimal number, or -1 if no such prime exists.

Original entry on oeis.org

-1, 231, 7315, 3241, 6, 12891, 22, 227, 127
Offset: 1

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Comments

The sequence can be extended to bases larger than 10 by concatenating the decimal equivalents of digits.
a(1) is -1 since no such primes are possible (the sequence in question is A362118). Proof. The number of ones in the resulting repunit is triangular and per A000217, 3 is the only prime triangular number, and per A004023, prime repunits must have prime indices.
If it exists, a(10) would be the index of the first prime in A007908. See A007908 for the latest information about the search for this prime.
a(10), ..., a(14) are respectively ?, 144, 307, ?, 25.
a(10) and a(13) are presently unknown. a(13) > 10000 if it is not -1.

Examples

			a(5) is 6: 12341011 (concatenate 1 though 6 in base 5) is a prime when interpreted as a decimal number.
		

Crossrefs

Sequences of concatenations: A362118 (base 1), A058935 (base 2), A360502 (base 3), A117640 (base 4), A362117 (base 5), A362119 (base 6), A007908 (base 10).
Cf. A376221.

Programs

  • Python
    from gmpy2 import is_prime
    from sympy.ntheory import digits
    from itertools import count, islice
    def c(base, s=""):
        if base == 1: yield from (s:=s+"1"*n for n in count(1))
        else:
            yield from (s:=s+"".join(map(str, digits(n, base)[1:])) for n in count(1))
    def a(n):
        if n == 1: return -1
        return next(k for k, t in enumerate(c(n), 1) if is_prime(int(t)))
Showing 1-3 of 3 results.