A360502 -id:A360502 - cp's OEIS frontend

A117640 Concatenation of first n numbers in base 4.

1, 12, 123, 12310, 1231011, 123101112, 12310111213, 1231011121320, 123101112132021, 12310111213202122, 1231011121320212223, 123101112132021222330, 12310111213202122233031

Offset: 1

Jonathan Vos Post, Apr 27 2006

Concatenation of the first n terms of A007090.

Base-4 analog of A058935.

N. J. A. Sloane, Table of n, a(n) for n = 1..35

Cf. A030373, A048436.

Other bases: A058935 (2), A360502 (3), A007908 (10).

Table[FromDigits[Flatten[Table[IntegerDigits[n,4],{n,k}]]],{k,15}] (* Harvey P. Dale, Jan 18 2023 *)

from gmpy2 import digits
def A117640(n): return int(''.join(digits(n,4) for n in range(1,n+1))) # Chai Wah Wu, Apr 19 2023

Edited by Jason Kimberley, Nov 27 2012

A360503 Numbers k such that A048435(k) is prime.

Original entry on oeis.org

2, 5, 82, 2546

Offset: 1

N. J. A. Sloane, Feb 17 2023

a(1), a(2), and a(3) were found by Kurt Foster, Oct 24 2015 (see A048435). a(4) was found by Jeff Peltier: see comments following the "Most Wanted Prime" video.

Brady Haran and N. J. A. Sloane, Most Wanted Prime, Numberphile video, December 2021.
Index entries for sequences related to Most Wanted Primes video

Cf. A007089, A048435, A360502.

A362118 a(n) = (10^(n*(n+1)/2)-1)/9.

Original entry on oeis.org

1, 111, 111111, 1111111111, 111111111111111, 111111111111111111111, 1111111111111111111111111111, 111111111111111111111111111111111111, 111111111111111111111111111111111111111111111, 1111111111111111111111111111111111111111111111111111111, 111111111111111111111111111111111111111111111111111111111111111111

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Apr 19 2023

nonn

Concatenate 1, 11, 111, ..., 11...1 (n ones). There are n*(n+1)/2 1's in a(n).
This is a kind of unary analog of A058935, A360502, A117640, etc.
When regarded as decimal numbers, which (if any) is the smallest prime?
Answer: All terms > 1 are composite, since 111 is composite, all triangular numbers > 3 are composite and a prime repunit must have a prime number of decimal digits (see A004023). - Chai Wah Wu, Apr 19 2023. [This result was independently obtained by Michael S. Branicky, see A362429. - N. J. A. Sloane, Apr 20 2023]
a(45) has more than 1000 digits, and so cannot be included in the b-file. - Jason Bard, Apr 12 2025

			a(3) = 111111 because 3(3+1)/2 = 6, and 111111 has 6 ones.

Jason Bard, Table of n, a(n) for n = 1..44

Cf. A000042, A004023, A058935, A360502, A117640, A007908.

A362118[n_]:=(10^(n(n+1)/2)-1)/9;Array[A362118,10] (* Paolo Xausa, Nov 27 2023 *)

def A362118(n): return 10**(n*(n+1)>>1)//9 # Chai Wah Wu, Apr 19 2023

a(n) = A000042(A000217(n)). - Jason Bard, Apr 12 2025

A360504 Concatenate the ternary strings for 1,2,...,n-1, n, n-1, ..., 2,1.

Original entry on oeis.org

1, 121, 121021, 1210111021, 12101112111021, 121011122012111021, 1210111220212012111021, 12101112202122212012111021, 1210111220212210022212012111021, 1210111220212210010110022212012111021, 1210111220212210010110210110022212012111021, 1210111220212210010110211010210110022212012111021

Offset: 1

N. J. A. Sloane, Feb 17 2023

If the terms are read as ternary strings and converted to base 10, we get A260853. For example, a(3) = 121021_3 = 439_10, which is A260853(3). This is a prime. What is the next prime term?

If the terms are read as decimal numbers, which of them are primes? a(3) = 121021_10 is a decimal prime, but what is the next one? It is a surprise that 121021 is a prime in both base 3 and base 10.

			To get a(3) we concatenate 1, 2, 10, 2, and 1, getting 121021.

Winston de Greef, Table of n, a(n) for n = 1..123
Index entries for sequences related to Most Wanted Primes video

Cf. A007089, A360502, A360503.

This is the ternary analog of A173426.

t:= n-> (l-> parse(cat(seq(l[-i], i=1..nops(l)))))(convert(n, base, 3)):
a:= n-> parse(cat(map(t, [$1..n, n-i$i=1..n-1])[])):
seq(a(n), n=1..12);  # Alois P. Heinz, Feb 17 2023

Table[FromDigits[Flatten[Join[IntegerDigits[#,3]&/@Range[n],IntegerDigits[#,3]&/@ Range[ n-1,1,-1]]]],{n,20}] (* Harvey P. Dale, Oct 01 2023 *)

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in list(range(1, n+1))+list(range(n-1, 0, -1))))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(): # generator of terms
    sf, sr = "", ""
    for n in count(1):
        sn = "".join(map(str, digits(n, 3)[1:]))
        sf += sn
        yield int(sf + sr)
        sr = sn + sr
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

A360506 Read A360505(n) as if it were a base-3 string and write it in base 10.

Original entry on oeis.org

1, 7, 34, 358, 4003, 43369, 456712, 4708240, 47754961, 1339156591, 39693785002, 1169411930926, 34213667699203, 995038950807565, 28790341783585180, 829295063367580492, 23793774263808446005, 680307709052882601259, 19390954850541496025998

Offset: 1

N. J. A. Sloane, Feb 17 2023

This has the same relationship to A360505 as A048435 does to A360502.
The primes in A048435 are in A360503. What are the primes in the present sequence?
Answer: The first primes are a(2) = 7, a(5) = 4003, a(13) = 34213667699203, a(57) and a(109). See A360507. - Rémy Sigrist, Feb 18 2023

			A360505(4) = 111021 and 111021_3 = 358_10 = a(4).

Winston de Greef, Table of n, a(n) for n = 1..407
Index entries for sequences related to Most Wanted Primes video

Cf. A048435, A028898, A360502, A360503, A360505, A360507.

a(n) = fromdigits(concat([digits(k, 3) | k <- Vecrev([1..n])]), 3) \\ Rémy Sigrist, Feb 18 2023

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(n, 0, -1)), 3)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Feb 19 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:="".join(map(str, digits(n, 3)[1:]))+s, 3) for n in count(1))
print(list(islice(agen(), 20))) # Michael S. Branicky, Feb 19 2023

from itertools import count, islice
def A360506_gen(): # generator of terms
    a, b, c = 3, 1, 0
    for i in count(1):
        if i >= a:
            a *= 3
        c += i*b
        yield c
        b *= a
A360506_list = list(islice(A360506_gen(),30)) # Chai Wah Wu, Nov 08 2023

a(n) = A028898(A360505(n)). - Rémy Sigrist, Feb 18 2023

More terms from Rémy Sigrist, Feb 18 2023

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

Michael S. Branicky and N. J. A. Sloane, Apr 19 2023

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

Cf. A362118, A058935, A360502, A117640, A007908.

A362117[n_]:=FromDigits[Flatten[IntegerDigits[Range[n],5]]];Array[A362117,20] (* Paolo Xausa, Nov 27 2023 *)

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

Michael S. Branicky and N. J. A. Sloane, Apr 19 2023

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.

Michael S. Branicky, Table of n, a(n) for n = 1..313

Cf. A362118, A058935, A360502, A117640, A362117, A007908.

A362119[n_]:=FromDigits[Flatten[IntegerDigits[Range[n],6]]];Array[A362119,20] (* Paolo Xausa, Nov 27 2023 *)

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

Michael S. Branicky and N. J. A. Sloane, Apr 19 2023

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.

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

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.

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)))

A360505 Concatenate the ternary strings for n, n-1, n-2, ..., 2, 1.

Original entry on oeis.org

1, 21, 1021, 111021, 12111021, 2012111021, 212012111021, 22212012111021, 10022212012111021, 10110022212012111021, 10210110022212012111021, 11010210110022212012111021, 11111010210110022212012111021, 11211111010210110022212012111021, 12011211111010210110022212012111021

Offset: 1

N. J. A. Sloane, Feb 17 2023

Similar to A360502, but here we count down from n to 1 in base 3 and concatenate the strings to get a(n).

This is the ternary analog of A000422.

Index entries for sequences related to Most Wanted Primes video

Cf. A000422, A007809, A360502, A360506.

a[n_] := FromDigits @ Flatten @ IntegerDigits[Range[n, 1, -1], 3]; Array[a, 15] (* Amiram Eldar, Feb 18 2023 *)

a(n) = strjoin(concat([digits(k, 3) | k <- Vecrev([1..n])])) \\ Rémy Sigrist, Feb 18 2023

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(n, 0, -1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:="".join(map(str, digits(n, 3)[1:]))+s) for n in count(1))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

cp's OEIS Frontend

A117640 Concatenation of first n numbers in base 4.

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A360503 Numbers k such that A048435(k) is prime.

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A362118 a(n) = (10^(n*(n+1)/2)-1)/9.

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A360504 Concatenate the ternary strings for 1,2,...,n-1, n, n-1, ..., 2,1.

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A360506 Read A360505(n) as if it were a base-3 string and write it in base 10.

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A362117 Concatenation of first n numbers in base 5.

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

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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.

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