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 1-6 of 6 results.

A331860 Numbers k such that R(k) + 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

6, 7, 12, 31, 58, 127, 454, 556, 558, 604, 2944, 8118, 12078, 16942, 26268, 45198
Offset: 1

Views

Author

M. F. Hasler, Jan 30 2020

Keywords

Comments

The corresponding primes are near-repunit primes, cf. A105992.
In base 10, R(k) + 10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 2 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 2 just left of the middle of the repunit of length k.
No term can be congruent to 2 (mod 3). - Chai Wah Wu, Feb 07 2020

Examples

			For n = 6,  R(6)  + 10^(3-1) = 111211 is prime.
For n = 7,  R(7)  + 10^(3-1) = 1111211 is prime.
For n = 12, R(12) + 10^(6-1) = 111111211111 is prime.

Links

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).
Cf. A331861 (variant with floor(n/2) instead of floor(n/2-1)), A331863 (variant with - (digit 0) instead of + (digit 2)).

Programs

  • PARI
    for(n=2,999,isprime(p=10^n\9+10^(n\2-1))&&print1(n","))

Extensions

a(8)-a(14) from Giovanni Resta, Jan 31 2020
a(15)-a(16) from Michael S. Branicky, Jul 23 2024

A331862 Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

3, 26, 186, 206, 258, 3486, 12602
Offset: 1

Views

Author

M. F. Hasler, Jan 30 2020

Keywords

Comments

The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
In base 10, R(n) - 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 0, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 0 is just left to the middle of the number.
There can't be an odd term > 3 because the corresponding palindrome factors as R((n-1)/2)*(10^((n+1)/2) + 1).
No term can be congruent to 1 mod 3. - Chai Wah Wu, Feb 07 2020

Examples

			For n = 3, R(n) - 10^floor(n/2) = 101 is prime.
For n = 26, R(n) - 10^floor(n/2) = 11111111111101111111111111 is prime.

Links

Crossrefs

Cf. A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A105992 (near-repunit primes), A138148 (Cyclops numbers with digits 0 & 1).
Cf. A331860 (variant with digit 2 instead of digit 0), A331863 (variant with floor(n/2-1) instead of floor(n/2)).

Programs

  • PARI
    for(n=0,9999,isprime(p=10^n\9-10^(n\2))&&print1(n","))

Extensions

a(6)-a(7) from Giovanni Resta, Jan 31 2020

A331864 Numbers k such that R(k) + 2*10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

2, 3, 5, 8, 9, 39, 78, 81, 155, 249, 387, 395, 510, 711, 1173, 1751, 10245
Offset: 1

Views

Author

M. F. Hasler, Jan 30 2020

Keywords

Comments

The corresponding primes are near-repunit primes, cf. A105992.
In base 10, R(k) + 2*10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 3 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 3 just left of the middle of the repunit of length k.
No term can be equivalent to 1 (mod 3). - Chai Wah Wu, Feb 07 2020

Examples

			For k = 2, R(2) + 2*10^(1-1) = 13 is prime.
For k = 3, R(3) + 2*10^(1-1) = 113 is prime.
For k = 5, R(5) + 2*10^(2-1) = 11131 is prime.
For k = 8, R(8) + 2*10^(4-1) = 11113111 is prime.

Links

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).
Cf. A331865 (variant with floor(n/2) instead of floor(n/2-1)), A331860, A331863 (variants with digit 2 resp. 0 instead of digit 3).

Programs

  • PARI
    for(n=2,999,isprime(p=10^n\9+2*10^(n\2-1))&&print1(n","))

Extensions

a(13)-a(16) from Daniel Suteu, Feb 01 2020
a(17) from Michael S. Branicky, Feb 03 2023

A331865 Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 12, 20, 39, 74, 78, 80, 84, 104, 195, 654, 980, 2076, 5940, 19637
Offset: 1

Views

Author

M. F. Hasler, Jan 30 2020

Keywords

Comments

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 2*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 3, and again floor(n/2) digits 1 (except for n=0). For odd n, this is a palindrome (a.k.a. wing prime, cf. A077779), for even n the digit 3 is just left to the middle of the number.
a(22) > 50000. - Michael S. Branicky, Feb 19 2025

Examples

			For n = 0, R(0) + 2*10^floor(0/2) = 2 is prime.
For n = 1, R(1) + 2*10^floor(1/2) = 3 is prime.
For n = 2, R(2) + 2*10^floor(2/2) = 31 is prime.
For n = 3, R(3) + 2*10^floor(3/2) = 131 is prime.
For n = 5, R(5) + 2*10^floor(5/2) = 11311 is prime.
For n = 6, R(6) + 2*10^floor(6/2) = 113111 is prime.

Links

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331860 & A331863 (variants with digit 2 resp. 0 instead of 3), A331864 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077779 (odd terms).

Programs

  • Mathematica
    Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 2*10^Floor[#/2]] &] (* Michael De Vlieger, Jan 31 2020 *)
  • PARI
    for(n=0,9999,isprime(p=10^n\9+2*10^(n\2))&&print1(n","))

Extensions

a(18)-a(20) from Giovanni Resta, Jan 30 2020
a(21) from Michael S. Branicky, Feb 19 2025

A331867 Numbers n for which R(n) + 3*10^floor(n/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

Original entry on oeis.org

68, 5252, 5494, 7102
Offset: 1

Views

Author

M. F. Hasler, Jan 30 2020

Keywords

Comments

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 3*10^floor(n/2-1) has ceiling(n/2) digits 1, one digit 4, and again floor(n/2-1) digits 1. For odd and even n, the digit 4 is just to the right of the middle of the number.
For odd n = 2m + 1, f(n) = R(n) + 3*10^floor(n/2-1) is divisible by 3, 7 or 13 when m is congruent 1 or 4, 3 or 5, resp. 0 or 2 (mod 6): there can't be an odd term.
For even n = 2m, f(n) is divisible by 3 or 7 when m is congruent to 0 or 3, resp. 1 or 2 (mod 6). When m = 6k + 4, then f(n) is prime for k = 5 and 437 (and no further k <= 600), and divisible by 23 or 53 when k is congruent to 10 (mod 11) resp. 3 (mod 13). When m = 6k + 5, f(n) is prime for k = 457 and 591 and no other value up to 600, and divisible by 23, 47, 53, 97, 163, 181, 859, ... for k congruent to 5 (mod 11), 11 (mod 23), 5 (mod 13), 0 (mod 32), 13 (mod 27), 26 (mod 30), 3 (mod 13), ..., respectively.
a(5) > 7272.

Examples

			For n = 2, R(2) + 3*10^floor(2/2-1) = 14 = 2*7 is not prime.
For n = 3, R(3) + 3*10^floor(3/2-1) = 114 = 2*3*19 is not prime.
For n = 4, R(4) + 3*10^floor(4/2) = 1141 = 7*163 is not prime.
For n = 5, R(5) + 3*10^floor(5/2) = 11141 = 13*857 is not prime.
For n = 68, R(68) + 3*10^floor(68/2) = 1...1141...1 is prime, with 34 digits '1' to the left of a digit '4' and 33 digits '1' to its right.

Links

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331860, A331863, A331864 (variants with digit 2, 0 resp. 3 instead of 4), A331866 (variant with floor(n/2) instead of floor(n/2-1)).

Programs

  • Mathematica
    Select[Range[2, 2500], PrimeQ[(10^# - 1)/9 + 3*10^Floor[#/2 - 1]] &]  (* corrected by Amiram Eldar, Feb 10 2020 *)
  • PARI
    for(n=2,9999,isprime(p=10^n\9+3*10^(n\2-1))&&print1(n","))

A331868 Numbers k for which R(k) + 4*10^floor(k/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

Original entry on oeis.org

4, 147, 270, 1288, 1551, 3427
Offset: 1

Views

Author

M. F. Hasler, Jan 30 2020

Keywords

Comments

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 4*10^floor(n/2-1) has ceiling(n/2) digits 1, one digit 5, and again floor(n/2-1) digits 1. For odd and even n as well, the digit 5 appears just to the right of the middle of the number.
a(7) > 10^4. - Daniel Suteu, Feb 10 2020
a(7) > 5*10^4. - Michael S. Branicky, Nov 02 2024

Examples

			For n = 4, R(4) + 4*10^floor(4/2-1) = 1151 is prime.
For n = 5, R(5) + 4*10^floor(5/2-1) = 11151 =  3^3*7*59 is not prime.
For n = 147, R(147) + 4*10^72 = 1(74)51(72) is prime, where (.) indicates how many times the preceding digit is repeated.

Links

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331863, A331860, A331864, A331867 (variants with digit 0, 2, 3 resp. 4 instead of 5), A331869 (variant with floor(n/2) instead of floor(n/2-1)).

Programs

  • Mathematica
    Select[Range[2, 2500], PrimeQ[(10^# - 1)/9 + 4*10^Floor[#/2 - 1]] &]
  • PARI
    for(n=2,9999,isprime(p=10^n\9+4*10^(n\2-1))&&print1(n","))

Extensions

a(6) from Daniel Suteu, Feb 10 2020
Showing 1-6 of 6 results.

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