A331862 -id:A331862 - cp's OEIS frontend

A331863 Numbers k such that R(k) - 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

8, 12, 17, 20, 24, 42, 1124, 1169, 1538, 7902, 27617, 29684

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
In base 10, R(k) - 10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 0 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 0 just right of the middle of the repunit of length k.
No term can be congruent to 1 (mod 3). - Chai Wah Wu, Feb 07 2020
a(13) > 50000. - Michael S. Branicky, Jul 23 2024

			For k = 8,  R(8)  - 10^(4-1) = 11110111 is prime.
For k = 12, R(12) - 10^(6-1) = 111111011111 is prime.
For k = 17, R(12) - 10^(8-1) = 11111111101111111 is prime.

Cf. A002275 (repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A138148 (Cyclop numbers with digits 0 & 1).

Cf. A331862 (variant with floor(n/2) instead of floor(n/2-1)), A331860 (variant with + (digit 2) instead of - (digit 0)).

for(n=2,9999,isprime(p=10^n\9-10^(n\2-1))&&print1(n","))

a(7)-a(10) from Giovanni Resta, Jan 31 2020

a(11)-a(12) from Michael S. Branicky, Jul 22 2024

A331861 Numbers n for which R(n) + 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

1, 6, 10, 18, 24, 4978

Offset: 1

M. F. Hasler, Jan 30 2020

The primes corresponding to the terms of the sequence are a subset of the near-repunit primes A105992.
In base 10, R(n) + 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 2, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 2 is just left to the middle of the number.
There cannot be an odd term > 1 since the corresponding palindrome factors as R((n+1)/2)*(10^((n-1)/2) + 1).
No term can be congruent to 2 mod 3. - Chai Wah Wu, Feb 07 2020

			For n = 1, R(n) + 10^floor(n/2) = 2 is prime.
For n = 6, R(n) + 10^floor(n/2) = 112111 is prime.
For n = 10, R(n) + 10^floor(n/2) = 1111211111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).

Cf. A331860 (variant with floor(n/2-1) instead of floor(n/2)), A331862 (variant with - (digit 0) instead of + (digit 2)).

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

a(6) from Daniel Suteu, Feb 01 2020

A331866 Numbers k for which R(k) + 3*10^floor(k/2) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

0, 2, 5, 7, 8, 10, 65, 91, 208, 376, 586, 2744, 3089, 19378, 20246

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subset of the near-repunit primes A105992 (at least when they have k > 2 digits).
In base 10, R(k) + 3*10^floor(k/2) has k digits all of which are 1 except for one digit 4 (for k > 0) located in the center (for odd k) or just to the left of it (for even k): i.e., there are ceiling(k/2)-1 digits 1 to the left and floor(k/2) digits 1 to the right of the digit 4. For odd k, this is a palindrome a.k.a. wing prime, cf. A077780, the subsequence of odd terms.
a(14) = 19378 was found by Amiram Eldar, verified to be the 14th term in collaboration with the author of the sequence and factordb.com. The term a(13) = 3089 corresponds to a certified prime (Ivan Panchenko, 2011, cf. factordb.com); a(12) and a(14) are only PRP as far as we know.

			For n = 0, R(0) + 3*10^floor(0/2) = 3 is prime.
For n = 2, R(2) + 3*10^floor(2/2) = 41 is prime.
For n = 5, R(5) + 3*10^floor(5/2) = 11411 is prime.
For n = 7, R(7) + 3*10^floor(7/2) = 1114111 is prime.
For n = 8, R(8) + 3*10^floor(8/2) = 11141111 is prime.

Chris Caldwell, The Prime Glossary, Near-repunit prime.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331862, A331861, A331865, A331869 (variants with digit 0, 2, 3 or 5 instead of 4), A331867 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077780 (odd terms).

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 3*10^Floor[#/2]] &]

for(n=0,9999,ispseudoprime(p=10^n\9+3*10^(n\2))&&print1(n","))

a(15) from Michael S. Branicky, Sep 24 2024

A331869 Numbers n for which R(n) + 4*10^floor(n/2) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

Original entry on oeis.org

1, 3, 4, 15, 76, 91, 231, 1363, 1714, 1942, 2497, 4963, 5379, 12397, 23224, 26395

Offset: 1

M. F. Hasler, Feb 09 2020

For n > 1, the corresponding primes are a subset of A105992: near-repunit primes.
In base 10, R(n) + 4*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 5, and again floor(n/2) digits 1, except for n = 0. For odd n, this is a palindrome (a.k.a. wing prime, cf. A077783: subsequence of odd terms), for even n the digit 5 is just left to the middle of the number.
See also the variant A331868 where the digit 5 is just to the right of the middle.

			For n = 1, R(1) + 4*10^floor(1/2) = 5 is prime.
For n = 3, R(3) + 4*10^floor(3/2) = 151 is prime.
For n = 4, R(4) + 4*10^floor(4/2) = 1511 is prime.
For n = 15, R(15) + 4*10^floor(15/2) = 111111151111111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331862, A331861, A331865, A331866 (variants with digit 0, 2, 3 or 4 instead of 5), A331868 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077783 (odd terms).

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 4*10^Floor[#/2]] &]

for(n=0,9999,ispseudoprime(p=10^n\9+4*10^(n\2))&&print1(n","))

a(12)-a(14) from Michael S. Branicky, Feb 03 2023

a(15)-a(16) from Michael S. Branicky, Apr 11 2023

cp's OEIS Frontend

A331863 Numbers k such that R(k) - 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

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A331861 Numbers n for which R(n) + 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

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A331866 Numbers k for which R(k) + 3*10^floor(k/2) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

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A331869 Numbers n for which R(n) + 4*10^floor(n/2) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

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