Colin M Ready - cp's OEIS frontend

User: Colin M Ready

Colin M Ready has authored 2 sequences.

A331807 a(n) is the smallest prime number p > n, not yet in the sequence, such that p is a palindrome when written in base n.

3, 13, 5, 31, 7, 71, 73, 109, 11, 199, 157, 313, 197, 241, 17, 307, 19, 419, 401, 463, 23, 599, 577, 701, 677, 757, 29, 929, 991, 1117, 1153, 1123, 1259, 1471, 37, 1481, 1483, 1873, 41, 1723, 43, 1979, 2069, 2161, 47, 2351, 2593, 2549, 2551, 2857, 53, 2969, 2917, 3191, 3137

Offset: 2

Colin M Ready, Feb 22 2020

Using a representation where the digits of the prime are written between "[" and "]" separated by commas with the base following the "" then by checking up to a base of 7000 (where the lowest prime palindrome is [1, 1]_7000):
1) Either the palindrome is [1, 1]_n where n is one less than a prime number, or [1, X, 1]_n where X << n, asymptotically.
2) A conjecture: No lowest primes need more than three digits.
3) The terms a(12) and a(30) differ from the similar sequence A331806 as these terms in A331806 are the same as the earlier terms a(3) and a(5).

			a(2)=3 which is 11 in binary, a(3)=13 which is 111 in ternary, a(4)=5 which is 11 in quaternary, a(16)=17 which is 11 in hexadecimal.
If we use the representation described earlier, then:
   a(2)  = 3    is [1, 1]_2,
   a(3)  = 13   is [1, 1, 1]_3,
   a(4)  = 5    is [1, 1]_4,
   a(11) = 199  is [1, 7, 1]_11,
   a(13) = 313  is [1, 11, 1]_13,
   a(16) = 17   is [1, 1]_16,
   a(48) = 2593 is [1, 6, 1]_48.

Chai Wah Wu, Table of n, a(n) for n = 2..10000

A331806 is a similar sequence where repeated terms are allowed.

Cf. A006093 (prime(n) - 1).

Array[Block[{p = Prime[PrimePi[#] + 1]}, While[! PalindromeQ@ IntegerDigits[p, #], p = NextPrime@ p]; p] &, 55, 2] (* Michael De Vlieger, Feb 25 2020 *)

A331806 a(n) is the least prime p > n which is palindromic in base n.

Original entry on oeis.org

3, 13, 5, 31, 7, 71, 73, 109, 11, 199, 13, 313, 197, 241, 17, 307, 19, 419, 401, 463, 23, 599, 577, 701, 677, 757, 29, 929, 31, 1117, 1153, 1123, 1259, 1471, 37, 1481, 1483, 1873, 41, 1723, 43, 1979, 2069, 2161, 47, 2351, 2593, 2549, 2551, 2857, 53, 2969, 2917, 3191, 3137

Offset: 2

Colin M Ready, Jan 27 2020

Using a representation where the digits of the prime are written between "[" and "]" separated by commas with the base following the "" then by checking up to a base of 7000 (where the lowest prime palindrome is [1, 1]_7000):
1) Either the palindrome is [1, 1]_n where n is one less than a prime number, or [1, X, 1]_n where X << n, asymptotically.
2) Many prime numbers occur more than once, e.g.,
    13 is [1, 1, 1]_3  and [1, 1]_12;
    71 is [1, 3, 1]_7  and [1, 1]_70;
  1471 is [1, 7, 1]_35 and [1, 1]_1470.

			a(2) = 3 which is 11 in binary, a(3) = 13 which is 111 in ternary, a(4) = 5 which is 11 in quaternary, a(16) = 17 which is 11 in hexadecimal.
If we use the representation described earlier, then:
  a(2)  = 3    is [1, 1]_2,
  a(3)  = 13   is [1, 1, 1]_3,
  a(4)  = 5    is [1, 1]_4,
  a(11) = 199  is [1, 7, 1]_11,
  a(13) = 313  is [1, 11, 1]_13,
  a(16) = 17   is [1, 1]_16,
  a(48) = 2593 is [1, 6, 1]_48.

Colin M Ready, Table of n, a(n) for n = 2..7000

Cf. A006093 (prime(n) - 1), A087952 (least prime > n^2 and congruent to 1 (mod n)).

Array[If[PrimeQ[# + 1], # + 1, Block[{p = If[PrimeQ@ #1, #1, Prime[#2 + 1]] & @@ {#, PrimePi[#]}}, While[! PalindromeQ@ IntegerDigits[p, #], Set[p, NextPrime@ p]]; p]] &, 55, 2] (* Michael De Vlieger, Jan 27 2020 *)

a(n) = {forprime(p=n+1, oo, my(d=digits(p, n)); if (Vecrev(d) ==d, return(p)););} \\ Michel Marcus, Jan 27 2020

a(p-1) = p for prime p > 2.

a(n) <= A087952(n) with equality if n+1 is not prime. - M. F. Hasler, Feb 27 2020

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User: Colin M Ready

A331807 a(n) is the smallest prime number p > n, not yet in the sequence, such that p is a palindrome when written in base n.

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