A016041 -id:A016041 - cp's OEIS frontend

A372142 a(n) is the smallest prime p such that there exist exactly n distinct primes q where q < p and the representation of p in base q is a palindrome.

2, 3, 31, 443, 23053, 86677, 11827763, 27362989, 755827199, 1306369439

Offset: 0

Tadayoshi Kamegai, Apr 21 2024

This is a special case of A372141.
It need not be the case that a(n) is a palindrome in base 2, as 23053 is a counterexample.
For p > 3, one only needs to check q such that q^2 + 1 <= p else p = cc_q = c*(q+1), not prime for c != 1 and q != 2. A similar argument shows that p cannot have an even number of digits in base q, else it would be divisible by (q+1). - Michael S. Branicky, Apr 21 2024

			a(5) = 86677, as it is palindromic in base 2, 107, 113, 151, and 233, and no smaller number satisfies the property.

Cf. A372141, A002385, A002113.

Cf. A016041, A007500, A077798.

from math import isqrt
from sympy import sieve
from sympy.ntheory import digits
from itertools import islice
def ispal(v): return v == v[::-1]
def f(p): return sum(1 for q in sieve.primerange(1, isqrt(p-1)+1) if ispal(digits(p, q)[1:]))
def agen():
    adict, n = {0:2, 1:3}, 0
    for p in sieve:
        v = f(p)
        if v >= n and v not in adict:
            adict[v] = p
            while n in adict:
                yield adict[n]; del adict[n]; n += 1
print(list(islice(agen(), 6))) # Michael S. Branicky, Apr 21 2024

a(6) from Jon E. Schoenfield, Apr 21 2024
a(7) from Michael S. Branicky, Apr 21 2024
a(8) from Michael S. Branicky, Apr 22 2024
a(9) from Michael S. Branicky, Apr 24 2024

A056145 Palindromic primes in bases 2 and 8.

Original entry on oeis.org

3, 5, 7, 73, 28807, 31727, 262657, 295433, 1311749, 1385621, 1478189, 1540157, 1543741, 1549501, 1551037, 1865159, 1932247, 2031599, 2067007, 2085247, 2087807, 83914757, 84663941, 85742021, 85808581, 88779413, 89420117, 89466197, 89924053, 90169301, 94971053, 94983341

Offset: 1

Robert G. Wilson v, Jul 29 2000

Cf. A016041 and A029976.

Do[ If[ PrimeQ[ n ], t=RealDigits[ n, 8 ][ [ 1 ] ]; If[ FromDigits[ t ]==FromDigits[ Reverse[ t ] ], s=RealDigits[ n, 2 ][ [ 1 ] ]; If[ FromDigits[ s ]==FromDigits[ Reverse[ s ] ], Print[ n ] ] ] ], {n, 1, 10^8} ]
Select[Prime[Range[155000]],PalindromeQ[IntegerDigits[#,2]] && PalindromeQ[ IntegerDigits[ #,8]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 23 2017 *)

More terms from Harvey P. Dale, Sep 23 2017

A155801 Nontrivial "Strobogrammatic" primes, the same "upside-down" in at least one base b with 2 <= b <= 10.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 31, 37, 43, 73, 101, 107, 127, 181, 257, 313, 443, 619, 757, 1093, 1193, 1297, 1453, 1571, 1619, 1787, 1831, 1879, 2801, 4889, 5113, 5189, 5557, 5869, 5981, 6211, 6827, 7607, 7759, 7919, 8191

Offset: 1

Jonathan Vos Post, Jan 27 2009

I have to say "nontrivial" because every nonnegative integer is strobogrammatic in base 1. Strobogrammatic binary primes == primes in A006995 == A016041. Strobogrammatic primes in base 3 = 13, 757, 1093, 9103, ... == primes strobogrammatic in bases 2 and 3. For bases 2 < k < 8 we have that every strobogrammatic prime in base k must also be strobogrammatic in base 2 and hence palindromic in base 2. Hence we have, for example, strobogrammatic base-4 primes = A056130 = "Palindromic primes in bases 2 and 4."

Strobogrammatic primes in base 5 = 31, 19531, 394501, 472631, ... == primes strobogrammatic in base 2 and base 5. Strobogrammatic primes base 6 = 7, 37, 43, 1297, 55987, ... == primes strobogrammatic in base 2 and base 6. Note that 1101011 (base 6) = 18881 (base 10) which is strobogrammatic base 10 but not prime base 6 nor 10 (though prime base 2). Strobogrammatic primes base 7 = 2801, 134807, this last being strobogrammatic prime in bases 2, 4 and 7. Strobogrammatic primes base 8 = 73, 262657, 295433, ... Strobogrammatic primes base 9 break the above pattern, as they can have the digit 8 and are A068188 (tetradic primes). Strobogrammatic primes base 10 == A007597. Except sometimes for the first element, these (for the same range of k) must all have an odd number of digits.

			5189 = 1101011 (base 6) which numeral string is the same upside-down (and backwards). 11, 101, 181 and 619 are strobogrammatic base 10, the conventional interpretation of the word.

Cf. A000040, A006995, A016041, A056130, A007597, A133207, A155584.

A000040 INTERSECTION A155584[1 < k < 11, n].

A211405 Prime numbers that in binary are palindromic and also have palindromic index.

Original entry on oeis.org

5, 17, 73, 127, 313, 1453, 22861, 28123, 296713, 309481, 2063947162127

Offset: 1

James G. Merickel, Feb 09 2013

This has been searched through 2^33.

The index of a(11) is 75558337841. a(12) > 2^44. - Giovanni Resta, Feb 12 2013

			The binary representations of the first three terms and their indices are 101 and 11, 10001 and 111, and 1001001 and 10101.

Cf. A046941, A016041.

a(11) from Giovanni Resta, Feb 12 2013

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A372142 a(n) is the smallest prime p such that there exist exactly n distinct primes q where q < p and the representation of p in base q is a palindrome.

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A056145 Palindromic primes in bases 2 and 8.

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A155801 Nontrivial "Strobogrammatic" primes, the same "upside-down" in at least one base b with 2 <= b <= 10.

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A211405 Prime numbers that in binary are palindromic and also have palindromic index.

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