A095741 -id:A095741 - cp's OEIS frontend

A016041 Primes that are palindromic in base 2 (but written here in base 10).

3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879, 4889, 5113, 5189, 5557, 5869, 5981, 6211, 6827, 7607, 7759, 7919, 8191, 17377, 18097, 18289, 19433, 19609, 19801, 21157, 22541, 22669, 22861, 23581, 24029

Offset: 1

Robert G. Wilson v

See A002385 for palindromic primes in base 10, and A256081 for primes whose binary expansion is "balanced" (see there) but not palindromic. - M. F. Hasler, Mar 14 2015

Number of terms less than 4^k, k=1,2,3,...: 1, 3, 5, 8, 11, 18, 30, 53, 93, 187, 329, 600, 1080, 1936, 3657, 6756, 12328, 23127, 43909, 83377, 156049, 295916, 570396, 1090772, 2077090, 3991187, 7717804, 14825247, 28507572, 54938369, 106350934, ..., partial sums of A095741 plus 1. - Robert G. Wilson v, Feb 23 2018, corrected by Jeppe Stig Nielsen, Jun 17 2023

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov, terms 1001..3000 from Michael De Vlieger)
Kevin S. Brown, On General Palindromic Numbers
Patrick De Geest, World!Of Palindromic Primes

Intersection of A000040 and A006995.
First row of A095749.
A095741 gives the number of terms in range [2^(2n), 2^(2n+1)].
Cf. A095730 (primes whose Zeckendorf expansion is palindromic), A029971 (primes whose ternary (base-3) expansion is palindromic).
Cf. A117697 (written in base 2), A002385, A194097, A256081.

[NthPrime(n): n in [1..5000] | (Intseq(NthPrime(n), 2) eq Reverse(Intseq(NthPrime(n), 2)))]; // Vincenzo Librandi, Feb 24 2018

lst = {}; Do[ If[ PrimeQ@n, t = IntegerDigits[n, 2]; If[ FromDigits@t == FromDigits@ Reverse@ t, AppendTo[lst, n]]], {n, 3, 50000, 2}]; lst (* syntax corrected by Robert G. Wilson v, Aug 10 2009 *)
pal2Q[n_] := Reverse[x = IntegerDigits[n, 2]] == x; Select[Prime[Range[2800]], pal2Q[#] &] (* Jayanta Basu, Jun 23 2013 *)
genPal[n_Integer, base_Integer: 10] := Block[{id = IntegerDigits[n, base], insert = Join[{{}}, {# - 1} & /@ Range[base]]}, FromDigits[#, base] & /@ (Join[id, #, Reverse@id] & /@ insert)]; k = 0; lst = {}; While[k < 100, AppendTo[lst, Select[ genPal[k, 2], PrimeQ]]; lst = Flatten@ lst; k++]; lst (* Robert G. Wilson v, Feb 23 2018 *)

is(n)=isprime(n)&&Vecrev(n=binary(n))==n \\ M. F. Hasler, Feb 23 2018

from sympy import isprime
def ok(n): return isprime(n) and (b:=bin(n)[2:]) == b[::-1]
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Apr 20 2024

Sum_{n>=1} 1/a(n) = A194097. - Amiram Eldar, Mar 19 2021

More terms from Patrick De Geest

A095759 Triangle T(row>=0, 0<= pos <=row) by rows: T(r,p) contains number of odd primes p in range [2^(r+1),2^(r+2)] for which A037888(p)=pos.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 2, 3, 0, 0, 0, 5, 2, 0, 0, 3, 4, 6, 0, 0, 0, 0, 15, 4, 4, 0, 0, 0, 3, 18, 15, 7, 0, 0, 0, 0, 0, 32, 20, 16, 7, 0, 0, 0, 0, 7, 33, 63, 24, 10, 0, 0, 0, 0, 0, 0, 63, 62, 88, 33, 9, 0, 0, 0, 0, 0, 12, 81, 135, 154, 56, 26, 0, 0, 0, 0, 0, 0, 0, 119, 150, 314, 197, 72, 20, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Jun 12 2004

			a(0) = T(0,0) = 1 as there is one prime 3 (11 in binary) in range ]2^1,2^2[ whose binary expansion is palindromic. a(1) = T(1,0) = 2 as there are two primes, 5 and 7 (101 and 111 in binary) in range ]2^2,2^3[ whose binary expansions are palindromic. a(2) = T(1,1) = 0, as there are no other primes in that range. a(3) = T(2,0) = 0, as there are no palindromic primes in range ]2^3,2^4[, but a(4) = T(2,1) = 2 as in the same range there are two primes 11 and 13 (1011 and 1101 in binary), whose binary expansion needs a flip of just one bit to become palindrome.

Row sums: A036378. Bisection of the leftmost diagonal: A095741. Next diagonals: A095753, A095754, A095755, A095756. Central diagonal (column): A095760. The rightmost nonzero terms from each row: A095757 (i.e. central diagonal and next-to-central diagonal interleaved). The penultimate nonzero terms from each row: A095758. Cf. also A095749, A048700-A048704, A095742.

A194097 Decimal expansion of the sum of the reciprocals of A016041 (primes that are binary palindromes).

Original entry on oeis.org

8, 1, 5, 7, 1, 0, 1, 9, 6, 2, 9, 0

Offset: 0

Kausthub Gudipati, Aug 15 2011

			Partial sum is 0.81571010760865175... if primes up to 40 binary digits are included; partial sum is 0.8157101534554485... if summed to 42 binary digits. - _R. J. Mathar_, Aug 27 2011
Partial sum is 0.81571019628323164... for primes up to 66 bits. - _Martin Raab_, Jan 13 2012

Ranta, Prime Curios!

Cf. A095741, A118064, A244162.

a(6)-a(11) from Martin Raab, Jan 13 2012

Edited by Jon E. Schoenfield, Nov 27 2016

A095731 Number of such primes p (A095730) such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045) and p's Zeckendorf-expansion A014417(p) is palindromic.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 3, 3, 0, 4, 8, 0, 15, 4, 0, 20, 42, 0, 44, 35, 0, 67, 147, 0, 231, 147, 0, 209, 538, 0, 833, 450, 0, 819, 2064, 0, 1701

Offset: 1

Antti Karttunen, Jun 12 2004

A. Karttunen, J. Moyer: C-program for computing the initial terms of this sequence

Cf. A095732, A095741.

A117773 Number of palindromic primes in base 2 with exactly n binary digits.

Original entry on oeis.org

0, 1, 2, 0, 2, 0, 3, 0, 3, 0, 7, 0, 12, 0, 23, 0, 40, 0, 94, 0, 142, 0, 271, 0, 480, 0, 856, 0, 1721, 0, 3099, 0, 5572, 0, 10799, 0, 20782, 0, 39468, 0, 72672, 0, 139867, 0, 274480, 0, 520376, 0, 986318, 0, 1914097, 0, 3726617, 0, 7107443, 0, 13682325, 0, 26430797, 0, 51412565, 0, 99204128, 0, 190457946, 0

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 2), i.e., by 3 in base 10, and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits, namely 11_2 = 3_{10}.

Chai Wah Wu, Table of n, a(n) for n = 1..76
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 34.
Eric Weisstein's World of Mathematics, Palindromic Prime.

Cf. A016041, A117697, A095741.

Array[If[And[OddQ[#], # > 1], 0, Count[Prime@ Range[PrimePi[2^#] - Boole[# == 1] + 1, PrimePi[2^(# + 1) - 1]], ?(PalindromeQ[IntegerDigits[#, 2]] &)]] &, 25, 0] (* _Michael De Vlieger, Sep 29 2023 *)

a(23)-a(40) from Donovan Johnson, Dec 02 2009

a(41)-a(66) from Martin Raab, Oct 20 2015

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A016041 Primes that are palindromic in base 2 (but written here in base 10).

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A095759 Triangle T(row>=0, 0<= pos <=row) by rows: T(r,p) contains number of odd primes p in range [2^(r+1),2^(r+2)] for which A037888(p)=pos.

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A194097 Decimal expansion of the sum of the reciprocals of A016041 (primes that are binary palindromes).

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A095731 Number of such primes p (A095730) such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045) and p's Zeckendorf-expansion A014417(p) is palindromic.

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A117773 Number of palindromic primes in base 2 with exactly n binary digits.

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