A194097 -id:A194097 - 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

A118031 Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.

Original entry on oeis.org

3, 3, 7, 0, 2, 8, 3, 2, 5, 9, 4, 9, 7, 3, 7, 3, 3, 2, 0, 4, 9, 2, 1, 5, 7, 2, 9, 8, 5, 0, 5, 5, 3, 1, 1, 2, 3, 0, 7, 1, 4, 5, 7, 7, 7, 9, 4, 5, 2, 7, 7, 8, 4, 9, 1, 3, 3, 5, 0, 6, 8, 9, 2, 5, 9, 8, 2, 5, 1, 9, 7, 6, 0, 3, 4, 9, 4, 7, 6, 7, 5, 8, 9, 7, 0, 3, 0, 1

Offset: 1

Martin Renner, May 11 2006

The sum using all palindromic numbers < 10^8 is 3.37000183240... Extrapolating using Wynn's epsilon method gives a value near 3.37018... - Eric W. Weisstein, May 14 2006

			3.3702832594973733204921572985...

Joseph Myers, Table of n, a(n) for n = 1..1001
Joseph Myers, Polynomial-time algorithm.
Michael Penn, Does this series converge??, YouTube video, 2021.
Radovan Potůček, Formulas for the Sums of the Series of Reciprocals of the Polynomial of Degree Two with Non-zero Integer Roots, Algorithms as a Basis of Modern Applied Mathematics, Studies in Fuzziness and Soft Computing book series (STUDFUZZ, Vol. 404) Springer (2021), 363-382.
Eric Weisstein's World of Mathematics, Palindromic Number.

Cf. A002113.

Similar sequences: A118064, A194097, A244162.

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits@ n}, If[ Union@ idn == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[ idn, Ceiling[l/2]]]] > FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[Join[Take[idn, Ceiling[l/2]], Reverse[Take[idn, Floor[l/2]]]]], idfhn = FromDigits[Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits@ idfhn, Drop[ Reverse[ IntegerDigits@ idfhn], Mod[l, 2]]]]]]]]; pal = 1; sm = 0; Do[ While[pal < 10^n + 1, sm = N[sm + 1/pal, 128]; pal = NextPalindrome@ pal]; Print[{n, sm}], {n, 0, 17}] (* Robert G. Wilson v, Oct 20 2010 *)

a(n) = Sum_{palindromes p>0} 1/p.

a(n) = Sum_{n>=2} 1/A002113(n).

Corrected by Eric W. Weisstein, May 14 2006
Corrected and extended by Robert G. Wilson v, Oct 20 2010
Corrected and extended by Joseph Myers, Jun 26 2014

A244162 Decimal expansion of the sum of the reciprocals of the binary palindromic numbers.

Original entry on oeis.org

2, 3, 7, 8, 7, 9, 5, 7, 0, 7, 5, 4, 1, 3, 6, 1, 0, 0, 2, 3, 3, 5, 3, 0, 1, 6, 8, 4, 4, 2, 8, 2, 5, 0, 6, 3, 2, 3, 5, 3, 2, 8, 0, 0, 8, 4, 4, 6, 5, 8, 5, 4, 5, 5, 9, 1, 8, 3, 8, 7, 8, 7, 2, 1, 0, 3, 2, 3, 4, 2, 0, 1, 6, 7, 8, 6, 5, 1, 5, 1, 4, 0, 8, 4, 6, 8, 6, 2

Offset: 1

Franklin T. Adams-Watters, Jun 21 2014

			2.3787957...

Joseph Myers, Table of n, a(n) for n = 1..301
Joseph Myers, Polynomial-time algorithm.
Phakhinkon Phunphayap, Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, (2019).

Cf. A006995, A194097, A118031.

First 8 terms from Lars Blomberg

Extended by Joseph Myers, Jun 26 2014

A118064 Decimal expansion of the sum of the reciprocals of the palindromic primes A002385 (Honaker's constant).

Original entry on oeis.org

1, 3, 2, 3, 9, 8, 2, 1, 4, 6, 8, 0, 6

Offset: 1

Martin Renner, May 11 2006

From Robert G. Wilson v, Nov 01 2010: (Start)
n \ sum to 10^n
1.267099567099567099567099567099567099567099567099567099567099567099567
1.320723244590290964212793334437872849720871258315369002493912638038324
1.323748402250648554164425746280035962754669829327727800040192015109270
1.323964105671202458016249150576217276147952428601889817773483085610332
1.323980718065525060936354534562000413901564393192688451911141729415146
1.323982026479475203850120990923294207966175748395470136325039323549015
1.323982136437462724794656629740867909978221153827990721566573347887836
1.323982145891606234777299440047139038371441916546100653011463101470839
1.323982146724859090645464845257681674740147563533254654075059843860490
1.323982146799188851138232927173756400348958236915409881890097448921521
1.323982146805857558347279363344557427339916178257233985191868031567947 (End)

			1.323982146806...

Carlos Rivera, Problems & Puzzles: Puzzle 056 - Honaker's Constant.
Eric Weisstein, Palindromic Prime.

Cf. A002385, A160910, A181442, A050251, A118031, A194097.

(* first obtain nextPalindrome from A007632 *) s = 1/11; c = 1; pp = 1; Do[ While[pp < 10^n, If[PrimeQ@ pp, c++; s = N[s + 1/pp, 64]]; pp = NextPalindrome@ pp]; If[ OddQ@ n, pp = 10^(n + 1); Print[{s, n, c}]], {n, 17}] (* Robert G. Wilson v, May 31 2009 *)
generate[n_] := Block[{id = IntegerDigits@n, insert = {{0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; sm = N[Plus @@ (1/{2, 3, 5, 7, 11}), 64]; k = 1; Do [While[k < 10^n, sm = N[sm + Plus @@ (1/Select[ generate@k, PrimeQ]), 128]; k++ ]; Print[{2 n + 1, sm}], {n, 9}] (* Robert G. Wilson v, Nov 01 2010 *)

Equals Sum_{p} 1/p, where p ranges over the palindromic primes.

Corrected by Eric W. Weisstein, May 14 2006
More terms from Robert G. Wilson v, Nov 01 2010
Entry revised by N. J. A. Sloane, May 05 2013

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

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A118031 Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.

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A244162 Decimal expansion of the sum of the reciprocals of the binary palindromic numbers.

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A118064 Decimal expansion of the sum of the reciprocals of the palindromic primes A002385 (Honaker's constant).

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