A050683
Number of nonzero palindromes of length n.
Original entry on oeis.org
9, 9, 90, 90, 900, 900, 9000, 9000, 90000, 90000, 900000, 900000, 9000000, 9000000, 90000000, 90000000, 900000000, 900000000, 9000000000, 9000000000, 90000000000, 90000000000, 900000000000, 900000000000, 9000000000000
Offset: 1
Cf.
A016116 for numbers of binary palindromes,
A016115 for prime palindromes.
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a:=[9,9];; for n in [3..30] do a[n]:=10*a[n-2]; od; a; # Muniru A Asiru, Oct 07 2018
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[9*10^Floor((n-1)/2): n in [1..30]]; // Vincenzo Librandi, Aug 16 2011
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seq(9*10^floor((n-1)/2),n=1..30); # Muniru A Asiru, Oct 07 2018
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With[{c=9*10^Range[0,20]},Riffle[c,c]] (* or *) LinearRecurrence[{0,10},{9,9},40] (* Harvey P. Dale, Dec 15 2013 *)
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A050683(n)=9*10^((n-1)\2) \\ M. F. Hasler, Nov 16 2008
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\\ using M. F. Hasler's is_A002113(n) from A002113
is_A002113(n)={Vecrev(n=digits(n))==n}
for(n=1,8,j=0;for(k=10^(n-1),10^n-1,if(is_A002113(k),j++));print1(j,", ")) \\ Hugo Pfoertner, Oct 03 2018
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is_palindrome(x)={my(d=digits(x));for(k=1,#d\2,if(d[k]!=d[#d+1-k],return(0)));return(1)}
for(n=1,8,j=0;for(k=10^(n-1),10^n-1,if(is_palindrome(k),j++));print1(j,", ")) \\ Hugo Pfoertner, Oct 02 2018
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a(n) = if(n<3, 9, 10*a(n-2)); \\ Altug Alkan, Oct 03 2018
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def A050683(n): return 9*10**(n-1>>1) # Chai Wah Wu, Jul 30 2025
A016115
Number of prime palindromes with n digits.
Original entry on oeis.org
4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, 42042, 0, 353701, 0, 3036643, 0, 27045226, 0, 239093865, 0, 2158090933, 0, 19742800564, 0, 180815391365, 0
Offset: 1
- James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.
- K. S. Brown, On General Palindromic Numbers
- Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 36.
- Patrick De Geest, World!Of Palindromic Primes
- Shyam Sunder Gupta, Palindromic Primes up to 10^19, Feb. 6, 2006.
- Shyam Sunder Gupta, Palindromic Primes up to 10^21, March 13, 2009.
- Shyam Sunder Gupta, Palindromic Primes up to 10^23, Oct. 4, 2013.
- Shyam Sunder Gupta, Palindromic Primes up to 10^25, Dec. 18, 2024.
- Eric Weisstein's World of Mathematics, Palindromic Prime.
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# A016115 Gets numbers of base-10 palindromic primes with exactly d digits, 1 <= d <= 13 (say), in the list "lis"
lis:=[4,1];
for d from 3 to 13 do
if d::even then
lis:=[op(lis),0];
else
m:= (d-1)/2:
Res2 := [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
ct:=0; for x in Res2 do if isprime(x) then ct:=ct+1; fi: od:
lis:=[op(lis),ct];
fi:
lprint(d,lis);
od:
lis; # N. J. A. Sloane, Oct 18 2015
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A016115[n_] := Module[{i}, If[EvenQ[n] && n > 2, Return[0]]; Return[Length[Select[Range[10^(n - 1), 10^n - 1], # == IntegerReverse[#] && PrimeQ[#] &]]]];
Table[A016115[n], {n, 6}] (* Robert Price, May 25 2019 *)
(* -OR- A less straightforward implementation, but more efficient in that the palindromes are constructed instead of testing every number in the range. *)
A016115[n_] := Module[{c, f, t0, t1},
If[n == 2, Return[1]];
If[EvenQ[n], Return[0]];
c = 0; t0 = 10^((n - 1)/2); t1 = t0*10;
For[f = t0, f < t1, f++,
If[n != 1 && MemberQ[{2,4,5,6,8}, Floor[f/t0]], f = f + t0 - 1; Continue[]];
If[PrimeQ[f*t0 + IntegerReverse[Floor[f/10]]], c++]]; Return[c]];
Table[A016115[n], {n, 1, 12}] (* Robert Price, May 25 2019 *)
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apply( {A016115(n)=if(n%2, (n<3)+vecsum([sum(k=i, i+n, (k*2-k%10)%3 && isprime(k*n+fromdigits(Vecrev(digits(k\10))))) | i<-[1, 3, 7, 9]*n=10^(n\2)]), n==2)}, [1..12]) \\ M. F. Hasler, Dec 19 2024
-
from sympy import isprime
from itertools import product
def pals(d, base=10): # all d-digit palindromes
digits = "".join(str(i) for i in range(base))
for p in product(digits, repeat=d//2):
if d > 1 and p[0] == "0": continue
left = "".join(p); right = left[::-1]
for mid in [[""], digits][d%2]: yield int(left + mid + right)
def a(n): return int(n==2) if n%2 == 0 else sum(isprime(p) for p in pals(n))
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Jun 23 2021
a(17) = 27045226 was found by Martin Eibl (M.EIBL(AT)LINK-R.de) and independently by
Warut Roonguthai and later confirmed by
Carlos Rivera, in June 1998.
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
-
(* 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 *)
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