A082768 -id:A082768 - cp's OEIS frontend

A082769 a(n) = smallest palindromic prime that begins with A082768(n), or 0 if no such number exists.

11, 3, 7, 919, 101, 11, 12421, 131, 14341, 151, 16061, 17471, 181, 191, 30103, 313, 32323, 33533, 34543, 353, 36263, 373, 383, 39293, 70207, 71317, 727, 73037, 74047, 757, 76367, 77377, 787, 797, 90709, 919, 929, 93139, 94049, 95959, 96269, 97379, 98389, 9902099, 1003001, 101, 1022201, 10301, 1043401, 10501, 10601, 1074701, 1082801, 1092901

Offset: 1

Amarnath Murthy, Apr 18 2003

Conjecture: no entry is zero.

Cf. A082768, A082770.

ispali := proc(n,b)
    local dgs,i ;
    dgs := convert(n,base,b) ;
    for i from 1 to nops(dgs)/2 do
        if op(i,dgs) <> op(-i,dgs) then
            return false;
        end if;
    end do:
    true;
end proc:
L082768 := [seq(A082768(n),n=1..200)] ; # use code in A082768
L082769 := [seq(0,n=1..200)] ;
for pi from 2 do
    p :=ithprime(pi) ;
    if ispali(p,10) then
        pdgs := convert(p,base,10) ;
        for sh from 0 do
            restp := add(op(i,pdgs)*10^(i-1),i=1..nops(pdgs)) ;
            for i from 1 to nops(L082768) do
                if op(i,L082768) = restp then
                    if op(i,L082769) = 0 then
                        L082769 := subsop(i=p,L082769) ;
                        print(L082769) ;
                    end if;
                end if;
            end do:
            # chop digits from palindromic prime starting at least signif
            pdgs := subsop(1=NULL,pdgs) ;
            if nops(pdgs) = 0 then
                break ;
            end if;
        end do:
    end if;
end do: # R. J. Mathar, Aug 27 2025

More terms from David Wasserman, Jul 28 2005

12 more terms from R. J. Mathar, Aug 27 2025

A082770 a(n) = smallest palindromic prime that begins with A082768(n) and contains more than twice the number of digits in A082768(n), or 0 if no such number exists.

Original entry on oeis.org

101, 313, 727, 919, 10301, 11311, 12421, 13331, 14341, 15451, 16061, 17471, 18181, 19391, 30103, 31013, 32323, 33533, 34543, 35053, 36263, 37273, 38083, 39293, 70207, 71317, 72227, 73037, 74047, 75557, 76367, 77377, 78487, 79397, 90709

Offset: 1

Amarnath Murthy, Apr 18 2003

Conjecture: no entry is zero. In most cases the number of digits required is 2k+1 where k is the number of digits in A082768(n). What is the first entry that requires more (than 2k+1) digits?

Answer to question: A082768(37) = 92, a(37) = 9200029. - David Wasserman, Jul 28 2005

Cf. A082768, A082769.

More terms from David Wasserman, Jul 28 2005

A045572 Numbers that are odd but not divisible by 5.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153

Offset: 1

Jeff Burch, Dec 11 1999

Contains the repunits R_n, (A000042 or A002275): For any m in the sequence (divisible by neither 2 nor 5), Euler's theorem (i.e., m | 10^m - 1 = 9*R_n) guarantees that R_n is always some multiple of m (see A099679) and thus forms a subsequence. - Lekraj Beedassy, Oct 26 2004
Inverse formula: n = 4*floor(a(n)/10) + floor((a(n) mod 10)/3) + 1. - Carl R. White, Feb 06 2008
Numbers ending with 1, 3, 7 or 9. - Lekraj Beedassy, Apr 04 2009
Complement of A065502. - Reinhard Zumkeller, Nov 15 2009
Union of evenish and oddish numbers, cf. A045797, A045798. - Reinhard Zumkeller, Dec 10 2011
Numbers k such that k^(4*j) mod 10 = 1, for any j. - Gary Detlefs, Jan 03 2012
Numbers coprime to 10. - Charles R Greathouse IV, Sep 05 2013
This is also the sequence of numbers such that all their divisors are the sum of the proper divisors of some number (see A001065 (sum of proper divisors) and A078923 (possible values of sigma(n)-n)). This is due to the fact that in the set of untouchable numbers (A005114) there are only 2 prime numbers (2 and 5) and all other terms are even composite. - Michel Marcus, Jun 14 2014
Numbers n for which A001589(n) is divisible by 5. - Bruno Berselli, Jun 18 2014
For a(n) > 1, positive integers x such that the decimal representation of 1/x is purely periodic after the decimal point (1/x is a repeating decimal with no non-repeating portion). - Doug Bell, Aug 05 2015
The asymptotic density of this sequence is 2/5. - Amiram Eldar, Oct 18 2020

			a(18) = 10*floor(17/4) + 2*floor( (4*(17 mod 4) + 1)/3 ) + 1
      = 10*4 + 2*floor( (4*(1)+1)/3 ) + 1
      = 40 + 2*floor(5/3) + 1
      = 40 + 2*1 + 1
      = 43.
G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Relative complement of A017329 in A005408.

Cf. A000035, A000042, A001065, A001589, A002275, A005114, A045797, A045798, A065502, A078923, A079998, A082768 (numbers that begin with 1, 3, 7 or 9), A085820, A099679.

a045572 n = a045572_list !! (n-1)
a045572_list = filter ((/= 0) . (`mod` 5)) a005408_list
-- Reinhard Zumkeller, Dec 10 2011

[ 2*n + 2*Floor((n-3)/4) + 1: n in [1..70] ]; // Vincenzo Librandi, Aug 01 2011

A045572:=n->2*n + 2*floor((n-3)/4) + 1; seq(A045572(n), n=1..50); # Wesley Ivan Hurt, Jun 14 2014

Flatten[Table[10n + {1, 3, 7, 9}, {n, 0, 19}]] (* Alonso del Arte, Jan 13 2012 *)
Select[2Range@78 -1, Mod[#, 5] > 0 &] (* Robert G. Wilson v, Apr 02 2017 *)
Map[(1/2*(5*# + Mod[3*# + 2, 4] - 4))&, Range[10^3]] (* Mikk Heidemaa, Nov 23 2017 *)

a(n)=10*(n>>2)+[-1,1,3,7][n%4+1] \\ Charles R Greathouse IV, Jul 31 2011

is(n)=gcd(n,10)==1 \\ Charles R Greathouse IV, Sep 05 2013

{a(n) = 2*n - 1 + (n+1) \ 4 * 2}; /* Michael Somos, Jun 15 2014 */

def A045572(n): return 2*(n+(n+1)//4) - 1 # Chai Wah Wu, Jan 08 2022

a(n) = 10*floor((n-1)/4) + 2*floor( (4*((n-1) mod 4) + 1)/3 ) + 1; a(n) = a(n-1) + 2 + 2*floor(((x+6) mod 10)/9). - Carl R. White, Feb 06 2008
a(n) = 2*n + 2*floor((n-3)/4) + 1. - Kenneth Hammond (weregoose(AT)gmail.com), Mar 07 2008
a(n) = -1 + 2*n + 2*floor((n+1)/4). - Kenneth Hammond (weregoose(AT)gmail.com), Mar 25 2008
From R. J. Mathar, Sep 22 2009: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1+x) * (x^2+1) * (x-1)^2). (End)
A000035(a(n))*(1 - A079998(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2009
a(n) = (10*n + 2*(-1)^(n*(n+1)/2) - (-1)^n - 5)/4. - Bruno Berselli, Nov 06 2011
G.f.: x * (1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x) * (1 - x^4)). - Michael Somos, Jun 15 2014
a(1 - n) = -a(n) for all n in Z. - Michael Somos, Jun 15 2014
0 = (a(n) - 2*a(n+1) + a(n+2)) * (a(n) - 4*a(n+2) + 3*a(n+3)) for all n in Z. - Michael Somos, Jun 15 2014
From Mikk Heidemaa, Nov 22 2017: (Start)
a(n) = (1/2)*(5*n + ((3*n + 2) mod 4) - 4);
a(n) = (1/4)*((-1)^(n + 1) + 10*n + 2*cos((n*Pi)/2) - 2*sin((n*Pi)/2) - 5);
a(n) = (1/4)*((-1)^(1 + n) + (1 - i)*exp(-(1/2)*i*n*Pi) + (1 + i)*exp(i*n*Pi/2) + 10*n - 5) (for n > 0), where i is the imaginary unit. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(10-2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 12 2021
E.g.f.: (2 + cos(x) + (5*x - 3)*cosh(x) - sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Dec 07 2022

A089743 Numbers that are the leading digits of a palindromic prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118

Offset: 1

Amarnath Murthy, Nov 19 2003

Conjecture: every number beginning with 1, 3, 7 or 9 begins infinitely many palindromic primes.

Cf. A082768, A089744.

Edited and extended by David Wasserman, Jul 28 2005

A054658 Primes beginning 1, 3, 7, 9 whose reversals are nonprimes.

Original entry on oeis.org

19, 103, 109, 127, 137, 139, 163, 173, 193, 197, 307, 317, 331, 349, 367, 379, 397, 719, 773, 911, 947, 977, 997, 1013, 1019, 1039, 1049, 1051, 1063, 1087, 1093, 1117, 1123, 1129, 1163, 1171, 1187, 1277, 1289, 1291, 1297, 1303, 1307, 1319, 1327, 1361

Offset: 1

Enoch Haga, Apr 18 2000

Or, primes whose reversals are composites ending in 1,3,7,9. - Lekraj Beedassy, Aug 02 2008

A subsequence of A143260. - Lekraj Beedassy, Aug 02 2008

			a(1)=19 because its reverse is a nonprime, 91.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A143260, A076056, A082768.

pbQ[p_]:=MemberQ[{1,3,7,9},IntegerDigits[p][[1]]]&&CompositeQ[IntegerReverse[p]]; Select[Prime[Range[300]],pbQ] (* Harvey P. Dale, Dec 02 2024 *)

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

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A082769 a(n) = smallest palindromic prime that begins with A082768(n), or 0 if no such number exists.

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A082770 a(n) = smallest palindromic prime that begins with A082768(n) and contains more than twice the number of digits in A082768(n), or 0 if no such number exists.

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A045572 Numbers that are odd but not divisible by 5.

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A089743 Numbers that are the leading digits of a palindromic prime.

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A054658 Primes beginning 1, 3, 7, 9 whose reversals are nonprimes.

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