A176371 Primes p such that reversal(p) - 13 is a square.

31, 41, 71, 83, 281, 311, 431, 479, 733, 751, 797, 2011, 2857, 3163, 4373, 4397, 4943, 7541, 7577, 7583, 9413, 9491, 20533, 20731, 20771, 24151, 24547, 24767, 26249, 28979, 31121, 41201, 41609, 43321, 43391, 43753, 45641, 49459, 49463, 49811, 49891

Offset

1,1

Comments

R(n) denotes the Reversal of a natural number n

List of all (p,N) for p < 10^6 - 1:

(*) for emirp pair (p,R(p)), (+) if square base N is a prime

(41,1), (71,2) (+) (*), (83,5) (+), (281,13) (+), (311,10) (*), (431,11) (+), (479,31) (+), (733,18) (*), (751,12) (*), (797,28),

(2011,33), (2857,87), (4373,61) (+), (4397,89) (+), (4943,59) (+), (7541,38), (7577,88) (*), (7583,62), (9413,56), (9491,44) (*), (20533,183), (20731,117), (20771,133), (24151,123), (24547,273), (24767,277) (+), (26249,307) (+), (28979,313) (+), (31121,110) (*), (41201,101) (+),

(41609,301), (43321,111), (43391,139) (+), (43753,189), (45641,121), (49459,309), (49463,191) (+), (49811,109), (49891,141), (71293,198) (*),

(73133,182), (73471,132), (73597,282) (*), (75521,112), (77611,108) (*), (77849,308), (77863,192) (*), (79613,178), (79841,122) (*), (83207,265),

(83231,115), (83243,185), (83299,315), (90031,114) (*), (92801,104), (96431,116) (*), (98057,274)

References

W. W. R. Ball, H. S. M.Coxeter: Mathematical Recreations and Essays, Dover Publications, 13th edition, 1987

O. Fritsche, R. Mischak and T. Krome: Verflixt und zugeknobelt, Mehr mathematische Raetselgeschichten, Rowohlt TB. Nr.62190, 2007

C. W. Trigg, Primes with Reverses That Are Powers, J. Rec. Math. 17, 1985

Links

Chai Wah Wu, Table of n, a(n) for n = 1..7605

Example

41 = prime(13), R(41) - 13 = 14 - 13 = 1^2, is a term.
71 = prime(20), 17 - 13 = 2^2, is a term.
83 = prime(23), 38 - 13 = 5^2, is a term.
797 = prime(139) = palindromic prime(18), N = 28^2, is also a term.
Note successive terms that are also consecutive primes: p(17) = 7577, p(18) = 7583, p(36) = 49459, p(37) = 49463, p(46) = 77849, p(47) = 77863.

Prog

(PARI) isok(n) = {if (! isprime(n), return (0)); d = digits(n); revn = sum(i=1, #d, d[i]*10^(i - 1)); issquare(revn-13); } \\ Michel Marcus, Aug 25 2013
(Python)
from sympy import isprime
A176371_list, i, j = [], 0, 13
while j < 10**10:
    p = int(str(j)[::-1])
    if j % 10 and isprime(p):
        A176371_list.append(p)
    j += 2*i+1
    i += 1
A176371_list = sorted(A176371_list) # Chai Wah Wu, Dec 17 2015

Crossrefs

Cf. A000040, A000290, A002385, A006567, A007488.

Sequence in context: A202286 A285805 A141180 * A284531 A040987 A040180

Adjacent sequences: A176368 A176369 A176370 * A176372 A176373 A176374

Keyword

base,nonn

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 16 2010

Extensions

Two more terms 31 and 3163 added by Michel Marcus, Aug 25 2013

Status

approved