A176838 - OEIS

%I #24 Sep 03 2021 13:54:57

%S 17,157,257,277,397,677,877,997,1217,1697,1997,2417,2777,3257,3517,

%T 3697,4157,4177,5077,5197,5897,6277,7417,7517,8377,9397,9497,9677,

%U 9857,11197,11597,12157,12457,12697,13397,13477,13877,14057,14197,15017,16477,17597,18097

%N Primes p such that p^3 = q//3 for a prime q, where "//" denotes concatenation.

%C Diophantine equation p^3 = 10 * q + 3 with side condition p and q prime. Necessarily the LSD for such primes p is e = 7 and the two least significant digit strings are "17", "57", "77" or "97".

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Theory, Applications, Generalizations, Cambridge University Press, 2003.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford University Press, 1980.

%D F. Padberg, Zahlentheorie und Arithmetik, Spektrum Akademie Verlag, Heidelberg - Berlin 1999.

%H Harvey P. Dale, <a href="/A176838/b176838.txt">Table of n, a(n) for n = 1..1000</a>

%e 17^3 = 4913 = prime(94)//3, 17 = prime(7) is the first term.

%e 157^3 = 3869893 = prime(32838)//3, 157 = prime(37) is the second term.

%p q:= n-> isprime(iquo(n^3, 10, 'd')) and d=3:

%p select(q, [ithprime(i)$i=1..3000])[];  # _Alois P. Heinz_, Sep 03 2021

%t Select[Range[7,20000,10],PrimeQ[#]&&PrimeQ[FromDigits[Most[IntegerDigits[ #^3]]]]&] (* _Harvey P. Dale_, Oct 03 2013 *)

%o (Python)

%o from sympy import isprime, primerange

%o def ok(p): q, r = divmod(p**3, 10); return r == 3 and isprime(q)

%o print(list(filter(ok, primerange(2, 18200)))) # _Michael S. Branicky_, Aug 31 2021

%o (PARI) isok(p) = if (isprime(p), my(v=divrem(p^3,10)); isprime(v[1]) && (v[2] == 3)); \\ _Michel Marcus_, Sep 03 2021

%Y Cf. A000040, A000578, A174979.

%K base,nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 27 2010