A176838 Primes p such that p^3 = q//3 for a prime q, where "//" denotes concatenation.

17, 157, 257, 277, 397, 677, 877, 997, 1217, 1697, 1997, 2417, 2777, 3257, 3517, 3697, 4157, 4177, 5077, 5197, 5897, 6277, 7417, 7517, 8377, 9397, 9497, 9677, 9857, 11197, 11597, 12157, 12457, 12697, 13397, 13477, 13877, 14057, 14197, 15017, 16477, 17597, 18097

Offset

1,1

Comments

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".

References

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

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

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

Links

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

Example

17^3 = 4913 = prime(94)//3, 17 = prime(7) is the first term.
157^3 = 3869893 = prime(32838)//3, 157 = prime(37) is the second term.

Maple

q:= n-> isprime(iquo(n^3, 10, 'd')) and d=3:
select(q, [ithprime(i)$i=1..3000])[];  # Alois P. Heinz, Sep 03 2021

Mathematica

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

Prog

(Python)
from sympy import isprime, primerange
def ok(p): q, r = divmod(p**3, 10); return r == 3 and isprime(q)
print(list(filter(ok, primerange(2, 18200)))) # Michael S. Branicky, Aug 31 2021
(PARI) isok(p) = if (isprime(p), my(v=divrem(p^3, 10)); isprime(v[1]) && (v[2] == 3)); \\ Michel Marcus, Sep 03 2021

Crossrefs

Cf. A000040, A000578, A174979.

Sequence in context: A160699 A034272 A142087 * A176622 A125380 A126538

Adjacent sequences: A176835 A176836 A176837 * A176839 A176840 A176841

Keyword

base,nonn

Author

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

Status

approved