|
| |
|
|
A104242
|
|
Primes which are the concatenation of two consecutive square numbers.
|
|
4
|
|
|
|
6481, 144169, 324361, 400441, 784841, 16001681, 23042401, 67246889, 77447921, 84648649, 92169409, 96049801, 1254412769, 1638416641, 1742417689, 1960019881, 2016420449, 4752447961, 5382454289, 5664457121, 5760058081, 6051661009
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
The second prime in this sequence, 144169, arises in the theory of modular forms, as observed by Hecke. On page 671 of Hecke (1937), Hecke works out the cusp forms of weight 24 and observes that the Hecke operators have eigenfunctions with Fourier coefficients in the quadratic field of discriminant 144169. Thanks to Jerrold B. Tunnell for this comment. See also the articles by Hida and Zagier. N. J. A. Sloane, Sep 13 2014
|
|
|
REFERENCES
|
E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, Math. Annalen, 114 (1937), 1-28; Werke pp. 644-671. See page 671.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The first term is 6481 which is a prime and is the concatenation of 64 and 81 which are two consecutive square numbers.
|
|
|
MAPLE
|
catn:= proc(a, b) 10^(1+ilog10(b))*a+b end proc:
R:= NULL: count:= 0:
for x from 2 by 2 while count < 100 do
y:= catn(x^2, (x+1)^2);
if isprime(y) then count:= count+1; R:= R, y; fi
od:
|
|
|
PROG
|
(Python)
from sympy import isprime
for n in range(1, 2000):
....x = int(str(n**2)+str((n+1)**2))
....if isprime(x):
|
|
|
CROSSREFS
|
A090738 gives the numbers n such that a(n) = (n^2 concatenated with (n+1)^2) is prime.
|
|
|
KEYWORD
|
base,nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
