A104242 Primes which are the concatenation of two consecutive square numbers.

6481, 144169, 324361, 400441, 784841, 16001681, 23042401, 67246889, 77447921, 84648649, 92169409, 96049801, 1254412769, 1638416641, 1742417689, 1960019881, 2016420449, 4752447961, 5382454289, 5664457121, 5760058081, 6051661009

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

Robert Israel, Table of n, a(n) for n = 1..10000

Haruzo Hida, Arithmetic of Weil numbers and Hecke fields.

Don Zagier, Elliptic Modular Forms and Their Applications

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:
R; # Robert Israel, May 19 2020

Prog

(Python)
from sympy import isprime
A104242_list = []
for n in range(1, 2000):
....x = int(str(n**2)+str((n+1)**2))
....if isprime(x):
........A104242_list.append(x) # Chai Wah Wu, Sep 13 2014

Crossrefs

A090738 gives the numbers n such that a(n) = (n^2 concatenated with (n+1)^2) is prime.

These are the primes in A246973. Cf. A104301, A246972.

Sequence in context: A087351 A104343 A207837 * A031842 A028544 A237245

Adjacent sequences: A104239 A104240 A104241 * A104243 A104244 A104245

Keyword

base,nonn

Author

Shyam Sunder Gupta, Apr 17 2005

Status

approved