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

Author

Shyam Sunder Gupta, Apr 17 2005

Keywords

• base
• nonn

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

Examples

			The first term is 6481 which is a prime and is the concatenation of 64 and 81 which are two consecutive square numbers.
		

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

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.

Programs

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

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