cp's OEIS Frontend

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

%I A104242 #27 May 22 2025 10:21:35
%S A104242 6481,144169,324361,400441,784841,16001681,23042401,67246889,77447921,
%T A104242 84648649,92169409,96049801,1254412769,1638416641,1742417689,
%U A104242 1960019881,2016420449,4752447961,5382454289,5664457121,5760058081,6051661009
%N A104242 Primes which are the concatenation of two consecutive square numbers.
%C A104242 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
%D A104242 E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, Math. Annalen, 114 (1937), 1-28; Werke pp. 644-671. See page 671.
%H A104242 Robert Israel, <a href="/A104242/b104242.txt">Table of n, a(n) for n = 1..10000</a>
%H A104242 Haruzo Hida, <a href="https://web.archive.org/web/20160428201334/http://www.math.ucla.edu/~hida/Sofia13Sum_sld.pdf">Arithmetic of Weil numbers and Hecke fields.</a>
%H A104242 Don Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf">Elliptic Modular Forms and Their Applications</a>
%e A104242 The first term is 6481 which is a prime and is the concatenation of 64 and 81 which are two consecutive square numbers.
%p A104242 catn:= proc(a,b) 10^(1+ilog10(b))*a+b end proc:
%p A104242 R:= NULL: count:= 0:
%p A104242 for x from 2 by 2 while count < 100 do
%p A104242   y:= catn(x^2,(x+1)^2);
%p A104242   if isprime(y) then count:= count+1; R:= R, y; fi
%p A104242 od:
%p A104242 R; # _Robert Israel_, May 19 2020
%o A104242 (Python)
%o A104242 from sympy import isprime
%o A104242 A104242_list = []
%o A104242 for n in range(1,2000):
%o A104242     x = int(str(n**2)+str((n+1)**2))
%o A104242     if isprime(x):
%o A104242         A104242_list.append(x) # _Chai Wah Wu_, Sep 13 2014
%Y A104242 A090738 gives the numbers n such that a(n) = (n^2 concatenated with (n+1)^2) is prime.
%Y A104242 These are the primes in A246973. Cf. A104301, A246972.
%K A104242 base,nonn
%O A104242 1,1
%A A104242 _Shyam Sunder Gupta_, Apr 17 2005