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A339978 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit square, a 2-digit square, a 3-digit square, ..., and an n-digit square, or 0 if there is no such prime.

%I A339978 #28 Dec 30 2020 07:46:55
%S A339978 0,449,981961,9819619801,981961980196721,981961980199856194481,
%T A339978 9819619801998569980018946081,981961980199856998001999824499740169,
%U A339978 981961980199856998001999824499980001989039601,9819619801998569980019998244999800019999508849977812321
%N A339978 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit square, a 2-digit square, a 3-digit square, ..., and an n-digit square, or 0 if there is no such prime.
%C A339978 If a(n) exists it has A000217(n)= n*(n+1)/2 digits.
%C A339978 All the terms end with 1 or 9.
%H A339978 David A. Corneth, <a href="/A339978/b339978.txt">Table of n, a(n) for n = 1..40</a> (first 16 terms from Michael S. Branicky)
%e A339978 a(1) = 0 because no 1-digit square {0, 1, 4, 9} is prime.
%e A339978 a(2) = 449 because 464, 481, 916, 925, 936, 949, 964, and 981 are not primes and 449, concatenation of 4 = 2^2 with 49 = 7^2, is prime.
%e A339978 a(4) = 9819619801, which is a prime is the concatenation of 9 = 3^2 with 81 = 9^2, then 961 = 31^2 and 9801 = 99^2. Observation, 9, 81, 961 and 9801 are the largest squares with respectively 1, 2, 3 and 4 digits.
%o A339978 (Python)
%o A339978 from sympy import isprime
%o A339978 from itertools import product
%o A339978 def a(n):
%o A339978   squares = [str(k*k) for k in range(1, int((10**n)**.5)+2)]
%o A339978   revsqrs = [[kk for kk in squares if len(kk)==i+1][::-1] for i in range(n)]
%o A339978   for t in product(*revsqrs):
%o A339978     intt = int("".join(t))
%o A339978     if isprime(intt): return intt
%o A339978   return 0
%o A339978 print([a(n) for n in range(1, 11)]) # _Michael S. Branicky_, Dec 25 2020
%Y A339978 Cf. A000290, A003618, A061433 (largest squares), A338968 (concatenate primes).
%K A339978 nonn,base
%O A339978 1,2
%A A339978 _Bernard Schott_, Dec 25 2020
%E A339978 a(5)-a(10) from _Michael S. Branicky_, Dec 25 2020