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A375380 Lexicographically earliest sequence S of distinct nonnegative integers such that 9 out of the last 10 digits of S always sum to a square.

%I A375380 #32 Aug 17 2024 12:12:39
%S A375380 1000000000,0,1,2,3,4,6,5,7,8,9,10,11,13,12,15,14,17,16,21,18,19,20,
%T A375380 27,22,25,23,30,24,26,31,28,29,32,34,33,36,38,35,37,40,39,41,42,43,44,
%U A375380 47,45,50,46,51,48,52,54,55,49,57,53,56,58,59,60,64,61,62,63
%N A375380 Lexicographically earliest sequence S of distinct nonnegative integers such that 9 out of the last 10 digits of S always sum to a square.
%C A375380 The smallest available 10-digit integer to start S with is a(1) = 1000000000.
%C A375380 The sequence is infinite as there are infinitely many integers whose 9 of the last 10 digits sum to a square.
%H A375380 Michael S. Branicky, <a href="/A375380/b375380.txt">Table of n, a(n) for n = 1..10000</a>
%e A375380 We start with a(1) = 1000000000:
%e A375380 S = 1000000000,
%e A375380 The last 10 digits of S are [1000000000]; we discard a 0 and the sum of the 9 remaining digits = 1 (a square). We try to extend S with 0:
%e A375380 S = 1000000000,0,
%e A375380 The last 10 digits of S are [0000000000]; we discard a 0 and the sum of the 9 remaining digits = 0 (a square). We try to extend S with 1:
%e A375380 S = 1000000000,0,1,
%e A375380 The last 10 digits of S are [0000000001]; we discard a 0 and the sum of the 9 remaining digits = 1 (a square). We try to extend S with 2:
%e A375380 S = 1000000000,0,1,2,
%e A375380 The last 10 digits of S are [0000000012]; we discard 2 and the sum of the 9 remaining digits = 1 (a square). We try to extend S with 3:
%e A375380 S = 1000000000,0,1,2,3,
%e A375380 The last 10 digits of S are [0000000123]; we discard 2 and the sum of the 9 remaining digits = 4 (a square). We try to extend S with 4:
%e A375380 S = 1000000000,0,1,2,3,4,
%e A375380 The last 10 digits of S are [0000001234]; we discard 1 and the sum of the 9 remaining digits = 9 (a square). We try to extend S with 5:
%e A375380 S = 1000000000,0,1,2,3,4,5,
%e A375380 The last 10 digits of S are [0000012345]; as no square can be reached, whatever we discard, we don’t extend S with 5. We try to extend S with 6:
%e A375380 S = 1000000000,0,1,2,3,4,6,
%e A375380 The last 10 digits of S are [0000012346]; we discard a 0 and the sum of the 9 remaining digits = 16 (a square).
%e A375380 Etc.
%o A375380 (Python)
%o A375380 from itertools import count, islice, combinations
%o A375380 def f(k, last10):
%o A375380     d = list(map(int, str(k)))
%o A375380     if len(d) >= 10: return d[-10:]
%o A375380     else: return last10[len(d):] + d
%o A375380 def c(k, last10):
%o A375380     for pick9 in combinations(f(k, last10), 9):
%o A375380         if sum(pick9) in {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}:
%o A375380             return True
%o A375380     return False
%o A375380 def agen(): # generator of terms
%o A375380     an, seen, last10, mink = 1000000000, set(), [], 0
%o A375380     while True:
%o A375380         yield an
%o A375380         seen.add(an)
%o A375380         last10 = f(an, last10)
%o A375380         an = next(k for k in count(mink) if k not in seen and c(k, last10))
%o A375380         while mink in seen: mink += 1
%o A375380 print(list(islice(agen(), 66))) # _Michael S. Branicky_, Aug 14 2024
%Y A375380 Cf. A352000.
%K A375380 base,nonn
%O A375380 1,1
%A A375380 _Eric Angelini_, Aug 14 2024
%E A375380 a(14) and beyond from _Michael S. Branicky_, Aug 14 2024