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A379831 Positions of records in A379857.

%I A379831 #36 Feb 09 2025 21:21:02
%S A379831 0,25,50,90,146,169,260,289,425,529,625,900,1156,1521,1681,2025,2500,
%T A379831 2704,3434,3600,4225,4624,4900,5625,7146,7225,8281,9409,10404,11236,
%U A379831 11881,13225,14400,15129,16900,18769,19600,21316,23409,25281,26896,28561,30625,32400,34969,36100,40000,41209,44944,47524
%N A379831 Positions of records in A379857.
%C A379831 This sequence is infinite in length.
%C A379831 Not all terms are squares, meaning that sometimes k=1 is not among the k values which set a new record.
%C A379831 Elements of this sequence are often, but not always, records of A033461.
%H A379831 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A379831 a(n) >> n^3. - _Charles R Greathouse IV_, Jan 06 2025
%e A379831 25 is a member of this sequence, as 25 = 5^2 = 3^2 + 4^2, meaning that it can be expressed as a sum of k squares for two values of k, in this case 1 and 2, which is more than any smaller value.
%e A379831 50 is a member of this sequence, as 50 = 7^2 + 1^2 = 5^2 + 4^2 + 3^2 = 6^2 + 3^2 + 2^2 + 1^2, giving 3 possible values of k, being 2, 3 and 4, more than any smaller value.
%e A379831 0 begins this sequence, as it is the sum of zero squares, so it has one possible value of k, being k = 0.
%o A379831 (Python)
%o A379831 MAXSQUARE = 500
%o A379831 possibleSums = {i: [[], []] for i in range(MAXSQUARE ** 2 + 1)}
%o A379831 possibleSums[0] = [[0], [0]]
%o A379831 for val in range(MAXSQUARE ** 2):
%o A379831     for posSquare in range(len(possibleSums[val][0])):
%o A379831         newSum = possibleSums[val][0][posSquare] + 1
%o A379831         curr = possibleSums[val][1][posSquare] + 1
%o A379831         while val + curr ** 2 <= MAXSQUARE ** 2:
%o A379831             nVal = val + curr ** 2
%o A379831             if newSum not in possibleSums[nVal][0]:
%o A379831                 possibleSums[nVal][0].append(newSum)
%o A379831                 possibleSums[nVal][1].append(curr)
%o A379831             else:
%o A379831                 index = possibleSums[nVal][0].index(newSum)
%o A379831                 if curr < possibleSums[nVal][1][index]:
%o A379831                     possibleSums[nVal][1][index] = curr
%o A379831             curr += 1
%o A379831 best = 0
%o A379831 for i in range(MAXSQUARE ** 2 + 1):
%o A379831     if len(possibleSums[i][0]) > best:
%o A379831         print(i)
%o A379831         best = len(possibleSums[i][0])
%o A379831 records = ()
%o A379831 best = 0
%o A379831 for i in range(MAXSQUARE ** 2 + 1):
%o A379831     sums = len(possibleSums[i][0])
%o A379831     if sums > best:
%o A379831         records += (i, )
%o A379831         best = sums
%Y A379831 Cf. A033461, A003995, A001156, A379857.
%K A379831 nonn
%O A379831 1,2
%A A379831 _Luke E. Holland_, Jan 03 2025