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See the Mathematical Reflections link for a proof that a(n) exists for all n.
a(4) > 300*10^6.
As shown below, 264575131106460 <= a(4) <= 13663784168010583.
The maximal sum of the last three digits of a square is 25, which occurs only when those digits are 889 (e.g., 83^2 = 6889), so since sumdigits(a(4)^2) = 256, the sum of the digits that precede the last three must be at least 256 - 25 = 231, thus a(4)^2 >= 69999999999999999999999999889 (call this number j), so a(4) >= ceiling(sqrt(j)) = 264575131106460. (This bound could easily be improved; e.g., since j is not a square, m^2 cannot be less than the smallest number greater than j whose digit sum is 256 and whose last three digits are 889, i.e., m^2 >= 78999999999999999999999999889, so m >= 281069386451104.)
The last digit of a square m^2 is maximized (at 9) iff the last digit of m is 3 or 7. The sum of the last two digits of m^2 is maximized (at 17) iff the last two digits are 89, which occurs iff the last two digits of m are 33, 83, 17, or 67. For k >= 3, it appears that the sum of the last k digits of m^2 is maximized (at 9k-2) iff the last k digits are all 9s except for the two 8s immediately before the final 9, which occurs iff the k-digit suffix of m takes one of eight values, as shown in the table below; the line extending upward from each suffix of more than one digit connects it to the suffix from which it inherits all but its first digit.
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k k-digit suffixes of m that maximize sumdigits(m^2 mod 10^k)
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1 3 7
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
2 33 83 17 67
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3 833 333 583 083 417 917 167 667
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4 1833 6833 0583 5583 9417 4417 8167 3167
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5 41833 91833 10583 60583 89417 39417 58167 08167
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6 041833 541833 010583 510583 989417 489417 958167 458167
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7 5041833 0041833 8010583 3010583 1989417 6989417 4958167 9958167
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. ... ... ... ... ... ... ... ...
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Since numbers m ending in these suffixes have squares m^2 such that sumdigits(m^2 mod 10^k) is maximized, their full digit sums sumdigits(m^2) tend to be larger than those of nearby numbers with other suffixes. A search over a range of prefixes using the 10-digit suffix 4168010583 found that m = 13663784168010583 has sumdigits(m^2) = 256, which yields an upper bound for a(4). (End)
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