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 A280798 a(n) is the smallest integer m such that sumdigits(m^2) = 4^n. 0
 1, 2, 13, 16667 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See the Mathematical Reflections link for a proof that a(n) exists for all n. a(4) > 300*10^6. From Jon E. Schoenfield, Jan 08 2017: (Start) 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. . k   k-digit suffixes of m that maximize sumdigits(m^2 mod 10^k) = =============================================================== 1               3                               7                / \                             / \               /   \                           /   \              /     \                         /     \             /       \                       /       \            /         \                     /         \           /           \                   /           \          /             \                 /             \ 2      33              83              17              67         |\              |\              |\              |\ 3     833 333         583 083         417 917         167 667         |\              |\              |\              |\ 4    1833 6833       0583 5583       9417 4417       8167 3167         |\              |\              |\              |\ 5   41833 91833     10583 60583     89417 39417     58167 08167         |\              |\              |\              |\ 6  041833 541833   010583 510583   989417 489417   958167 458167         |\              |\              |\              |\ 7 5041833 0041833 8010583 3010583 1989417 6989417 4958167 9958167         |\              |\              |\              |\ .     ... ...         ... ...         ... ...         ... ... . 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) LINKS Mathematical Reflections, Solution to Problem J307, Issue 5, 2015, p. 1. EXAMPLE a(1)=2 since 2^2=4 with sum of digits 4. PROG (PARI) a(n) = my(k=1); while (sumdigits(k^2) != 4^n, k++); k; CROSSREFS Cf. A000302, A007953, A061912. Sequence in context: A306206 A111016 A176947 * A118912 A027680 A241569 Adjacent sequences:  A280795 A280796 A280797 * A280799 A280800 A280801 KEYWORD nonn,base,more AUTHOR Michel Marcus, Jan 08 2017 STATUS approved

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Last modified May 30 12:03 EDT 2020. Contains 334724 sequences. (Running on oeis4.)