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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

Table of n, a(n) for n=0..3.

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.)