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A293832 Smallest of a run of n consecutive integers in A061910, i.e., for which the sum of digits of their square is again a square. 3
1, 1, 1, 9, 9, 9, 9, 46045846, 302260461719025 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(9) > 10^11. It can be expected that a(n) exists for all n, at an order of magnitude a(n) ~ 10^m with m/log_10(m) ~ n.

A061910 considers only positive integers, and here we also consider this case. If 0 were allowed, then the first 4 terms would be 0.

LINKS

Table of n, a(n) for n=1..9.

User 'mathlove', Numbers n such that the digit sum of n² is a square, Mathematics StackExchange, Jan. 6, 2015.

EXAMPLE

For k = 1, 2 and 3, k^2 is a single-digit number and thus equal to its sum of digits, which therefore is a square. Therefore a(n) = 1 starts the first run of n consecutive integers with this property, for n = 1, 2 and 3.

However, the square of k = 4 has digit sum 7 which is not a square, and the same is the case for k = 5, 7 and 8. (Only k = 6 would have the required property.)

The consecutive integers { 9, 10, 11, 12, 13, 14, 15 } have squares 81, 100, 121, 169, 196, 225 which all have a digit sum (9, 1, 4, 16, 16 and 9) which is a square. Therefore a(n) = 9 starts the first run of n consecutive integers with this property, for n = 4 through 7.

(Actually, 10^(3m-2)^2-1 starts a run of 7 such numbers, for any m >= 1.)

The first run of 8 such numbers is (46045846, ..., 46045853), whence a(8) = 46045846.

PROG

(PARI) A293832(n, k=0)=until(, forstep(j=n, 1, -1, issquare(sumdigits((k+j)^2))||next(2+!k+=j)); return(k+1))

CROSSREFS

Cf. A007953, A061910, A068833.

Sequence in context: A245429 A242893 A275485 * A277223 A144586 A141557

Adjacent sequences:  A293829 A293830 A293831 * A293833 A293834 A293835

KEYWORD

nonn,base,hard,more

AUTHOR

M. F. Hasler, Oct 16 2017

EXTENSIONS

a(9) from Giovanni Resta, Aug 27 2018

STATUS

approved

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Last modified April 5 13:26 EDT 2020. Contains 333241 sequences. (Running on oeis4.)