login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226087 Number of values k in base n for which the sum of digits of k = sqrt(k). 2
1, 4, 2, 3, 3, 6, 2, 2, 2, 5, 2, 6, 2, 5, 5, 2, 2, 4, 2, 6, 6, 4, 2, 5, 2, 4, 2, 6, 2, 11, 2, 2, 6, 4, 5, 6, 2, 4, 6, 5, 2, 11, 2, 6, 5, 4, 2, 6, 2, 4, 6, 5, 2, 4, 5, 5, 6, 4, 2, 13, 2, 4, 4, 2, 5, 11, 2, 5, 6, 11, 2, 5, 2, 4, 6, 6, 6, 11, 2, 5, 2, 4, 2, 12, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Values of k in base n have at most 3 digits. Proof: Because sqrt(k) increases faster than the digit sum of k, only numbers with d digits meeting the condition d*(n-1)>=n^(d/2) are candidate fixed points. Using numeric methods, d<3 for n>6. and since there are no fixed points of four or more digits in bases 2 through 5, there are no fixed points in any base with more than 3 digits.

From the above, it can be shown that for three-digit fixed points of the form xyz, x <= 6; also x<=4 for n>846. These theoretical upper limits are statistically unlikely, and in fact of the 86356 solutions in bases 2 to 10000, only 6.5% of them begin with 2, and none begin with 3 through 6.

LINKS

Christian N. K. Anderson, Table of n, a(n) for n = 2..10000

EXAMPLE

For a(16)=5 the solutions are the square numbers {1, 36, 100, 225, 441} because in base 16 they are written as {1, 24, 64, E1, 1B9} and

sqrt(1) = 1

sqrt(36) = 6 = 2+4

sqrt(100) = 10 = 6+4

sqrt(225) = 15 = 14+1, and

sqrt(441) = 21 = 1+11+9

PROG

(R) sapply(2:16, function(n) sum(sapply((1:(n^ifelse(n>6, 1.5, 2)))^2, function(x) sum(inbase(x, n))==sqrt(x))))

CROSSREFS

Cf. A226224.

Cf. digital sums for digits at various powers: A007953, A003132, A055012,A055013, A055014, A055015.

Sequence in context: A079623 A177864 A090112 * A140396 A118945 A016692

Adjacent sequences:  A226084 A226085 A226086 * A226088 A226089 A226090

KEYWORD

nonn,base

AUTHOR

Christian N. K. Anderson and Kevin L. Schwartz, May 25 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 06:18 EST 2019. Contains 330016 sequences. (Running on oeis4.)