OFFSET
1,3
COMMENTS
It is interesting that the graph of this sequence appears almost identical as the maximum value of n increases by factors of 10. Compare the graph of the b-file (having numbers up to 10^6) with the plot of the terms up to 10^8. - T. D. Noe, Apr 28 2012
If iterated digit sum (A010888, A056992) is used instead of just digit sum (A007953, A004159), we get A090570 of which this sequence is a subset. - Jeppe Stig Nielsen, Feb 18 2015
Hare, Laishram, & Stoll show that this sequence (indeed, even its subsequence A254066) is infinite. In particular for each k in {846, 847, 855, 856, 864, 865, 873, ...} there are infinitely many terms in this sequence not divisible by 10 that have digit sum k. - Charles R Greathouse IV, Aug 25 2015
There are infinitely many n such that both n and n+1 are in the sequence. This includes A002283. - Robert Israel, Aug 26 2015
LINKS
Zak Seidov, Table of n, a(n) for n = 1..8354 (to a(n) = 10^6)
Code Golf StackExchange, Equality in the sum of digits, coding challenge started Mar 11 2016.
K. G. Hare, S. Laishram, and T. Stoll, The sum of digits of n and n^2, International Journal of Number Theory 7:7 (2011), pp. 1737-1752.
T. D. Noe, Plot of terms up to 10^8
FORMULA
EXAMPLE
Digit sum of 9 = 9 9^2 = 81, 8+1 = 9 digit sum of 145 = 1+4+5 = 10 145^2 = 21025, 2+1+0+2+5 = 10 digit sum of 954 = 9+5+4 = 18 954^2 = 910116, 9+1+0+1+1+6 = 18. - Florian Roeseler (hazz_dollazz(AT)web.de), May 03 2010
MAPLE
sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if sd(n) = sd(n^2) then n else end if end proc; seq(a(n), n = 0 .. 1400); # Emeric Deutsch, May 11 2010
select(t -> convert(convert(t, base, 10), `+`)=convert(convert(t^2, base, 10), `+`),
[seq(seq(9*i+j, j=0..1), i=0..1000)]); # Robert Israel, Aug 26 2015
MATHEMATICA
Select[Range[0, 1200], Total[IntegerDigits[#]]==Total[IntegerDigits[ #^2]]&] (* Harvey P. Dale, Jun 14 2011 *)
PROG
(Haskell)
import Data.List (elemIndices)
import Data.Function (on)
a058369 n = a058369_list !! (n-1)
a058369_list =
elemIndices 0 $ zipWith ((-) `on` a007953) [0..] a000290_list
-- Reinhard Zumkeller, Aug 17 2011
(PARI) is(n)=sumdigits(n)==sumdigits(n^2) \\ Charles R Greathouse IV, Aug 25 2015
(Magma) [n: n in [0..1200] |(&+Intseq(n)) eq (&+Intseq(n^2))]; // Vincenzo Librandi, Aug 26 2015
(Python)
def ds(n): return sum(map(int, str(n)))
def ok(n): return ds(n) == ds(n**2)
def aupto(nn): return [m for m in range(nn+1) if ok(m)]
print(aupto(1189)) # Michael S. Branicky, Jan 09 2021
CROSSREFS
KEYWORD
base,easy,nice,nonn
AUTHOR
G. L. Honaker, Jr., Dec 17 2000
EXTENSIONS
Edited by N. J. A. Sloane, May 30 2010
STATUS
approved