A058369 Numbers k such that k and k^2 have same digit sum.

0, 1, 9, 10, 18, 19, 45, 46, 55, 90, 99, 100, 145, 180, 189, 190, 198, 199, 289, 351, 361, 369, 379, 388, 450, 451, 459, 460, 468, 495, 496, 550, 558, 559, 568, 585, 595, 639, 729, 739, 775, 838, 855, 900, 954, 955, 990, 999, 1000, 1098, 1099, 1179, 1188, 1189

Offset: 1

Author

G. L. Honaker, Jr., Dec 17 2000

Keywords

• base
• easy
• nice
• nonn

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

Examples

			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
		

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

Crossrefs

Cf. A002283, A007953, A058370, A058852, A077436, A254066.

Cf. A147523 (number of numbers in each decade).

Subsequence of A090570.

Programs

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

Magma

[n: n in [0..1200] |(&+Intseq(n)) eq (&+Intseq(n^2))]; // Vincenzo Librandi, Aug 26 2015

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 *)

PARI

is(n)=sumdigits(n)==sumdigits(n^2) \\ Charles R Greathouse IV, Aug 25 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

Formula

A007953(a(n)) = A004159(a(n)). - Reinhard Zumkeller, Apr 25 2009

Extensions

Edited by N. J. A. Sloane, May 30 2010