A058852 -id:A058852 - cp's OEIS frontend

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

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

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

			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

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

Cf. A002283, A007953, A058370, A058852, A077436, A254066.
Cf. A147523 (number of numbers in each decade).
Subsequence of A090570.

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

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

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

Select[Range[0,1200],Total[IntegerDigits[#]]==Total[IntegerDigits[ #^2]]&] (* Harvey P. Dale, Jun 14 2011 *)

is(n)=sumdigits(n)==sumdigits(n^2) \\ Charles R Greathouse IV, Aug 25 2015

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

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

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

A028553 Numbers k such that k*(k+3) is a palindrome.

Original entry on oeis.org

0, 1, 8, 28, 66, 88, 211, 298, 671, 2126, 2998, 28814, 29369, 29998, 63701, 212206, 212671, 299998, 636776, 2122206, 2861419, 2999998, 9443423, 21341691, 28862883, 29999998, 212325206, 289053683, 294127328, 294174669, 299999998, 2134473706, 2946920844, 2999999998

Offset: 1

Patrick De Geest

Also: numbers k such that the sum of the first k even composites is palindromic. Sequence is 4 + 6 + 8 + 10 + 12 + 14 + ... + z. For values of z see A058851. (Comment added by author 12/2000.)

All numbers of the form 3*10^j - 2 for j >= 0 are terms. For n > 1, a(n) mod 10 is one of {1,3,4,6,8,9}. - Chai Wah Wu, Feb 20 2021

Chai Wah Wu, Table of n, a(n) for n = 1..40
Patrick De Geest, Palindromic Quasipronics
Patrick De Geest, Palindromic Sums 2
Erich Friedman, What's Special About This Number? (See entries 298, 2126, 2998.)

Cf. A058852, A028554.

(Sqrt[4#+9]-3)/2&/@Select[Table[k(k+3),{k,0,3*10^6}],PalindromeQ] (* The program generates the first 22 terms of the sequence. *) (* Harvey P. Dale, Oct 03 2023 *)

n, m, A028553_list = 0, 0, []
while n < 10**12:
    s = str(m)
    if s == s[::-1]:
        A028553_list.append(n)
    m += 2*(n+2)
    n += 1 # Chai Wah Wu, Feb 20 2021

More terms from Chai Wah Wu, Feb 20 2021

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

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A028553 Numbers k such that k*(k+3) is a palindrome.

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A058370 Primes p such that p and p^2 have same digit sum.

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