A114135 -id:A114135 - cp's OEIS frontend

A254066 Primitive numbers n such that the sums of the digits of n and n^2 coincide.

1, 9, 18, 19, 45, 46, 55, 99, 145, 189, 198, 199, 289, 351, 361, 369, 379, 388, 451, 459, 468, 495, 496, 558, 559, 568, 585, 595, 639, 729, 739, 775, 838, 855, 954, 955, 999, 1098, 1099, 1179, 1188, 1189, 1198, 1269, 1468, 1485, 1494, 1495, 1585, 1738, 1747

Offset: 1

Nikhil Mahajan, Jan 25 2015

Members of A058369 not congruent to 0 (mod 10).
This sequence is to A058369 what A114135 is to A111434.
Hare, Laishram, & Stoll show that this sequence is infinite. In particular for each k in {846, 847, 855, 856, 864, 865, 873, ...} there are infinitely many terms in this sequence with digit sum k. - Charles R Greathouse IV, Aug 25 2015

			9 is in the sequence because the digit sum of 9^2 = 81 is 9.
18 is in the sequence because the digit sum of 18^2 = 324 is 9, same as the digit sum of 18.

Nikhil Mahajan, Table of n, a(n) for n = 1..10000
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.

Cf. A058369, A114135, A111434.

Subsequence of A090570.

[n: n in [1..1000] | &+Intseq(n) eq &+Intseq(n^2) and not IsZero(n mod 10)]; // Bruno Berselli, Jan 29 2015

Select[Range[1000],!Divisible[#,10]&&Total[IntegerDigits[#]] == Total[ IntegerDigits[#^2]]&] (* Harvey P. Dale, Dec 27 2015 *)

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

list(lim)=my(v=List()); forstep(n=1,lim,[8, 9, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 9], if(sumdigits(n)==sumdigits(n^2), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Aug 26 2015

[n for n in [0..1000] if sum(n.digits())==sum((n^2).digits()) and n%10!=0] # Tom Edgar, Jan 27 2015

More terms from Harvey P. Dale, Dec 27 2015

A111434 Numbers k such that the sums of the digits of k, k^2 and k^3 coincide.

Original entry on oeis.org

0, 1, 10, 100, 468, 585, 1000, 4680, 5850, 5851, 5868, 10000, 28845, 46800, 58500, 58510, 58680, 58968, 100000, 288450, 468000, 585000, 585100, 586800, 589680, 1000000, 2884500, 4680000, 5850000, 5851000, 5868000, 5896800, 10000000

Offset: 1

Giovanni Resta, Nov 21 2005

The sequence is clearly infinite, since we can add trailing zeros. Is the subset of values not ending in 0 infinite too (see A114135)?

			468 is in the sequence since 468^2 = 219024 and 468^3 = 102503232 and we have 18 = 4+6+8 = 2+1+9+0+2+4 = 1+0+2+5+0+3+2+3+2.
5851 is in the sequence because 5851, 34234201 (= 5851^2) and 200304310051 (=5851^3) all have digital sum 19.

David A. Corneth, Table of n, a(n) for n = 1..1124 (using the b-file in A114135).

Cf. A011557, A058369, A070276.

s:=proc(n) local nn: nn:=convert(n,base,10): sum(nn[j],j=1..nops(nn)): end: a:=proc(n) if s(n)=s(n^2) and s(n)=s(n^3) then n else fi end: seq(a(n),n=0..1000000); # Emeric Deutsch, May 13 2006

SumOfDig[n_]:=Apply[Plus, IntegerDigits[n]]; Do[s=SumOfDig[n]; If[s==SumOfDig[n^2] && s==SumOfDig[n^3], Print[n]], {n, 10^6}]
Select[Range[0,10000000],Length[Union[Total/@IntegerDigits[{#,#^2,#^3}]]] == 1&] (* Harvey P. Dale, Apr 26 2014 *)

b-file Corrected by David A. Corneth, Jul 22 2021

A257787 Numbers n such that the sum of the digits of n to some power divided by the sum of the digits equal n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 48, 415, 231591, 3829377463694454, 56407086228259246207394322684

Offset: 1

Pieter Post, May 08 2015

The first nine terms are trivial, but then the terms become very rare. It appears that this sequence is finite.

			37 = (3^3+7^3)/(3+7).
231591 = (2^7+3^7+1^7+5^7+9^7+1^7)/(2+3+1+5+9+1).

Cf. A061209, A115518, A111434, A114135, A130680, A257784, A257768.

def moda(n,a):
    kk = 0
    while n > 0:
        kk= kk+(n%10)**a
        n =int(n//10)
    return kk
def sod(n):
    kk = 0
    while n > 0:
        kk= kk+(n%10)
        n =int(n//10)
    return kk
for a in range (1, 10):
    for c in range (1, 10**6):
        if c*sod(c)==moda(c, a):
            print (a,c, moda(c,a),sod(c))

a(14) from Giovanni Resta, May 09 2015

a(15) from Chai Wah Wu, Nov 30 2015

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A254066 Primitive numbers n such that the sums of the digits of n and n^2 coincide.

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A111434 Numbers k such that the sums of the digits of k, k^2 and k^3 coincide.

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A257787 Numbers n such that the sum of the digits of n to some power divided by the sum of the digits equal n.

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Python

Extensions