A111434 -id:A111434 - cp's OEIS frontend

A114135 Primitive numbers n such that the sums of the digits of n, n^2 and n^3 coincide (cf. A111434).

1, 468, 585, 5851, 5868, 28845, 58968, 21688965, 29588877, 37848897, 49879981, 58577797, 79898994, 79958368, 79979698, 89757468, 109699677, 159699969, 468957888, 479597652, 479896587, 480749985, 494899398, 497349981, 498678256

Offset: 1

Giovanni Resta and Robert G. Wilson v, Nov 21 2005

Members of A111434 not congruent to 0 (mod 10). If k is a member of A111434 then so is 10^e*k.
The authors have calculated all members below 10^11.
The number of members less than 10^n {n=0..11}: 0,1,1,3,5,7,7,7,16,34,57,125.
Number of members congruent to k (mod 10): 0,7,1,0,2,23,8,20,49,15. But more interesting, number of members are congruent to k (mod 9): 66,59,0,0,0,0,0,0,0.
A007953(n) == n mod 9.  Since 0 and 1 are the only k in [0,1,...8] with k == k^2 mod 9, all terms are congruent to 0 or 1 mod 9. - Robert Israel, Jan 26 2015

Toshitaka Suzuki and Nikhil Mahajan, Table of n, a(n) for n = 1..600 (first 325 terms from Toshitaka Suzuki)

Cf. A007953, A111434, A005188.

sod[n_] := Plus @@ IntegerDigits@n; lst = {}; Do[ If[(Mod[n, 9] == 0 || Mod[n, 9] == 1) && Mod[n, 10] != 0 && sod@n == sod[n2] == sod[n3], AppendTo[lst, n]], {n, 108/2}]; lst
Select[Range[5*10^8],Length[Union[Total/@IntegerDigits/@{#,#^2,#^3}]]==1 && Mod[#,10]!=0&] (* Harvey P. Dale, Jul 07 2020 *)

isok(n) = (n % 10) && ((sd=sumdigits(n)) == sumdigits(n^2)) && (sd == sumdigits(n^3)); \\ Michel Marcus, Jan 20 2015

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

Original entry on oeis.org

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

A257768 Numbers m such that for some power k, m is the sum of d + d^k as d runs through the digits of m.

Original entry on oeis.org

12, 18, 30, 90, 666, 870, 960, 1998, 7816, 42648, 119394, 302034, 360522, 1741752, 12051036, 909341082, 931186956, 1136424308, 1145082306, 8390370196, 49388550660, 52927388760, 100552730520, 41845367362266, 51671446297908, 245917854035004, 607628544623816, 858683110606660, 4023730658941192

Offset: 1

Pieter Post, May 07 2015

The power k of most terms in this sequence is equal to or one more or one less than the number of digits in the term. One exception is 302034: 302034 = 3^9 + 0^9 + 2^9 + 0^9 + 3^9 + 4^9 + 3+0+2+0+3+4.

			666 = (6+6+6) + (6^3 + 6^3 + 6^3).
7816 = (7+8+1+6) + (7^4 + 8^4 + 1^4 + 6^4).
360522 = (3+6+0+5+2+2) + (3^7 + 6^7 + 0^7 + 5^7 + 2^7 + 2^7).

Cf. A115518, A130680, A111434.

mmax:= 10:  # to get all terms < 10^mmax
Res:= NULL:
score:= (c,p) -> add(c[i+1]*(i+i^p), i=0..9):
for m from 2 to mmax do
comps:= convert(map(`-`,combinat:-composition(10+m,10),[1$10]),list):
for c in comps do
  cL:= [seq(i$c[i+1], i=0..9)];
  if max(c[3..-1]) = 0 then slim:= 0 else slim:= 10^m fi;
  for p from 1 do
    s:= score(c,p);
    L:= sort(convert(s,base,10));
    if L = cL then Res:= Res,s; break fi;
    if s >= slim then break fi;
  od:
od:
od:
sort([Res]); # Robert Israel, May 08 2015

# WARNING: this prints numbers in the sequence, but not in increasing order.
def moda(n,a):
    kk = 0
    while n > 0:
        kk= kk+(n%10)**a
        n = n//10
    return kk
def sod(n):
    kk = 0
    while n > 0:
        kk += n % 10
        n = n//10
    return kk
for a in range (1, 10):
    for c in range (10, 10**6):
        if c == moda(c,a)+sod(c):
            print(c, end=",")

a(14)-a(29) from Giovanni Resta, May 08 2015

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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A114135 Primitive numbers n such that the sums of the digits of n, n^2 and n^3 coincide (cf. A111434).

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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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A257768 Numbers m such that for some power k, m is the sum of d + d^k as d runs through the digits of m.

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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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