A257784 -id:A257784 - cp's OEIS frontend

A258484 Numbers m such that m equals a fixed number raised to the powers of the digits.

1, 10, 12, 100, 101, 111, 1000, 1010, 1033, 1100, 2112, 4624, 10000, 10001, 11101, 20102, 31301, 100000, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101100, 101110, 101111, 101121, 110000, 110001, 110010, 110100, 110110, 110111, 111000

Offset: 1

Pieter Post, May 31 2015

Let m = abcde... and z is a fixed radix -> m = z^a +z^b +z^c +z^d +z^e...

A number m made of k ones and h zeros is a member if m-h is divisible by k. Several other large members exist, including 12095925296900865188 (base = 113) and 115330163577499130079377256005 (base = 1500). - Giovanni Resta, Jun 01 2015

			12 = 3^1 + 3^2;
31301 = 25^3 + 25^1 + 25^3 + 25^0 + 25^1;
595968 = 4^5 + 4^9 + 4^5 + 4^9 + 4^6 + 4^8;
13177388 = 7^1 + 7^3 + 7^1 + 7^7 + 7^7 + 7^3 + 7^8 + 7^8.

Giovanni Resta, Table of n, a(n) for n = 1..956 (terms < 10^12)
Giovanni Resta, Table of a(1..956) values and corresponding bases

Cf. A005188, A023052, A257784, A257766, A257787, A257814.

okQ[v_] := Block[{b, d=IntegerDigits@ v, y, t}, t = Last@ Tally@ Sort@d; b = Floor[ (v/t[[2]]) ^ (1/t[[1]])]; While[(y = Total[b^d]) > v, b--]; v==y]; Select[Range[10^5],okQ] (* Giovanni Resta, Jun 01 2015 *)

for(n=1,10^5,d=digits(n);for(m=1,n,s=sum(i=1,#d,m^d[i]);if(s==n,print1(n,", ");break);if(s>n,break))) \\ Derek Orr, Jun 12 2015

def moda(n,a,m):
    kk = 0
    while n > 0:
        na=int(n%m)
        kk= kk+a**na
        n =int(n//m)
    return kk
for c in range (1, 10**8):
    for a in range (1,20):
        if  c==moda(c,a,10):
            print (a,c)

More terms from Giovanni Resta, Jun 01 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

A257860 Numbers n such that a digit of n to the power k plus the sum of the other digits of n equals n, where k is a positive integer.

Original entry on oeis.org

1, 89, 132, 264, 518, 739, 2407, 6579, 8200, 8201, 8202, 8203, 8204, 8205, 8206, 8207, 8208, 8209, 32780, 32781, 32782, 32783, 32784, 32785, 32786, 32787, 32788, 32789, 59060, 59061, 59062, 59063, 59064, 59065, 59066, 59067, 59068, 59069, 78145, 524300, 524301, 524302, 524303, 524304, 524305, 524306, 524307, 524308, 524309, 531459, 823567, 2097178

Offset: 1

Pieter Post, May 11 2015

There are numbers that come in groups of 10, like 8200, 32780 and 524300. But there are also a few stand-alone numbers. Like 531459 (=5+3+1+4+5+9^6).

It is easy to generate large terms in the sequence, for example, 9^104+409 and 9^1047+4561 are the smallest terms with 100 and 1000 digits, respectively. - Giovanni Resta, May 12 2015

			89 is in the sequence because 89 = 8+9^2.
2407 is in the sequence because 2407 = 2+4+0+7^4.
8202 is in the sequence because 8202 = 8+ 2^13 +0+2, also 8202 = 8+2+0+2^13.

Cf. A061209, A130680, A257766, A257768, A257784,

Cf. A007953.

import Data.List (nub); import Data.List.Ordered (member)
a257860 n = a257860_list !! (n-1)
a257860_list = 1 : filter f [1..] where
   f x = any (\d -> member (x - q + d) $ ps d) $ filter (> 1) $ nub ds
         where q = sum ds; ds = (map (read . return) . show) x
   ps x = iterate (* x) (x ^ 2)
-- Reinhard Zumkeller, May 12 2015

def sod(n):
    kk = 0
    while n > 0:
        kk= kk+(n%10)
        n =int(n//10)
    return kk
for i in range (1,10**7):
    for j in range(1,len(str(i))+1):
        k=(i//(10**(j-1)))%10
        for m in range (2,30):
            if i==sod(i)+k**m-k:
                print (i)

One more term and some missing data added by Reinhard Zumkeller, May 12 2015

A334249 Numbers k such that (k - digitsum(k))(k + digitsum(k)) contains k as a substring.

Original entry on oeis.org

11, 88, 101, 448, 673, 776, 1001, 2879, 3553, 9537, 10001, 14651, 36559, 49056, 51073, 54116, 59600, 100001, 505025, 998999, 1000001, 4115964, 5050250, 5133355, 10000001, 10050125, 19349727, 26550976, 33726078, 35792647, 42349456, 43605459, 50050025, 66952741, 88027284, 88819024, 100000001, 105124922

Offset: 1

Scott R. Shannon, May 05 2020

All numbers of the form 10^n + 1 (for n > 0) are in the sequence.

			11 is a term as digitsum(11) = 2 and (11 - 2)(11 + 2) = 117, which contains '11' as a substring.
9537 is a term as digitsum(9537) = 24 and (9537 - 24)(9537 + 24) = 90953793, which contains '9537' as a substring.

Giovanni Resta, Table of n, a(n) for n = 1..95

Cf. A007953, A257784, A125526, A082208, A225780.

A257969 Numbers m such that the sum of the digits (sod) of m, m^2, m^3, ..., m^9 are in arithmetic progression: sod(m^(k+1)) - sod(m^k) = f for k=1..8.

Original entry on oeis.org

1, 10, 100, 1000, 7972, 10000, 53941, 79720, 100000, 134242, 539410, 698614, 797200, 1000000, 1342420, 5394100, 6986140, 7525615, 7972000, 9000864, 10000000, 10057054, 13424200, 15366307, 17513566, 20602674, 23280211, 24716905, 25274655, 25665559, 32083981, 34326702, 34446204, 34534816

Offset: 1

Pieter Post, May 15 2015

All powers of 10 are terms of this sequence.
If m is a term, then so is 10*m.
Number of terms < 10^k for k >= 1: 1, 2, 3, 5, 8, 13, 20, 62.

			7972 is in the sequence, because the difference between the successive sum-of-digit values is 15:
  sod(7972) = 25;
  sod(7972^2) = 40;
  sod(7972^3) = 55;
  sod(7972^4) = 70;
  sod(7972^5) = 85;
  sod(7972^6) = 100;
  sod(7972^7) = 115;
  sod(7972^8) = 130;
  sod(7972^9) = 145;
  sod(7972^10) = 178, where the increment is no longer 15.
But there are seven numbers below 10^9 with a longer sequence (namely, 134242, 23280211, 40809168, 46485637, 59716223, 66413917, and 97134912) where sod(m^(k+1)) - sod(m^k) = f for k=1..9.
  sod(134242) = 16;
  sod(134242^2) = 40;
  sod(134242^3) = 64;
  sod(134242^4) = 88;
  sod(134242^5) = 112;
  sod(134242^6) = 136;
  sod(134242^7) = 160;
  sod(134242^8) = 184;
  sod(134242^9) = 208;
  sod(134242^10) = 232;
  sod(134242^11) = 283, where the increment is no longer 24.

Cf. A061209, A115518, A257784, A258722.

fQ[n_] := Block[{g}, g[x_] := Power[x, #] & /@ Range@ 9; Length@ DeleteDuplicates@ Differences[Total[IntegerDigits@ #] & /@ g@ n] == 1]; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Jun 12 2015 *)
Select[Range[35*10^6],Length[Union[Differences[Total/@IntegerDigits[ #^Range[9]]]]] ==1&] (* Harvey P. Dale, Aug 23 2017 *)

isok(n) = {my(osod = sumdigits(n^2)); my(f = osod - sumdigits(n)); for (k=3, 9, my(nsod = sumdigits(n^k)); if (nsod - osod != f, return (0)); osod = nsod;); return (1);} \\ Michel Marcus, May 28 2015

{m : sod(m^(k+1)) - sod(m^k) = f for k=1..8}.

Corrected and extended by Harvey P. Dale, Aug 23 2017

Edited by Jon E. Schoenfield, Mar 01 2022

A270542 Numbers of the form (pq)^2, where p is the number of digits of n (A055642) and q is the sum of the digits of n (A007953).

Original entry on oeis.org

0, 1, 2704, 5184, 7744

Offset: 1

José de Jesús Camacho Medina, Mar 18 2016

			2704 is a term because 2704 = [4*(2+7+0+4)]^2;
5184 is a term because 5184 = [4*(5+1+8+4)]^2.

Cf. A007953, A055642, A257784.

Position[ Table[ IntegerLength[k] Sum[( Floor[k/10^n] - 10 Floor[k/10^(n + 1)]), {n, 0, IntegerLength@ k^2}] - k, {k, 1, 10^6}], 0] // Flatten = {1, 1232, 4100, 268542}
Select[Range[10^3]^2,With[{id=IntegerDigits[#]},#==(Length[id]*Plus@@id)^2]&] (* Ray Chandler, Apr 01 2016 *)

isok(n) = d = digits(n); (#d*vecsum(d))^2 == n; \\ Michel Marcus, Mar 26 2016

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A258484 Numbers m such that m equals a fixed number raised to the powers of the digits.

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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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A257860 Numbers n such that a digit of n to the power k plus the sum of the other digits of n equals n, where k is a positive integer.

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A334249 Numbers k such that (k - digitsum(k))(k + digitsum(k)) contains k as a substring.

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A257969 Numbers m such that the sum of the digits (sod) of m, m^2, m^3, ..., m^9 are in arithmetic progression: sod(m^(k+1)) - sod(m^k) = f for k=1..8.

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A270542 Numbers of the form (pq)^2, where p is the number of digits of n (A055642) and q is the sum of the digits of n (A007953).

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