A101337 -id:A101337 - cp's OEIS frontend

A113018 {Digital root of n} raised to the power {number of digits of n}.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1

Offset: 0

Alexandre Wajnberg, Jan 03 2006

Fixed points are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 81.

			a(9) = 9^1 = 9;
a(19) = (1 + 9 = 10 => 1 + 0) = 1^2 =  1;
a(85) = (8 + 5 = 13 => 1 + 3) = 4^2 = 16.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A010888, A055642.

Cf. A101337, A113019.

A113018 := proc(n) if(n=0)then return 0:fi: return (((n-1) mod 9) + 1)^length(n): end: seq(A113018(n), n=0..100); # Nathaniel Johnston, May 05 2011

Join[{0},Table[ResourceFunction["AdditiveDigitalRoot"][n]^IntegerLength[n],{n,73}]] (* James C. McMahon, Jun 14 2024 *)

a(n) = A010888(n)^A055642(n). - Michel Marcus, Feb 10 2019

A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212, 223123410

Offset: 1

Paolo P. Lava, Mar 13 2019

1, 2, 3, 4, 5, 6, 7, 8, 9, 370 and 407 are also equal to the determinant of the circulant matrix formed by their decimal digits.

			     | 2 6 1 |
perm | 1 2 6 | = 2*2*2 + 6*6*6 + 1*1*1 + 1*2*6 + 6*1*2 + 2*6*1 = 261.
     | 6 1 2 |
.
     | 2 2 3 1 2 3 4 1 0 |
     | 0 2 2 3 1 2 3 4 1 |
     | 1 0 2 2 3 1 2 3 4 |
     | 4 1 0 2 2 3 1 2 3 |
perm | 3 4 1 0 2 2 3 1 2 | = 223123410
     | 2 3 4 1 0 2 2 3 1 |
     | 1 2 3 4 1 0 2 2 3 |
     | 3 1 2 3 4 1 0 2 2 |
     | 2 3 1 2 3 4 1 0 2 |

Eric Weisstein's World of Mathematics, Permanent
Eric Weisstein's World of Mathematics, Circulant Matrix

Cf. A219324, A219327, A306662, A306593, A306714.

Up to n=110 the permanent of the circulant matrix of the digits of n is equal to A101337 but from n=111 on it can differ.

with(linalg): P:=proc(q) local a, b, c, d, i, j, k, n, t;
for n from 1 to q do d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do b:=[op(b), c[j]]; od;
c:=b; t:=[op(t), c]; od; if n=permanent(t)
then print(n); fi; od; end: P(10^7);

mpd(n) = {my(d = digits(n)); matpermanent(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));}
isok(n) = mpd(n) == n; \\ Michel Marcus, Mar 14 2019

from sympy import Matrix
A306853_list = []
for n in range(1,10**6):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).per():
        A306853_list.append(n) # Chai Wah Wu, Oct 18 2021

a(15) from Vaclav Kotesovec, Aug 19 2021

A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 5, 10, 17, 4, 5, 8, 13, 20, 9, 10, 13, 18, 25, 16, 17, 20, 25, 32, 1, 2, 9, 28, 65, 2, 3, 10, 29, 66, 9, 10, 17, 36, 73, 28, 29, 36, 55, 92, 65, 66, 73, 92, 129, 8, 9, 16, 35, 72, 9, 10, 17, 36, 73, 16, 17, 24, 43, 80, 35, 36, 43, 62, 99, 72, 73, 80, 99, 136, 27

Offset: 1

Francesco A. Catalanotti, Oct 26 2022

			For n = 13_10 = 23_5 (2 digits in base 5): a(13) = 2^2 + 3^2 = 13.
For n = 73_10 = 243_5 (3 digits in base 5): a(73) = 2^3 + 4^3 + 3^3 = 99.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A357954, A010346, A110592.

Cf. in base 10: A157714, A101337, A151544.

f:= proc(n) local L,d,i;
  L:= convert(n,base,5);
  d:= nops(L);
  add(L[i]^d,i=1..d)
end proc:
map(f,[$1..100]); # Robert Israel, Oct 26 2023

a[n_] := Total[IntegerDigits[n, 5]^IntegerLength[n, 5]]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

a(n) = my(d=digits(n, 5)); sum(k=1, #d, d[k]^#d); \\ Michel Marcus, Oct 29 2022

from sympy.ntheory.factor_ import digits
def A357143(n):
    t = len(s:=digits(n,5)[1:])
    return sum(d**t for d in s) # Chai Wah Wu, Oct 31 2022

a(n) = Sum_{i=1..A110592(n)} d(i)^A110592(n), where d(i) is the i-th digit of n in base 5.

Corrected and extended by Michel Marcus, Oct 29 2022

A113010 {Number of digits of n} raised to the power of {the sum of the digits of n}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

Offset: 0

Alexandre Wajnberg, Jan 03 2006

n=1 and 32 are two fixed points. Are there any others?

There are no other fixed points less than 10^1000. - Chai Wah Wu, Feb 28 2019

Cf. A101337.

def A113010(n):
    return len(str(n))**sum(int(d) for d in str(n)) # Chai Wah Wu, Feb 28 2019

a(ijk...) [m digits ijk...] = m^(i+j+k+...)

A308105 Numbers m such that m is greater than the sum of the k-th powers of its digits, where k is the number of digits of m.

Original entry on oeis.org

10, 11, 12, 13, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 80, 81, 82, 83, 84, 90, 91, 92, 93, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134

Offset: 1

Bernard Schott, May 13 2019

These integers are called "nombres résistants" on the French site Diophante.

There exists a smallest number M_0 such that every number >= M_0 is a term of this sequence. This integer has 60 digits: M_0 = 102 * 10^57. So 102 * 10^57 - 1 is not "résistant" (proof in the link).

			34 - (3^2 + 4^2) = 9 so 34 is a term.
126 - (1^3 + 2^3 + 6^3) = -99 and 126 is not a term.

Robert Israel, Table of n, a(n) for n = 1..10000
Diophante, A367. Les entiers font de la résistance, Oct. 2017 (in French).

Cf. A000027, A101337, A005188.

sol:=[];v:=[];digit:=[]; m:=1;
for u in [1..150] do
        digit:=Intseq(u);
             for i in [1..#digit] do v[i]:=digit[i]^#digit; end for;
             if u-&+v gt 0 then sol[m]:=u; m:=m+1; end if;
end for;
sol; // Marius A. Burtea, May 13 2019

filter:= proc(n) local L,m,t;
  L:= convert(n,base,10);
  m:= nops(L);
  n > add(t^m,t=L)
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 21 2019

Select[Range[140], # - Total[IntegerDigits[#]^IntegerLength[#]] > 0 &] (* Michael De Vlieger, Jun 09 2019 *)

isok(n) = { my(d=digits(n), nb=#d); n > sum(k=1, #d, d[k]^nb);} \\ Michel Marcus, May 19 2019

Numbers m such that m - A101337(m) > 0.

A330125 Positive integers whose digit-power sum is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 14, 16, 21, 23, 25, 27, 32, 38, 41, 45, 49, 52, 54, 56, 58, 61, 65, 72, 78, 83, 85, 87, 94, 101, 110, 111, 113, 115, 122, 124, 128, 131, 139, 142, 146, 148, 151, 155, 164, 166, 182, 184, 193, 199, 212, 214, 218, 221, 223, 227, 232, 236, 238, 241

Offset: 1

Manan Shah, Dec 01 2019

Let M be an N-digit positive integer with digits (base 10) d_1, d_2, d_3, ..., d_N. If Sum_{i = 1..N} (d_i)^N is prime, then M is part of this sequence.
Numbers k such that A101337(k) is prime.
Both A139749 and A178357 are similar and match the first several terms of this sequence, but the digit powers are different. Additionally, perhaps a more interesting sequence is the subsequence of primes: 2, 3, 5, 7, 11, 23, 41, 61, 83.

			The first four terms are the single-digit primes; a(5) = 11 since 1^2 + 1^2 = 2, which is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Math Misery, Of Probes and Primes

Cf. A101337, A139749, A178357.

filter:= proc(n) local L,d,t;
   L:= convert(n,base,10);
   d:= nops(L);
   isprime(add(t^d, t=L))
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 17 2023

Select[Range[250], (d = IntegerDigits[#]; PrimeQ@ Total[d^Length[d]]) &] (* Giovanni Resta, Dec 02 2019 *)

isok(n) = {my(d = digits(n)); isprime(sum(k=1, #d, d[k]^#d));} \\ Michel Marcus, Dec 05 2019

More terms from Giovanni Resta, Dec 02 2019

A353138 Sum of (the number of digits in n to the power (each digit in n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 5, 6, 8, 12, 20, 36, 68, 132, 260, 516, 9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 17, 18, 20, 24, 32, 48, 80, 144, 272, 528, 33, 34, 36, 40, 48, 64, 96, 160, 288, 544, 65, 66, 68, 72, 80, 96

Offset: 1

William Halpin, Apr 26 2022

In base 10, 1 and 4624 are the only numbers where a(n)=n (conjectured).

a(n) < n when n > 2.1*(10^10).

			a(3) = 1^3 = 1;
a(164) = 3^1 + 3^6 + 3^4 = 813;
a(4624) = 4^4 + 4^6 + 4^2 + 4^4 = 4624.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055642, A101337.

f:= proc(n) local m,L,t;
  L:= convert(n,base,10);
  m:= nops(L);
  add(m^t, t=L)
end proc:
map(f, [$1..100]);

a(n) = my(d=digits(n)); vecsum(vector(#d, k, #d^d[k])); \\ Michel Marcus, Apr 27 2022

def a(n): s = str(n); return sum(len(s)**int(d) for d in s)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Apr 26 2022

a(n) = Sum_{k=1..A055642(n)} (A055642(n))^(floor(n*10^(1-k)) mod 10).

A357954 Integers k that are periodic points for some iterations of k->A357143(k).

Original entry on oeis.org

1, 2, 3, 4, 13, 18, 28, 118, 194, 289, 338, 353, 354, 419, 489, 528, 609, 769, 1269, 1299, 2081, 4890, 4891, 9113, 18575, 18702, 20759, 35084, 1874374, 338749352, 2415951874

Offset: 1

Francesco A. Catalanotti, Oct 22 2022

Given the function A357143(k), a number k is a term of the sequence if there exists a j such that A357143^j(k) = k, where j is the number of iterations applied.
The sequence is finite.
Proof: A357143(k) < k for all big enough k. g(k) = A110592(k)*4^A110592(k) is clearly an upper bound of A357143(k). Hence k > g(k) -> k > A357143(k), therefore every periodic point must be in an interval [s;t] such that for every k in [s;t] k <= g(k). Limit_{k->oo} g(k)/k = 0; now using the little-o definition we can show that there always exists a certain k_0 such that, for every k > k_0, k > g(k). The conclusion is that there must exist a finite number of intervals [s;t] and, consequently, a finite number of periodic points.
Every term k of the sequence is a periodic point (either a perfect digital invariant or a sociable digital invariant) for the function A357143(k).
The longest cycle needs 6 iterations to end: [489, 609, 769, 1269, 1299, 2081].

			k=9113 is a fixed point (perfect digital invariant) for the reiterated function A357143(k):
    9113_10 = 242423_5 (a 6-digit number in base 5);
    A357143(9113) = 2^6 + 4^6 + 2^6 + 4^6 + 2^6 + 3^6 = 9113.
k=18702 is a sociable digital invariant for the reiterated function A357143(k), requiring 2 iterations:
  1st iteration:
    18702_10 = 1044302_5 (a 7-digit number in base 5);
    A357143(18702) = 1^7 + 0^7 + 4^7 + 4^7 + 3^7 + 0^7 + 2^7 = 35084;
  2nd iteration:
    35084_10 = 2110314_5 (also a 7-digit number in base 5);
    A357143(35084) = 2^7 + 1^7 + 1^7 + 0^7 + 3^7 + 1^7 + 4^7 = 18702.

Wikipedia, Perfect digital invariant

Cf. A357143 , A010346 (fixed points), A110592 (exponents p(k)).

Cf. A157714 (base-10 sociable digital invariants), A101337 (A357143(k) base 10), A151544 (base-10 periodic points).

A281724 Numbers k with the property that (sum of digits of k) times (number of digits of k) equals (sum of {each digit of k} raised to the power {number of digits in k}).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 102, 120, 201, 210

Offset: 1

José de Jesús Camacho Medina, Jan 28 2017

Numbers with the property that q-p = 0, where p = (sum of digits of k) times (number of digits of k) (A110805) and q = (sum of {each digit of k} raised to the power {number of digits in k}) (A101337).
From David Consiglio, Jr., Jan 29 2017: (Start)
The sequence is finite and 210 is the last term. Proof:
1. Let Y = length(k).
2. The maximum value of digit_sum(k) is 9Y.
3. Since p = Y*digit_sum(k), p has a maximum value of 9Y^2.
4. 2^Y > 9Y^2 for all Y > 9. Therefore q > p for all numbers longer than 9 digits that contain any digits > 1.
   a. Example: 1,000,000,002. q = 1^5 + 8*0^5 + 2^10 = 1025. The largest p value for a 10-digit number would be for 9,999,999,999 which has p = 10*(9*10) = 900. Since q > all possible p values at this size, any term of this sequence must either have fewer than 10 digits or contain only 0's and 1's as digits.
5. Now we can consider only numbers with 0 and 1 digits.
6. Let Z = number of 1 digits in a number k.
7. q = Z because q = Z*1^Y + (Y-Z)*0^Y = Z.
8. p = YZ because the sum of the digits is equal to the number of 1's.
9. Z = YZ only in the case of Y = 1. Thus, the only term of this sequence that contains only 0's and 1's has a length of only 1 digit. Thus, k = 1 is in this sequence.
10. Therefore a candidate number must have fewer than 10 digits if it contains a digit 2 or larger, and must have fewer than 2 digits if it does not. All numbers in this range have been checked, and no additional values of k with q = p have been found.
Thus the sequence is finite. (End)

			    9 is a term because             9^1 = 1*(9);
   20 is a term because       2^2 + 0^2 = 2*(2 + 0);
  210 is a term because 2^3 + 1^3 + 0^3 = 3*(2 + 1 + 0).

Cf. A101337, A110805.

W = Array[Total[IntegerDigits[#]^IntegerLength[#]]&, 100]-Table[IntegerLength[n] * Total[IntegerDigits[n]], {n, 100}]; Flatten[Position[W,0]]

A334601 Positive integers m such that sum of cubes of the digits of m, t=A055012(m), is a multiple of m (m/A055012(m) is an integer >= 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 27, 37, 48, 153, 370, 371, 407, 459

Offset: 1

Zak Seidov, May 07 2020 and May 12 2020

Corresponding values of t: 1, 8, 27, 64, 125, 216, 343, 512, 729, 72, 351, 370, 576, 153, 370, 371, 407, 918 (first 9 terms are all cubes).
Corresponding values of t/m: 1, 4, 9, 16, 25, 36, 49, 64, 81, 3, 13, 10, 12, 1, 1, 1, 1, 2 (first 9 terms are all squares).
The subsequence of numbers m such that sum of cubes of its digits is equal to m is A046197 \ {0}. - Bernard Schott, May 11 2020

			m = 459, t = 4^3 + 5^3 + 9^3 = 918, t/m = 2.

Cf. A005188, A046197, A046459, A055012, A055642, A101337, A306354, A306360, A306361, A329291.

Select[Range[500], Divisible[Plus @@ (IntegerDigits[#]^3), #] &] (* Amiram Eldar, May 11 2020 *)

isok(m) = my(d=digits(m)); sum(k=1, #d, d[k]^3) % m == 0; \\ Michel Marcus, May 14 2020

cp's OEIS Frontend

A113018 {Digital root of n} raised to the power {number of digits of n}.

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A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

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A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

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A113010 {Number of digits of n} raised to the power of {the sum of the digits of n}.

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A308105 Numbers m such that m is greater than the sum of the k-th powers of its digits, where k is the number of digits of m.

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Magma

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A330125 Positive integers whose digit-power sum is a prime.

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A353138 Sum of (the number of digits in n to the power (each digit in n)).

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