A005188 -id:A005188 - cp's OEIS frontend

A061862 Powerful numbers (2a): a sum of nonnegative powers of its digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 254, 258, 262, 263, 264, 267, 283, 332, 333, 334, 347, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 472, 518, 538, 598, 629, 635, 653, 675, 730, 731, 732

Offset: 1

Erich Friedman, Jun 23 2001

Zero digits cannot be used in the sum. - N. J. A. Sloane, Aug 31 2009

More precisely, digits 0 do not contribute to the sum, in contrast to A134703 where it is allowed to use 0^0 = 1. - M. F. Hasler, Nov 21 2019

			43 = 4^2 + 3^3; 254 = 2^7 + 5^3 + 4^0 = 128 + 125 + 1.
209 = 2^7 + 9^2.
732 = 7^0 + 3^6 + 2^1.

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.

Different from A007532 and A134703, which are variations.

a061862 n = a061862_list !! (n-1)
a061862_list = filter f [0..] where
   f x = g x 0 where
     g 0 v = v == x
     g u v = if d <= 1 then g u' (v + d) else v <= x && h 1
             where h p = p <= x && (g u' (v + p) || h (p * d))
                   (u', d) = divMod u 10
-- Reinhard Zumkeller, Jun 02 2013

f[ n_ ] := Module[ {}, a=IntegerDigits[ n ]; e=g[ Length[ a ] ]; MemberQ[ Map[ Apply[ Plus, a^# ] &, e ], n ] ] g[ n_ ] := Map[ Take[ Table[ 0, {n} ]~Join~#, -n ] &, IntegerDigits[ Range[ 10^n ], 10 ] ] For[ n=0, n >= 0, n++, If[ f[ n ], Print[ n ] ] ]

If n = d_1 d_2 ... d_k in decimal then there are integers m_1 m_2 ... m_k >= 0 such that n = d_1^m_1 + ... + d_k^m_k.

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

Original entry on oeis.org

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

A151543 Smallest precursor of n-th cycle in the "Recurring Digital Invariant Variant" problem.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 59, 108, 119, 136, 138, 147, 177, 389, 407, 559, 709, 999, 1118, 1157, 1346, 4479, 11227, 12399, 22779, 30489, 100666, 127779, 577999, 677779, 1000259, 1001458, 1007889, 1035889, 1124577, 1188888

Offset: 1

N. J. A. Sloane, May 15 2009 based on email from Eric Angelini, Feb 18 2009

The problem is the following:
a) choose a number N
b) let k be the number of digits in N
c) raise each digit of N to the k-th power and add the results
d) call the new number N and repeat
Example:
a) 14 = N
b) k = 2
c) 1^2 + 4^2 = 17
d) 17 = N
e) k = 2
f) 1^2 + 7^2 = 50
g) 50 = N
... etc.
Here is the trajectory of 14:
-> 1^2 + 4^2 = 17
-> 1^2 + 7^2 = 50
-> 5^2 + 0^2 = 25
-> 2^2 + 5^2 = 29
-> 2^2 + 9^2 = 85
-> 8^2 + 5^2 = 89
-> 8^2 + 9^2 = 145
-> 1^3 + 4^3 + 5^3 = 190
-> 1^3 + 9^3 + 0^3 = 730
-> 7^3 + 3^3 + 0^3 = 370
-> 3^3 + 7^3 + 0^3 = 370 (fixed point)
The question is, what are the cycles that appear in the trajectories?
The following table of the first 34 cycles (arranged in order of the smallest precursor) was calculated by Hans Havermann:
The format for each cycle is:
Index {the smallest precursor (the current sequence), the cycle length, {the cycle itself with the smallest element of the cycle first - see A151544}}:
{ 1, 1, {1}}
{ 2, 1, {2}}
{ 3, 1, {3}}
{ 4, 1, {4}}
{ 5, 1, {5}}
{ 6, 1, {6}}
{ 7, 1, {7}}
{ 8, 1, {8}}
{ 9, 1, {9}}
{ 14, 1, {370}}
{ 59, 3, {160, 217, 352}}
{ 108, 1, {153}}
{ 119, 1, {371}}
{ 136, 2, {136, 244}}
{ 138, 10, {259, 862, 736, 586, 853, 664, 496, 1009, 6562, 3233}}
{ 147, 14, {18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413}}
{ 177, 2, {58618, 76438}}
{ 389, 6, {2929, 13154, 4394, 7154, 3283, 4274}}
{ 407, 1, {407}}
{ 559, 3, {282595, 824963, 845130}}
{ 709, 1, {8208}}
{ 999, 2, {2178, 6514}}
{ 1118, 4, {10933, 59536, 73318, 50062}}
{ 1157, 12, {5908997, 17347727, 23131558, 17571846, 30442597, 49340036, 44870531, 23070276, 13216291, 44733413, 5981093, 11743403}}
{ 1346, 1, {1634}}
{ 4479, 1, {9474}}
{ 11227, 1, {54748}}
{ 12399, 1, {32164049651}}
{ 22779, 1, {92727}}
{ 30489, 1, {93084}}
{100666, 12, {1680387, 5299971, 15250704, 6611844, 2689794, 12783081, 39326052, 45130596, 45579685, 68505765, 27073124, 11602212}}
{127779, 1, {548834}}
{577999, 1, {4210818}}
{677779, 3, {2767918, 8807272, 5841646}}
{1000259, 1, {9926315}}
{1001458, 6, {2191663, 5345158, 2350099, 9646378, 8282107, 5018104}}
{1007889, 1, {9800817}}
{1035889, 2, {8139850, 9057586}}
{1124577, 1, {1741725}}
{1188888, 1, {24678051}}
{2055779, 2, {2755907, 6586433}}
{2566699, 1, {472335975}}
{4888888, 10, {180450907, 564207094, 440329717, 468672187, 369560719, 837322786, 359260756, 451855933, 527799103, 857521513}}
{10135679, 1, {24678050}}
{10146899, 1, {146511208}}
{10233389, 1, {88593477}}
{10266888, 7, {1139785743, 5136409024, 3559173428, 4863700423, 1418899523, 9131926726, 7377037502}}
{14489999, 3, {180975193, 951385123, 525584347}}
{14788889, 1, {912985153}}
{20248999, 1, {534494836}}
{155999999, 2, {277668893, 756738746}}
Any number < 10^9 will fall into one of these 51 cycles.
The name "Recurring Digital Invariant Variant" was suggested by Mensanator on the rec.puzzles web site.

Hans Havermann, Table of n, a(n) for n = 1..51
Eric Angelini, A Recurring Digital Invariant Variant
E. Angelini, A Recurring Digital Invariant Variant [Cached, with permission]

Cf. A005188, A151544, A157714.

A306354 a(n) = gcd(n, A101337(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 12, 1, 2, 9, 4, 1, 6, 1, 4, 3, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 9

Offset: 1

Ctibor O. Zizka, Feb 09 2019

A101337(n) / n = r, r an integer, gives A306360. A101337(n) / n = 1 gives A005188. n / A101337(n) = s, s an integer, gives A306361. The motivation for this sequence was the question as to which numbers n have the property A101337(n) / n = r and the property n / A101337(n) = s?

			For n = 24, a(24) = gcd(24, 2*2 + 4*4) = gcd(24,20) = 4, thus a(24) = 4;
for n = 153, a(153) = gcd(153, 1*1*1 + 5*5*5 + 3*3*3) = gcd(153,153) = 153, thus a(153) = 153.

Cf. A005188, A066750, A101337.

Array[GCD[#1, Total[#2^Length[#2]]] & @@ {#, IntegerDigits@ #} &, 90] (* Michael De Vlieger, Feb 09 2019 *)

a(n) = my(d=digits(n)); gcd(n, sum(i=1, #d, d[i]^#d)); \\ Michel Marcus, Feb 12 2019

from math import gcd
def A306354(n): return gcd(n,sum(int(d)**len(str(n)) for d in str(n))) # Chai Wah Wu, Jan 26 2022

A016087 First nontrivial or multidigital Armstrong number to base n.

Original entry on oeis.org

5, 28, 13, 99, 10, 20, 41, 153, 61, 29, 17, 244, 113, 342, 40, 80, 181, 2413, 26, 97, 53, 11080, 313, 5425, 90, 180, 421, 68, 37, 205, 109, 356, 613, 5489, 160, 85, 761, 10413, 261, 353, 50, 637, 1013, 292, 104, 500, 1201, 1025, 1301, 541, 281, 127295, 178

Offset: 3

Robert G. Wilson v

Also called narcissistic numbers or pluperfect digital invariants.

a(90) > 10^8 if it exists at all. - Robert Israel, Aug 24 2015

Jinyuan Wang, Table of n, a(n) for n = 3..269 (terms 3..89 from Tim Johannes Ohrtmann, terms 91..269 from Robert Israel)
Dik T. Winter, Armstrong numbers

Cf. A005188, A068072, A259347.

f:= proc(n) local k,L,m;
  for k from n+1 do
    L:= convert(k,base,n);
    m:= nops(L);
    if add(t^m,t=L) = k then return(k) fi
  od
end proc:
map(f, [$3..89]); # Robert Israel, Aug 24 2015

Do[k = n + 1; While[l = IntegerDigits[k, n]; Apply[Plus, l^Length[l]] != k, k++]; Print[k], {n, 3, 75}]

Edited and extended by Robert G. Wilson v, Feb 28 2002

A229381 The Simpsons's perfect number, Mersenne prime, and narcissistic number.

Original entry on oeis.org

8128, 8191, 8208

Offset: 1

Joe Sondow and Jonathan Sondow, Sep 23 2013

The perfect number 8128, Mersenne prime 8191, and narcissistic number 8208 appeared together on the screen in the "Marge and Homer Turn a Couple Play" episode of Season 17 of The Simpsons.

S. Singh, The Simpsons and Their Mathematical Secrets, Bloomsbury Publishing, London, 2013, pp. 93-96, ISBN 9781408835302 / 9781408843734.

C. Goff, Animating Popular Mathematics: "The Simpsons and Their Mathematical Secrets", AMS Notices, Vol. 62, No. 1 (2015), 40-44.
Simon Singh, The Simpsons' secret formula: it's written by maths geeks, The Guardian, 21 September 2013.

Cf. A000396, A000668, A005188.

a(1) = A000396(4), a(2) = A000668(5), a(3) = A005188(15).

A306361 Numbers k divisible by A101337(k) (narcissistic function).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, 100, 110, 111, 153, 200, 221, 370, 371, 407, 500, 702, 1000, 1010, 1011, 1020, 1100, 1101, 1110, 1121, 1122, 1634, 2000, 2322, 4104, 5000, 8208, 9474, 10000, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11022, 11100, 11122, 11220, 12012, 12110, 12210, 12320, 14550

Offset: 1

Ctibor O. Zizka, Feb 10 2019

A005188 is a subsequence of this sequence.
Numbers in A007088 with either 3 or 9 ones are terms of this sequence. - Chai Wah Wu, Feb 26 2019
For all N in A007088 we have A101337(N) = A007953(N) = number of digits '1'; whenever this equals 2^k*5^m (k, m >= 0) and N ends in max(k,m) '0's, then N is also in this sequence. - M. F. Hasler, Nov 18 2019

			For k = 20, 20 / (2^2 + 0^2) = 5;
for k = 221, 221 / (2^3 + 2^3 + 1^3) = 13.

Cf. A005188, A007088, A101337, A306354, A306360.

isok(n) = frac(n/A101337(n)) == 0; \\ Michel Marcus, Feb 11 2019

select( is_A306361=t->!(t%A101337(t)), [0..9999]) \\ M. F. Hasler, Nov 18 2019

A243025 Fixed points of the transform n = d_(k)10^(k-1) + d_(k-1)10^(k-2) + ... + d_(2)*10 + d_(1) -> Sum_{i=1..k-1}{d_(i)^d(i+1)}+d(k)^d(1) (A243023).

Original entry on oeis.org

1, 4155, 4355, 1953504, 1954329, 522169982

Offset: 1

Paolo P. Lava, May 29 2014

Subset of A243023.

This sequence is finite by using the same argument that Armstrong numbers (A005188) are finite. - Robert G. Wilson v, Jun 01 2014

			1^1 = 1.
5^5 + 5^1 + 1^4 + 4^5 = 4155.
5^5 + 5^3 + 3^4 + 4^5 = 4355.
4^0 + 0^5 + 5^3 + 3^5 + 5^9 + 9^1 + 1^4 = 1953504.
9^2 + 2^3 + 3^4 + 4^5 + 5^9 + 9^1 + 1^9 = 1954329.

Cf. A243023, A243024.

Cf. A005188, A003321.

with(numtheory): P:=proc(q) local a,b,k,ok,n; for n from 10 to q do a:=[]; b:=n;
while b>0 do a:=[op(a),b mod 10]; b:=trunc(b/10); od; b:=0; ok:=1; for k from 2 to nops(a)
do if a[k-1]=0 and a[k]=0 then ok:=0; break; else b:=b+a[k-1]^a[k]; fi; od;
if ok=1 then if n=(b+a[nops(a)]^a[nops(1)]) then print(n);
fi; fi; od; end: P(10^10);

fQ[n_] := Block[{r = Reverse@ IntegerDigits@ n}, n == Plus @@ (r^RotateLeft@ r)]; k = 1; lst = {}; While[k < 1000000001, If[ fQ@ k, AppendTo[ lst, k]; Print@ k]; k++] (* Robert G. Wilson v, Jun 01 2014 *)

Added a(1) as 1 and a(6) by Robert G. Wilson v, Jun 01 2014

A257814 Numbers n such that k times the sum of the digits (d) to the power k equal n, so n=k*sum(d^k), for some positive integer k, where k is smaller than sum(d^k).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 50, 298, 130004, 484950, 3242940, 4264064, 5560625, 36550290, 47746195, 111971979, 129833998, 9865843497, 46793077740, 767609367921, 4432743262896, 42744572298532, 77186414790914, 99320211963544, 99335229415136, 456385296642870

Offset: 1

Pieter Post, May 10 2015

The first nine terms are trivial, but then the terms become scarce. The exponent k must be less than the "sum of the digits" raised to the k-th power, otherwise there will be infinitely many terms containing 1's and 0's, like 11000= 5500*(1^5500+1^5500+0^5500+0^5500+0^5500). It appears this sequence is finite, because there is a resemblance with the Armstrong numbers (A005188).

			50 = 2*(5^2+0^2);
484950 = 5*(4^5+8^5+4^5+9^5+5^5+0^5).

Chai Wah Wu, Table of n, a(n) for n = 1..54 (terms < 10^32, n = 1..53 from Giovanni Resta)

Cf. A005188, A023052, A257768.

sdk(d, k) = sum(j=1, #d, d[j]^k);
isok(n) = {d = digits(n); k = 1; while ((val=k*sdk(d,k)) != n, k++; if (val > n, return (0))); k < sdk(d,k);} \\ Michel Marcus, May 30 2015

def mod(n,a):
    kk = 0
    while n > 0:
        kk= kk+(n%10)**a
        n =int(n//10)
    return kk
for a in range (1, 10):
    for c in range (1, 10**7):
        if c==a*mod(c,a) and a

a(16)-a(28) from Giovanni Resta, May 10 2015

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

Original entry on oeis.org

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

cp's OEIS Frontend

A061862 Powerful numbers (2a): a sum of nonnegative powers of its digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A151543 Smallest precursor of n-th cycle in the "Recurring Digital Invariant Variant" problem.

Views

Author

Keywords

Comments

Links

Crossrefs

A306354 a(n) = gcd(n, A101337(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

A016087 First nontrivial or multidigital Armstrong number to base n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A229381 The Simpsons's perfect number, Mersenne prime, and narcissistic number.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A306361 Numbers k divisible by A101337(k) (narcissistic function).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

A243025 Fixed points of the transform n = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + ... + d_(2)*10 + d_(1) -> Sum_{i=1..k-1}{d_(i)^d(i+1)}+d(k)^d(1) (A243023).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A243025 Fixed points of the transform n = d_(k)10^(k-1) + d_(k-1)10^(k-2) + ... + d_(2)*10 + d_(1) -> Sum_{i=1..k-1}{d_(i)^d(i+1)}+d(k)^d(1) (A243023).