A055013 -id:A055013 - cp's OEIS frontend

A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 0

R. Muller

Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).
Also the fixed point of the morphism 0 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -> {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - Robert G. Wilson v, Jul 27 2006
For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - Hieronymus Fischer, Jun 17 2007
It appears that a(n) is the position of 10*n in the ordered set of numbers obtained by inserting/placing one digit anywhere in the digits of n (except a zero before 1st digit). For instance, for n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where 20 is at position 2, so a(2) = 2. - Michel Marcus, Aug 01 2022
Also the total number of beads required to represent n on a Russian abacus (schoty). - P. Christopher Staecker, Mar 31 2023
a(n) / a(2n) <= 5 with equality iff n is in A169964, while a(n) / a(3n) is unbounded, since if n = (10^k + 2)/3, then a(n) = 3*k+1, a(3n) = 3, so a(n) / a(3n) = k + 1/3 -> oo when k->oo (see Diophante link). - Bernard Schott, Apr 29 2023
Also the number of symbols needed to write number n in Egyptian numerals for n < 10^7. - Wojciech Graj, Jul 10 2025

			a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38.
Krassimir Atanassov, On Some of the Smarandache's Problems, 1999.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, Vol. 18 (2011), #P178.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Diophante, A1762, Des chiffres à la moulinette (in French).
Ernesto Estrada and Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.
A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données (French) Acta Arith., Vol. 13 (1967/1968), pp. 259-265. MR0220693 (36 #3745)
Christian Mauduit and András Sárközy, On the arithmetic structure of sets characterized by sum of digits properties J. Number Theory, Vol. 61, No. 1 (1996), pp. 25-38. MR1418316 (97g:11107)
Christian Mauduit and András Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith., Vol. 81, No. 2 (1997), pp. 145-173. MR1456239 (99a:11096)
Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.
Kerry Mitchell, Spirolateral image for this sequence . [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Mathematische Semesterberichte, Vol. 49 (2002), pp. 209-226.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme.
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. pp. 205-206).
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.
Wikipedia, Digit sum.
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A169964, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).
Bisections: A004092, A004155.
For n + digsum(n) see A062028.

a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n',r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011

[ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Mar 17 2011

Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
Total/@IntegerDigits[Range[0,90]] (* Harvey P. Dale, May 10 2016 *)
DigitSum[Range[0, 100]] (* Requires v. 14 *) (* Paolo Xausa, May 17 2024 *)

a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10))) \\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n>9, n=divrem(n, 10); n[2]+a(n[1]), n)

a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\ M. F. Hasler, Mar 22 2011

a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:

a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\ M. F. Hasler, May 10 2015
/* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by M. F. Hasler, Nov 09 2018] */

a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018

def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014

def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021

(0 to 99).map(.toString.map(.toInt - 48).sum) // Alonso del Arte, Sep 15 2019

"Recursive version for general bases. Set base = 10 for this sequence."
digitalSum: base
| s |
base = 1 ifTrue: [^self].
(s := self // base) > 0
  ifTrue: [^(s digitalSum: base) + self - (s * base)]
  ifFalse: [^self]
"by Hieronymus Fischer, Mar 24 2014"

A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

a(A051885(n)) = n.
a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.
a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)
From Hieronymus Fischer, Jun 17 2007: (Start)
G.f. g(x) = Sum_{k > 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).
a(n) = n - 9*Sum_{10 <= k <= n} Sum_{j|k, j >= 10} floor(log_10(j)) - floor(log_10(j-1)). (End)
From Hieronymus Fischer, Jun 25 2007: (Start)
The g.f. can be expressed in terms of a Lambert series, in that g(x) = (x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k >= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k > 1 is a power of 10, else b(k) = 0.
G.f.: g(x) = (Sum_{k > 0} (1 - 9*c(k))*x^k)/(1-x), where c(k) = Sum_{j > 1, j|k} floor(log_10(j)) - floor(log_10(j-1)).
a(n) = n - 9*Sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
From Hieronymus Fischer, Oct 06 2007: (Start)
a(n) <= 9*(1 + floor(log_10(n))), equality holds for n = 10^m - 1, m > 0.
lim sup (a(n) - 9*log_10(n)) = 0 for n -> oo.
lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)
a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008
a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - Reinhard Zumkeller, Apr 25 2009
a(n) mod 2 = A179081(n). - Reinhard Zumkeller, Jun 28 2010
a(n) <= 9*log_10(n+1). - Vladimir Shevelev, Jun 01 2011
a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011
a(n) = Sum_{k >= 0} A031298(n, k). - Philippe Deléham, Oct 21 2011
a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - Hieronymus Fischer, Mar 24 2014
Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

More terms from Hieronymus Fischer, Jun 17 2007

Edited by Michel Marcus, Nov 11 2013

A055014 Sum of 5th powers of digits of n.

Original entry on oeis.org

0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 1, 2, 33, 244, 1025, 3126, 7777, 16808, 32769, 59050, 32, 33, 64, 275, 1056, 3157, 7808, 16839, 32800, 59081, 243, 244, 275, 486, 1267, 3368, 8019, 17050, 33011, 59292, 1024, 1025, 1056, 1267, 2048

Offset: 0

Henry Bottomley, May 31 2000

Fixed points are listed in A052464. - M. F. Hasler, Apr 12 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Chikawa, K. Iséki, T. Kusakabe, and K. Shibamura, Computation of cyclic parts of Steinhaus problem for power 5, Acta Arithmetica 7 (1962), 253-254. [From _Don Knuth_, Sep 07 2015]
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012, A055013.
Cf. A007953, A055017, A076313, A076314.
Cf. A052464.

[0] cat [&+[d^5: d in Intseq(n)]: n in [1..45]]; // Bruno Berselli, Feb 01 2013

A055014 := proc(n)
   add(d^5,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Jul 08 2012

Total/@(IntegerDigits[Range[50]]^5)  (* Harvey P. Dale, Jan 22 2011 *)
Table[Sum[DigitCount[n][[i]] i^5, {i, 9}], {n, 0, 45}] (* Bruno Berselli, Feb 01 2013 *)

A055014(n)=sum(i=1, #n=digits(n), n[i]^5) \\ M. F. Hasler, Apr 12 2015

a(n) = Sum_{k>=1} (floor(n/10^k) - 10*floor(n/10^(k+1)))^5. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n) + k^5, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007

A052455 Fixed points for operation of repeatedly replacing a number with the sum of the fourth power of its digits.

Original entry on oeis.org

0, 1, 1634, 8208, 9474

Offset: 1

Henry Bottomley, Mar 15 2000

This is row n=4 in A252648. - M. F. Hasler, Apr 12 2015

			a(2)=1634 since 1^4+6^4+3^4+4^4=1+1296+81+256=1634

Cf. A023052, A046074, A046197, A052464, A124068, A124069, A226970, A003321.

for(n=0,10^5,A055013(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

a(n) = A055013(a(n)). - M. F. Hasler, Apr 12 2015

A055015 Sum of 6th powers of digits of n.

Original entry on oeis.org

0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1, 2, 65, 730, 4097, 15626, 46657, 117650, 262145, 531442, 64, 65, 128, 793, 4160, 15689, 46720, 117713, 262208, 531505, 729, 730, 793, 1458, 4825, 16354, 47385, 118378, 262873

Offset: 0

Henry Bottomley, May 31 2000

The only fixed points (n = 0, 1 and 548834) are listed in row 6 of A252648. - M. F. Hasler, Apr 12 2015

Michel Marcus, Table of n, a(n) for n = 0..999
Index entries for Colombian or self numbers and related sequences

Cf. A003132.
Cf. A003132, A055012, A055013, A005514.
Cf. A007953, A055017, A076313, A076314.

[0] cat [&+[d^6: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

for n from 0 to 3 do seq(n^6+j^6, j=0..9 ); od; # Zerinvary Lajos, Nov 06 2006

Table[Sum[DigitCount[n][[i]] i^6, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A055015(n)=sum(i=1,#n=digits(n),n[i]^6) \\ M. F. Hasler, Apr 12 2015

a(n) = Sum_{k>0} (floor(n/10^k) - 10*floor(n/10^(k+1)))^6. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n) + k^6, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007

A169665 Numbers divisible by the sum of 4th powers of their digits.

Original entry on oeis.org

1, 10, 100, 102, 110, 111, 1000, 1010, 1011, 1020, 1100, 1101, 1110, 1121, 1122, 1634, 2000, 2322, 4104, 5000, 8208, 9474, 10000, 10010, 10011, 10100, 10101, 10110, 10200, 10412, 11000, 11001, 11010, 11100, 11210, 11220, 12502, 12521, 14758

Offset: 1

Michel Lagneau, Apr 05 2010

			12521 is a term since 1^4 + 2^4 + 5^4 + 2^4 + 1^4 = 659, and 12521 = 19*659;
89295 is a term since 8^4 + 9^4 + 2^4 + 9^4 + 5^4 = 17859, and 89295 = 5*17859.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit.

Cf. A034087, A034088, A055013, A169666.

A:= proc(n) add(d^4, d=convert(n, base, 10)) ; end proc: for n from 1 to 200000 do:if irem( n,A(n))=0 then printf(`%d, `,n):else fi:od:

Select[Range[15000], Divisible[#, Plus @@ (IntegerDigits[#]^4)] &] (* Amiram Eldar, Jan 31 2021 *)

Numbers k such that A055013(k) | k.

A226224 The largest value of k in base n for which the sum of digits of k = sqrt(k).

Original entry on oeis.org

1, 25, 9, 64, 100, 144, 49, 64, 81, 225, 121, 441, 169, 441, 441, 256, 289, 324, 361, 1296, 1296, 484, 529, 1089, 625, 676, 729, 2401, 841, 2601, 961, 1024, 3025, 1156, 2500, 4096, 1369, 1444, 4356, 3136, 1681, 4900, 1849, 5929, 3025, 2116, 2209, 6561, 2401

Offset: 2

Christian N. K. Anderson and Kevin L. Schwartz, May 31 2013

There are no values of k in base n with more than 3 digits. Proof: such a value with d digits would need to meet the criterion d*(n-1)>=sqrt(n)^d which establishes an upper limit of 4 digits for 2<=n<=6 and 3 for n>6. Because there are no four digit values of k in bases 2 through 6, k has a maximum of three digits in all bases.
Because k must be a square, there are only sqrt(n)^3 possible values in any base.
From the above, it can be shown that for three-digit fixed points of the form xyz, x <= 6; also x<=4 for n>846. These theoretical upper limits are statistically unlikely, and in fact of the 86356 solutions in bases 2 to 10000, only 6.5% of them begin with 2, and none begin with 3 through 6.
a(n)=1 iff A226087(n)=1. Conjecture: this occurs exactly once -- in base 2.

			For a(16) the solutions are the square numbers {1, 36=6^2, 100=10^2, 225=15^2, 441=21^2} because in base 16 they are written as {1, 24, 64, E1, 1B9} and 1 = 1, 6 = 2+4, 10 = 6+4, 15 = 14+1, and 21 = 1+11+9.

Christian N. K. Anderson, Table of n, a(n) for n = 2..10000
Christian N. K. Anderson, Table of base, max(k), number of k, and all values of k for base 2 to 10000.

Cf. A007953, A003132, A055012, A055013, A055014, A055015, A226087.

for(n in 2:500) cat("Base",n,":",which(sapply((1:(ifelse(n>6,7,1)*n^ifelse(n>6,1,2)))^2, function(x) sum(inbase(x,n))==sqrt(x)))^2, "\n")

A123253 Sum of 7th powers of digits of n.

Original entry on oeis.org

0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 128, 129, 256, 2315, 16512, 78253, 280064, 823671, 2097280, 4783097, 2187, 2188, 2315, 4374, 18571, 80312, 282123

Offset: 0

Zerinvary Lajos, Nov 06 2006

Fixed points are listed in A124068 = row n=7 of A252648. - M. F. Hasler, Apr 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A003132, A055012, A055013, A055014, A055015, A210840.

[0] cat [&+[d^7: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

A123253 := proc(n)
        add(d^7,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Jan 16 2013

Table[Sum[DigitCount[n][[i]] i^7, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A123253(n)=sum(i=1,#n=digits(n),n[i]^7) \\ M. F. Hasler, Apr 12 2015

A210767 Numbers whose digit sum as well as sum of the 4th powers of the digits is a prime.

Original entry on oeis.org

11, 12, 14, 16, 21, 23, 25, 29, 32, 34, 38, 41, 43, 47, 52, 58, 61, 67, 74, 76, 83, 85, 89, 92, 98, 101, 102, 104, 106, 110, 111, 113, 119, 120, 131, 133, 140, 146, 160, 164, 166, 179, 191, 197, 201, 203, 205, 209, 210, 223, 230, 232, 250, 269, 290, 296, 302

Offset: 1

Jonathan Vos Post, May 10 2012

This is to the exponent 4 as A182404 is to the exponent 2.

			21 is in the sequence because sum of digits 2+1= 3 is prime, and sum of the 4th powers of the digits 2^4+1^4=17 is a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007953, A055013, A182404.

Select[Range[350],AllTrue[{Total[IntegerDigits[#]],Total[ IntegerDigits[ #]^4]},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 01 2019 *)

dspow(n,b,k)=my(s);while(n,s+=(n%b)^k;n\=b);s
select(n->isprime(sumdigits(n))&&isprime(dspow(n,10,4)), vector(10^3, i, i)) \\ Charles R Greathouse IV, May 11 2012

{n such that A055013(n) and A007953(n) are both primes}.

A210840 Sum of the 8th powers of the digits of n.

Original entry on oeis.org

0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 1, 2, 257, 6562, 65537, 390626, 1679617, 5764802, 16777217, 43046722, 256, 257, 512, 6817, 65792, 390881, 1679872, 5765057, 16777472, 43046977, 6561, 6562, 6817, 13122, 72097, 397186

Offset: 0

Jonathan Vos Post, May 10 2012

This is to exponent 8 as A007953 is to exponent 0, A003132 is to exponent 2, and A055013 is to exponent 4. The subsequence of primes (for n = 11, 12, 14, 21, 41, ...) begins 2, 257, 65537, 65537.

			a(12) = 1^8 + 2^8 = 257.

Cf. A007953, A003132, A055013.

[0] cat [&+[d^8: d in Intseq(n)]: n in [1..35]]; // Bruno Berselli, Feb 01 2013

Table[Total[IntegerDigits[n]^8], {n, 0, 100}] (* T. D. Noe, May 18 2012 *)
Table[Sum[DigitCount[n][[i]] i^8, {i, 9}], {n, 0, 35}] (* Bruno Berselli, Feb 01 2013 *)

A362954 Numbers k such that k + the sum of the fourth powers of its digits is again a fourth power.

Original entry on oeis.org

0, 6047, 7518, 8127, 12207, 25247, 50000, 71966, 77326, 89582, 156156, 376189, 384624, 384640, 599611, 611356, 700158, 794139, 796715, 800558, 1172829, 1329051, 1329324, 1329340, 1488080, 1492525, 1862190, 2546894, 2547885, 5295852, 5302286, 5755548, 6244080, 6246510, 7291980, 7869294

Offset: 0

Will Gosnell and M. F. Hasler, May 09 2023

Karl-Heinz Hofmann, Visualization of n = 0 to 4.

Cf. A000583 (4th powers), A055013 (sum of 4th powers of decimal digits of n).

Cf. A362953 (the same for 3rd powers).

select( {is(n,p=4)=ispower(vecsum([d^p|d<-digits(n)])+n,p)}, [0..10^5])

aupto   = 7869300
A362954 = []
A000583 = set(fp**4 for fp in range(0, int(aupto**(1/4)+3)))
for n in range(0, aupto+1):
    if n + sum(int(digit)**4 for digit in str(n)) in A000583: A362954.append(n)
print(A362954) # Karl-Heinz Hofmann, Jun 02 2023

cp's OEIS Frontend

A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

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A055014 Sum of 5th powers of digits of n.

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A052455 Fixed points for operation of repeatedly replacing a number with the sum of the fourth power of its digits.

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A055015 Sum of 6th powers of digits of n.

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A169665 Numbers divisible by the sum of 4th powers of their digits.

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A226224 The largest value of k in base n for which the sum of digits of k = sqrt(k).

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