A055013 -id:A055013 - cp's OEIS frontend

A169662 Numbers divisible by the sum of their digits, and by the sum of their digits squared, by the sum of their digits cubed and by the sum of 4th powers of their digits.

1, 10, 100, 110, 111, 1000, 1010, 1011, 1100, 1101, 1110, 2000, 5000, 10000, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11100, 20000, 50000, 55000, 100000, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101100

Offset: 1

Michel Lagneau, Apr 05 2010

The numbers such that all digits are nonzero are rare (see the subsequence A176194).

			1121211 is a term since 1^4 + 1^4 + 2^4 + 1^4 + 2^4 + 1^4 + 1^4 = 37 and 1121211 = 37*30303 ; 1^3 + 1^3 + 2^3 + 1^3 + 2^3 + 1^3 + 1^3 = 21 and 1121211 = 21*53391 ; 1^2 + 1^2 + 2^2 + 1^2 + 2^2 + 1^2 + 1^2 = 13 and 1121211 = 13* 86247 ; 1 + 1 + 2 + 1 + 2 + 1 + 1 = 9 and 1121211 = 9*124579.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005349, A034087, A034088 and A169665.

Cf. A072081, A117562, A176194.

isA169662 := proc(n)
        dgs := convert(n,base,10) ;
        if (n mod ( add(d,d=dgs) ) = 0)  and (n mod (add(d^2,d=dgs) )) =0 and (n mod (add(d^3,d=dgs))) =0 and (n mod (add(d^4,d=dgs))) = 0 then
                true;
        else
                false;
        end if;
end proc:
for i from 1 to 110000 do
        if isA169662(i) then
                printf("%d,",i) ;
        end if;
end do: # R. J. Mathar, Nov 07 2011

q[n_] := And @@ Divisible[n, Plus @@@ Transpose @ Map[#^Range[4] &, IntegerDigits[n]]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 31 2021 *)

{n : A007953(n)|n and A003132(n)|n and A055012(n)| n and A055013(n)| n}.

A176194 Numbers with no zero digits divisible by the sum of the k-th powers of their digits, for each k = 1,2,3,4.

Original entry on oeis.org

1, 111, 1121211, 11243232, 12132432, 12413232, 22331232, 23111352, 23411232, 24113232, 41223312, 42131232, 44662464, 111111111, 112452144, 114251424, 135964224, 211412544, 246134592, 313212312, 332131212, 382941675, 416283624, 442114512, 523173456, 671635575, 979652772

Offset: 1

Michel Lagneau, Apr 11 2010

For the numbers divisible by the sum of k-th powers of digits including 0, see A169662. The numbers such that the digits are > 0 are rare.

			For n = 246134592 we obtain :
2^4 + 4^4 + 6^4 + 1^4 + 3^4 + 4^4 + 5^4 + 9^4 + 2^4 = 9108, and 246134592 = 9108*27024 ;
2^3 + 4^3 + 6^3 + 1^3 + 3^3 + 4^3 + 5^3 + 9^3 + 2^3 = 1242, and 246134592 = 1242*198176 ;
2^2 + 4^2 + 6^2 + 1^2 + 3^2 + 4^2 + 5^2 + 9^2 + 2^2 = 192, and 246134592 = 192*1281951 ;
2 + 4 + 6 + 1 + 3 + 4 + 5 + 9 + 2 = 36, and 246134592 = 36*6837072.

Amiram Eldar, Table of n, a(n) for n = 1..464

Intersection of A052382 and A169662.

Cf. A034087, A072081, A117562, A034087, A072081, A117562, A034088, A034088.

with(numtheory):for n from 2 to 500000000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:s4:=0:p:=1:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:p:=p*u:s2:=s2+u^2:s3:=s3+u^3:s4:=s4+u^4: od:if irem(n,s1)=0 and irem(n,s2)=0 and irem(n,s3)=0 and irem(n,s4)=0 and p<>0 then print(n):else fi:od:

A007953 (n)|n and A003132(n)|n and A055012 (n)| n and A055013 (n)| n and all digits < > 0.

a(1)-a(2) and more terms add by Amiram Eldar, Apr 20 2023

A226087 Number of values k in base n for which the sum of digits of k = sqrt(k).

Original entry on oeis.org

1, 4, 2, 3, 3, 6, 2, 2, 2, 5, 2, 6, 2, 5, 5, 2, 2, 4, 2, 6, 6, 4, 2, 5, 2, 4, 2, 6, 2, 11, 2, 2, 6, 4, 5, 6, 2, 4, 6, 5, 2, 11, 2, 6, 5, 4, 2, 6, 2, 4, 6, 5, 2, 4, 5, 5, 6, 4, 2, 13, 2, 4, 4, 2, 5, 11, 2, 5, 6, 11, 2, 5, 2, 4, 6, 6, 6, 11, 2, 5, 2, 4, 2, 12, 5

Offset: 2

Christian N. K. Anderson and Kevin L. Schwartz, May 25 2013

Values of k in base n have at most 3 digits. Proof: Because sqrt(k) increases faster than the digit sum of k, only numbers with d digits meeting the condition d*(n-1) >= n^(d/2) are candidate fixed points. d < 3 for n > 6, and since there are no fixed points of four or more digits in bases 2 through 5, there are no fixed points in any base with more than 3 digits.

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.

			For a(16)=5 the solutions are the square numbers {1, 36, 100, 225, 441} because in base 16 they are written as {1, 24, 64, E1, 1B9} and
  sqrt(1) = 1
  sqrt(36) = 6 = 2+4
  sqrt(100) = 10 = 6+4
  sqrt(225) = 15 = 14+1, and
  sqrt(441) = 21 = 1+11+9.

Christian N. K. Anderson, Table of n, a(n) for n = 2..10000

Cf. A226224.

Cf. digital sums for digits at various powers: A007953, A003132, A055012, A055013, A055014, A055015.

sapply(2:16,function(n) sum(sapply((1:(n^ifelse(n>6,1.5,2)))^2, function(x) sum(inbase(x,n))==sqrt(x))))

A226352 Number of integers k in base n whose squared digits sum to sqrt(k).

Original entry on oeis.org

1, 3, 2, 2, 1, 1, 4, 2, 1, 2, 3, 6, 1, 6, 3, 3, 1, 2, 2, 3, 2, 4, 4, 4, 2, 9, 2, 4, 2, 3, 1, 3, 3, 3, 3, 1, 2, 4, 5, 4, 1, 6, 1, 5, 2, 5, 2, 5, 4, 1, 3, 5, 1, 5, 2, 5, 1, 7, 3, 2, 2, 7, 3, 2, 2, 4, 3, 2, 1, 3, 3, 6, 3, 3, 2, 1, 2, 5, 3, 4, 1, 4, 1, 3, 2, 3, 1

Offset: 2

Christian N. K. Anderson and Kevin L. Schwartz, Jun 04 2013

Any d-digit number in base n meeting the criterion must also meet the condition d*(n-1)^2 < n^(d/2). Numerically, it can be shown this limits the candidate values to squares < 22*n^4. The larger values are statistically unlikely, and in fact the largest value of k in the first 1000 bases is ~9.96*n^4 in base 775.

			In base 8, the four solutions are the values {1,16,256,2601}, which are written as {1,20,400,5051} in base 8 and
sqrt(1)    =  1 = 1^2;
sqrt(16)   =  4 = 2^2 + 0^2;
sqrt(256)  = 16 = 4^2 + 0^2 + 0^2;
sqrt(2601) = 51 = 5^2 + 0^2 + 5^2 + 1^2,

Christian N. K. Anderson, Table of n, a(n) for n = 2..1000

Cf. A226353.
Cf. A226353, A226224.
Cf. digital sums for digits at various powers: A007953, A003132, A055012, A055013, A055014, A055015.

inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }
for(n in 2:50) cat("Base", n, ":", which(sapply((1:(4.7*n^2))^2, function(x) sum(inbase(x, n)^2)==sqrt(x)))^2, "\n")

A226353 Largest integer k in base n whose squared digits sum to sqrt(k).

Original entry on oeis.org

1, 49, 169, 36, 1, 1, 2601, 1089, 1, 8836, 33489, 44100, 1, 149769, 128164, 96721, 1, 156816, 1225, 40804, 12321, 831744, 839056, 1149184, 1737124, 3655744, 407044, 1890625, 2208196, 1089, 1, 1466521, 6125625, 2235025, 2832489, 1, 3759721, 6885376, 8844676

Offset: 2

Christian N. K. Anderson and Kevin L. Schwartz, Jun 04 2013

Any d-digit number in base n meeting the criterion must also meet the condition d*(n-1)^2 < n^(d/2). Numerically, it can be shown this limits the candidate values to squares < 22*n^4. The larger values are statistically unlikely, and in fact the largest value of k in the first 1000 bases is ~9.96*n^4 in base 775.

a(n)=1 iff A226352(n)=1.

			In base 8, the four solutions are the values {1,16,256,2601}, which are written as {1,20,400,5051} in base 8 and
sqrt(1)   = 1  = 1^2
sqrt(16)  = 4  = 2^2+0^2
sqrt(256) = 16 = 4^2+0^2+0^2
sqrt(2601)= 51 = 5^2+0^2+5^2+1^2

Christian N. K. Anderson, Table of n, a(n) for n = 2..1000
Christian N. K. Anderson, Table of base, all solutions in base 10, and all solutions in base n for bases 2 to 1000.

Cf. A226352, A226224.

Cf. digital sums for digits at various powers: A007953, A003132, A055012, A055013, A055014, A055015.

inbase=function(n,b) { x=c(); while(n>=b) { x=c(n%%b,x); n=floor(n/b) }; c(n,x) }
for(n in 2:50) cat("Base",n,":",which(sapply((1:(4.7*n^2))^2,function(x) sum(inbase(x,n)^2)==sqrt(x)))^2,"\n")

A055207 Sum of n-th powers of digits of n.

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 1, 2, 4097, 1594324, 268435457, 30517578126, 2821109907457, 232630513987208, 18014398509481985, 1350851717672992090, 1048576, 2097153, 8388608, 94151567435, 281474993487872, 298023223910507557

Offset: 0

Henry Bottomley, Jun 19 2000

			a(12) = 1^12 + 2^12 = 1 + 4096 = 4097.

Seiichi Manyama, Table of n, a(n) for n = 0..1048

Cf. A045503, A003132, A055012, A055013, A055014.

a:= n-> add(i^n, i=convert(n, base, 10)):
seq(a(n), n=0..29);  # Alois P. Heinz, Dec 18 2022

Join[{1},Table[Total[IntegerDigits[n]^n],{n,25}]] (* Harvey P. Dale, Jul 16 2011 *)

A217532 Numbers k such that sum of 4th power of digits of k equals the sum of prime divisors of k.

Original entry on oeis.org

210, 1210, 2100, 12100, 21000, 21021, 36522, 63141, 89195, 92029, 112132, 116010, 118461, 121000, 149851, 203013, 203202, 210000, 214456, 303142, 304341, 313230, 323723, 331401, 351760, 416213, 441532, 524371, 534656, 574915, 610171, 654560, 897648, 999643

Offset: 1

Michel Lagneau, Oct 05 2012

Numbers k such that A055013(k) = A008472(k).

			210 = 2*3*5*7 is in the sequence because 2^4 + 1^4 + 0^4 = 2 + 3 + 5 + 7 = 17.

Cf. A055013, A008472.

Rest[Select[Range[1000000], Total[Transpose[FactorInteger[#]][[1]]]==Total[IntegerDigits[#]^4]&]]

A260598 Numbers n such that the sum of the divisors of n equals the fourth power of the sum of the digits of n.

Original entry on oeis.org

1, 510, 11235, 12243, 14223, 136374, 142494, 145266, 148614, 163158, 171465, 181815, 214863, 240963, 246507, 323976, 397182, 404994, 1548798

Offset: 1

Michael Savoric, Aug 05 2015

Let n be a k-digit number. Then, sigma(n) >= 10^(k-1) and (9*k)^4 >= sum_of_digits(n)^4. So, n must be less than 10^9. - Hiroaki Yamanouchi, Aug 29 2015

			510 is in the sequence, since (1 + 2 + 3 + 5 + ... + 255 + 510) = (5 + 1 + 0)^4.

Cf. A000203, A000583, A007953, A055013.

[n: n in [1..3*10^6] | DivisorSigma(1,n) eq (&+Intseq(n)^4)]; // Vincenzo Librandi, Aug 29 2015

n = 10000000;
list = {};
x = 1;
While[x <= n,
  If[Total[Divisors[x]] == Total[IntegerDigits[x]]^4,
   AppendTo[list, x]];
  x = x + 1
  ];
list

isok(n) = sigma(n) == sumdigits(n)^4; \\ Michel Marcus, Aug 06 2015

A000583(A007953(a(n))) = A000203(a(n)).

A360422 Numbers k such that k^2 + (sum of fourth powers of the digits of k^2) is a square.

Original entry on oeis.org

0, 89, 137, 6985

Offset: 1

Robert Israel and Will Gosnell, Feb 06 2023

Since A055013(k) < 9^4*(1+log_10(k)) and k^2 + 9^4*(1+log_10(k^2)) < (k+1)^2 for k > 32918, one need only search up to k = 32918.

			a(3) = 137 is a term because 137^2 = 18769 and 18769 + 1^4 + 8^4 + 7^4 + 6^4 + 9^4 = 182^2.

Cf. A055013, A360424.

select(n -> issqr(n^2 + add(t^4, t = convert(n^2,base,10))), [$0..32918]);

from math import isqrt
def sq(n): return isqrt(n)**2 == n
def ok(n): return sq(n**2 + sum(int(d)**4 for d in str(n**2)))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Feb 12 2023

A055208 Table read by ascending antidiagonals: T(n,k) (n >= 1, k >= 1) is the sum of k-th powers of digits of n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 1, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 2, 1, 729, 4096, 16807, 46656, 78125

Offset: 1

Henry Bottomley, Jun 19 2000

			T(34,2) = 3^2 + 4^2 = 25.
From _Georg Fischer_, Mar 01 2022: (Start)
Array T(n, k) (n >= 1, k >= 1) begins:
  1,   1,   1,   1, ...
  2,   4,   8,  16, ...
  3,   9,  27,  64, ...
  4,  16,  64, 256, ...
  ...
(End)

Cf. A051128. Rows include A003132, A055012, A055013, A055014. Columns include A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A000012, A007395, A000051, A034472, A052539, A034474.

Definition clarified by Georg Fischer, Mar 01 2022

cp's OEIS Frontend

A169662 Numbers divisible by the sum of their digits, and by the sum of their digits squared, by the sum of their digits cubed and by the sum of 4th powers of their digits.

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A176194 Numbers with no zero digits divisible by the sum of the k-th powers of their digits, for each k = 1,2,3,4.

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A226087 Number of values k in base n for which the sum of digits of k = sqrt(k).

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A226352 Number of integers k in base n whose squared digits sum to sqrt(k).

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A226353 Largest integer k in base n whose squared digits sum to sqrt(k).

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R

A055207 Sum of n-th powers of digits of n.

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A217532 Numbers k such that sum of 4th power of digits of k equals the sum of prime divisors of k.

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A260598 Numbers n such that the sum of the divisors of n equals the fourth power of the sum of the digits of n.

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