A046197 -id:A046197 - cp's OEIS frontend

A046156 Limit set for operation of repeatedly replacing a number with the sum of the cubes of its digits.

0, 1, 55, 133, 136, 153, 160, 217, 244, 250, 352, 370, 371, 407, 919, 1459

Offset: 1

Richard C. Schroeppel

Range of A165330; A165330(a(n))=a(n); A165331(a(n))=0. - Reinhard Zumkeller, Sep 17 2009

lst = {}; k = 0; While[k < 1500, a = NestWhile[Plus @@ (IntegerDigits@ #^3) &, k, Unequal, All]; If[FreeQ[lst, a], AppendTo[lst, a]]; k++]; Sort@ lst (* Robert G. Wilson v, Jan 19 2006, revised Jan 03 2015 *)
Table[Nest[Total[IntegerDigits[#]^3]&,n,30],{n,0,1500}]//Union (* Harvey P. Dale, Aug 04 2018 *)

A165333 Numbers that eventually reach the fixed point 370 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

7, 19, 34, 37, 43, 58, 67, 70, 73, 76, 85, 88, 91, 109, 118, 124, 139, 142, 145, 148, 154, 157, 166, 169, 175, 178, 181, 184, 187, 190, 193, 196, 214, 223, 226, 232, 241, 247, 259, 262, 268, 274, 277, 286, 295, 304, 307, 319, 322, 334, 340, 343, 346, 355, 358

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 370;

Subsequence of A031179 and of A016777; a(n) mod 3 = 1.

			a(3)=34: 34 -> 3^3+4^3=91 -> 9^3+1=730 -> 7^3+3^3+0=370.

Cf. A031179, A046197, A035504, A008585, A165334, A165335.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ ,All]]; Select[Range[358], Last[Trajectory[#]] == 370&] (* Ant King, May 24 2013 *)

A165334 Numbers that eventually reach the fixed point 371 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 50, 53, 56, 59, 62, 65, 68, 71, 80, 83, 86, 92, 95, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185, 188, 191

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 371;
Subsequence of A031179;
complement of A165335 with respect to A016789; a(n) mod 3 = 2.

			a(10)=29: 29 -> 2^3+9^3=737 -> 2*7^3+3^3=713 -> 7^3+1+3^3=371.

Cf. A031179, A046197, A035504, A008585, A165333.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ ,All]]; Select[Range[191], Last[Trajectory[#]]==371 &] (* Ant King, May 24 2013 *)

A165335 Numbers that eventually reach the fixed point 407 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

47, 74, 77, 89, 98, 407, 449, 470, 494, 578, 587, 668, 686, 704, 707, 740, 758, 770, 785, 788, 809, 857, 866, 875, 878, 887, 890, 908, 944, 980, 1124, 1139, 1142, 1148, 1157, 1175, 1178, 1184, 1187, 1193, 1214, 1241, 1319, 1367, 1376, 1391, 1412, 1418, 1421

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 407.

			a(4)=89: 89 -> 8^3+9^3=1241 -> 1+2^3+4^3+1=74 -> 7^3+4^3=407.

Subsequence of A031179.

Cf. A046197, A035504, A008585, A165333.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f,n,UnsameQ,All]]; Select[Range[1421], Last[Trajectory[#]]==407 &] (* Ant King, May 24 2013 *)
Select[Range[1500],FixedPoint[Total[IntegerDigits[#]^3]&,#,100]==407&] (* Harvey P. Dale, Apr 17 2020 *)

Complement of A165334 with respect to A016789; a(n) mod 3 = 2.

A225534 Numbers whose sum of cubed digits is prime.

Original entry on oeis.org

11, 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, 245, 254, 256, 263, 265, 269, 272, 278, 281, 283, 287, 289, 296, 298, 311, 319, 322, 326, 328, 335

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

Note that 11 is the only two-digit number in the sequence.

a(n) ~ n. For 414 < n < 10000, 6.38*n - 528 provides an estimate of a(n) to within 6%.

			139 is in the sequence because 1^3 + 3^3 + 9^3 = 757, which is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A197039, A055012, A046459, A225535.
Cf. A165330, A046197.
Cf. A046704, A023194.

Select[Range[350],PrimeQ[Total[IntegerDigits[#]^3]]&] (* Harvey P. Dale, Mar 16 2016 *)

digcubesum<-function(x) sum(as.numeric(strsplit(as.character(x),split="")[[1]])^3); library(gmp);
which(sapply(1:1000,function(x) isprime(digcubesum(x))>0))

A194025 Number of fixed points under iteration of sum of cubes of digits in base b.

Original entry on oeis.org

1, 2, 9, 3, 4, 7, 6, 8, 5, 8, 5, 5, 3, 3, 24, 3, 2, 9, 2, 3, 16, 5, 2, 20, 2, 2, 7, 9, 3, 14, 2, 6, 8, 4, 10, 12, 2, 8, 8, 7, 2, 12, 4, 5, 17, 5, 4, 27, 6, 5, 10, 4, 2, 11, 9, 5, 9, 6, 3, 25, 5, 6, 24, 5, 4, 17, 5, 5, 9, 10, 1, 15, 4, 3, 13, 3, 5, 19, 4, 13, 7

Offset: 2

Martin Renner, Aug 22 2011

If b >= 2 and n >= 2*b^3, then S(n,3,b) < n. For each positive integer n, there is a positive integer m such that S^m(n,3,b) < 2*b^3. (Grundman/Teeple, 2001, Lemma 8 and Corollary 9.)
From Christian N. K. Anderson, May 23 2013: (Start)
1 is considered a fixed point in all bases, 0 is not.
In order for a number with d digits in base n to be a fixed point, it must satisfy the condition d*(n-1)^3 < n^d, which can only occur when d < 4 for n > 2. Because all binary numbers are "happy" (become 1 under iteration), there are no fixed points with more than 4 digits in any base. It can further be demonstrated that all 4-digit solutions begin with 1 in base n.
Unlike the number of fixed points under iteration of sum of squares of digits (A193583), this sequence contains many even numbers, and its histogram converges to a smooth distribution (approximately gamma(2.64,2.8); see "histogram" in links). (End)

			In the decimal system all integers go to (1); (153); (370); (371); (407) or (55, 250,133); (136, 244); (160, 217, 352); (919, 1459) under the iteration of sum of cubes of digits, hence there are five fixed points, two 2-cycles and two 3-cycles. Therefore a(10) = 5.

Christian N. K. Anderson, Table of n, a(n) for n = 2..1000
Christian N. K. Anderson, Histogram of a(n)
H. G. Grundman and E. A. Teeple, Generalized Happy Numbers, Fibonacci Quarterly 39 (2001), nr. 5, p. 462-466.

Cf. A193594, A194281.
Solutions for a(10): A046197.
Largest of the a(n) fixed points: A226026.
Related sequences for sum of squared digits: A193583, A209242.

S:=proc(n,p,b) local Q,k,N,z; Q:=[n]; for k from 1 do N:=Q[k]; z:=convert(sum(N['i']^p,'i'=1..nops(N)),base,b); if not member(z,Q) then Q:=[op(Q),z]; else Q:=[op(Q),z]; break; fi; od; return Q; end:
a:=proc(b) local F,i,A,Q,B,C; A:=[]: for i from 1 to 2*b^3 do Q:=S(convert(i,base,b),3,b); A:={op(A),Q[nops(Q)]}; od: F:={}: for i from 1 while nops(A)>0 do B:=S(A[1],3,b); C:=[seq(B[i],i=1..nops(B)-1)]: if nops(C)=1 then F:={op(F),op(C)}: fi: A:=A minus {op(B)}; od: return(nops(F)); end:
# Martin Renner, Aug 24 2011

#See A226026 for an optimized version
inbase=function(n, b) { x=c(); while(n>=b) { x=c(n%%b, x); n=floor(n/b) }; c(n, x) }; yn=rep(NA, 30)
for(b in 2:30) yn[b]=sum(sapply(1:(2*b^3), function(x) sum(inbase(x, b)^3))==1:(2*b^3)); yn # Christian N. K. Anderson, Jun 08 2013

def A194025(n):
    # inefficient but straightforward
    return len([i for i in (1..2*n**3) if i==sum(d**3 for d in i.digits(base=n))]) # D. S. McNeil, Aug 23 2011

A225535 Numbers whose cubed digits sum to a cube, and have more than one nonzero digit.

Original entry on oeis.org

168, 186, 345, 354, 435, 453, 534, 543, 618, 681, 816, 861, 1068, 1086, 1156, 1165, 1516, 1561, 1608, 1615, 1651, 1680, 1806, 1860, 3045, 3054, 3405, 3450, 3504, 3540, 4035, 4053, 4305, 4350, 4503, 4530, 5034, 5043, 5116, 5161, 5304, 5340, 5403, 5430, 5611

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

			5^3 + 6^3 + 1^3 + 1^3 = 343, which is 7^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A225534 (cubed digits sum to a prime), A197039 (square), A046459. A055012.
Cf. A165330 (cube cycle), A046197 (cubic fixed points), A000578 (cubes).
Cf. A052034 (squared digits sum to a prime), A028839, A117685.
Cf. A164882 (n such that sum of the cubes of the digits of n^3 is perfect cube). - Zak Seidov, May 21 2013

fQ[n_] := Module[{d = IntegerDigits[n]}, Count[d, 0] + 1 < Length[d] && IntegerQ[Total[d^3]^(1/3)]]; Select[Range[5611], fQ] (* T. D. Noe, May 19 2013 *)

y=rep(0,10000); len=0; x=0; library(gmp);
digcubesum<-function(x) sum(as.numeric(unlist(strsplit(as.character(as.bigz(x)),split="")))^3);
iscube<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(factorize(x)))%%3==0));
nonzerodig<-function(x) sum(strsplit(as.character(x),split="")[[1]]!="0");
which(sapply(1:6000,function(x) nonzerodig(x)>1 & iscube(digcubesum(x))))

A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

Original entry on oeis.org

1, 10, 2, 6, 5, 8, 3, 7, 5, 6, 3, 10, 2, 3, 3, 2, 4, 6, 2, 6, 3, 4, 2, 7, 5, 10, 2, 9, 2, 9, 2, 6, 3, 5, 3, 6, 3, 5, 5, 7, 2, 2, 4, 9, 6, 9, 5, 7, 2, 3, 2, 4, 2, 3, 6, 4, 5, 4, 2, 4, 4, 4, 3, 7, 3, 6, 3, 4, 3, 3, 4, 3, 4, 5, 3, 4, 5, 5, 3, 3, 2, 3, 2, 4, 3, 8, 3, 5, 2, 7, 3

Offset: 0

M. F. Hasler, Apr 14 2015

Row lengths of the table A252648.

For a number with d digits, the sum of n-th powers cannot exceed d*9^n, but the number is not less than 10^(d-1). Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10).

			a(0)=1 because 1 is the only number equal to the sum of 0th powers of its digits.
a(1)=10 because { 0, 1, ... 9 } are the only numbers equal to the sum of their digits (taken to the power 1).
a(2)=2 because 0 and 1 are the only numbers equal to the sum of the squares of their digits.
a(3)=6 because { 0, 1, 153, 370, 371, 407 } is the set of all numbers equal to the sum of the 3rd powers of their digits, cf. A046197.
For more examples, see the table A252648.

Don Knuth, Table of n, a(n) for n = 0..172
Table of a(n) for n=0..172 [From _Don Knuth_, Sep 09 2015]

Cf. A252648, A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Reap@ For[n = 0, n < 6, n++, Sow@ Length@ Select[Range[0, 10^(n + 1)], Plus @@ (IntegerDigits[#]^n) == # &]] // Flatten // Rest (* Michael De Vlieger, Apr 14 2015 *)

a(n) >= 2 for all n > 0, since 0 and 1 are digital invariants for any power n > 0.

a(10)-a(90) from Don Knuth, Sep 09 2015

A226026 Maximum fixed points under iteration of sum of cubes of digits in base n.

Original entry on oeis.org

1, 17, 62, 118, 251, 250, 433, 1052, 407, 1280, 2002, 1968, 793, 3052, 5614, 1456, 5337, 5939, 2413, 5615, 20217, 11648, 11080, 31024, 5425, 1737, 28027, 26846, 17451, 33535, 10261, 64019, 23552, 44937, 30086, 84870, 17353, 55243, 48824, 108936, 58618, 87977

Offset: 2

Kevin L. Schwartz and Christian N. K. Anderson, May 23 2013

1 is considered a fixed point in all bases, 0 is not.
a(n)=1 iff A194025(n)=1.
In order for a number with d digits in base n to be a fixed point, it must satisfy the condition d*(n-1)^32. Because all binary numbers are "happy" (become 1 under iteration), there are no fixed points with more than 4 digits in any base.
Furthermore, 4-digit solutions of the form x0mm or xmmm (where m is n-1) represent extreme values of sum of cubed digits, and so 4-digit numbers can only be solutions if xn^3+n^2-1<=2n^3+x^3. For x=2 this reduces to n<=3, so any 4-digit solution must begin with 1 in bases above 3.

			In base 5, the numbers 1, 28 and 118 are written as 1, 103, and 433. The sum of the cubes of their digits are 1, 1+0^3+3^3=28, and 4^3+3^3+3^3=118. There are no other solutions, so a(5)=118.

Christian N. K. Anderson, Table of n, a(n) for n = 2..1000
Christian N. K. Anderson, Table of base, maximum fixed point, number of fixed points, and all fixed points for base 2 to 1000.

Number of fixed points in base n: A194025.
All fixed points in base 10: A046197.
Cf. A193583, A209242.

inbase=function(n,b) { x=c(); while(n>=b) { x=c(n%%b,x); n=floor(n/b) }; c(n,x) }
yfp=vector("list",100)
for(b in 2:100) { fp=c()
    for(w in 0:1) for(x in 1:b-1) for(y in 1:b-1) if((u1=w^3+x^3+y^3)<=(u2=w*b^3+x*b^2+y*b) & u1+b^3>u2+b-1)
        if(length((z=which((1:b-1)*((1:b-1)^2-1)==u2-u1)-1))) fp=c(fp,u2+z)
    yfp[[b]]=fp[-1]
    cat("Base",b,":",fp,"\n")
}

A226063 Number of fixed points in base n for the sum of the fourth power of its digits.

Original entry on oeis.org

1, 1, 3, 4, 1, 1, 7, 3, 4, 3, 1, 2, 1, 7, 2, 2, 1, 4, 2, 6, 2, 3, 1, 3, 1, 11, 3, 3, 2, 2, 7, 4, 1, 4, 3, 1, 3, 4, 1, 2, 2, 2, 3, 4, 2, 2, 1, 2, 1, 2, 1, 2, 4, 3, 3, 2, 2, 1, 3, 2, 5, 2, 9, 2, 1, 2, 1, 1, 3, 2, 2, 1, 2, 5, 1, 5, 5, 4, 2, 5, 3, 2, 2, 3, 3, 1, 2

Offset: 2

Kevin L. Schwartz and Christian N. K. Anderson, May 24 2013

All fixed points in base n have at most 5 digits. Proof: In order to be a fixed point, a number with d digits in base n must meet the condition n^d <= d*(n-1)^4, which is only possible for d < 5.

For 5-digit numbers vwxyz in base n, only numbers where v*n^4 + n^3 - 1 <= v^4 + 3*(n-1)^4 or v*n^4 + n^4 - 1 <= v^4 + 4*(n-1)^4 are possible fixed points. v <= 2 for n <= 250.

			For a(8)=7, the solutions are {1,16,17,256,257,272,273}. In base 8, these are written as {1, 20, 21, 400, 401, 420, 421}. Because 1^4 = 1, 2^4 + 0^4 = 16, 2^4 + 1^4 = 17, 4^4 + 0^4 + 0^4 = 256, etc., these are the fixed points in base 8.

Christian N. K. Anderson, Table of n, a(n) for n = 2..250

Cf. A226064 (greatest fixed point).
Cf. A052455 (fixed points in base 10).
Cf. A023052, A046074, A046197.

inbase=function(n,b) { x=c(); while(n>=b) { x=c(n%%b,x); n=floor(n/b) }; c(n,x) }
yn=rep(NA,20)
for(b in 2:20) yn[b]=sum(sapply(1:(1.5*b^4),function(x) sum(inbase(x,b)^4))==1:(1.5*b^4)); yn

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A046156 Limit set for operation of repeatedly replacing a number with the sum of the cubes of its digits.

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A165333 Numbers that eventually reach the fixed point 370 under "x -> sum of cubes of digits of x" (see A055012).

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A165334 Numbers that eventually reach the fixed point 371 under "x -> sum of cubes of digits of x" (see A055012).

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A165335 Numbers that eventually reach the fixed point 407 under "x -> sum of cubes of digits of x" (see A055012).

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Formula

A225534 Numbers whose sum of cubed digits is prime.

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A194025 Number of fixed points under iteration of sum of cubes of digits in base b.

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Maple

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A225535 Numbers whose cubed digits sum to a cube, and have more than one nonzero digit.

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A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

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