cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

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

Views

Author

Reinhard Zumkeller, Sep 17 2009

Keywords

Comments

A165330(a(n)) = 407.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

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

Keywords

Comments

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%.

Examples

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

Links

Crossrefs

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

Programs

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

A165339 Numbers that eventually reach the cycle 919-1459 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

49, 94, 199, 337, 373, 379, 397, 409, 478, 487, 490, 733, 739, 748, 784, 793, 847, 874, 904, 919, 937, 940, 973, 991, 1099, 1129, 1147, 1174, 1192, 1219, 1237, 1273, 1291, 1327, 1339, 1369, 1372, 1393, 1396, 1399, 1417, 1459, 1468, 1471, 1486, 1495, 1549
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 17 2009

Keywords

Comments

A165330(a(n)) = 919;
Subsequence of A165336.

Examples

			a(1)=49: 49 -> 4^3+9^3=793 -> 7^3+9^3+3^3=1099 -> 1+0+2*9^3=1459 -> 919 -> 1459 ... .

Crossrefs

Cf. A154820, A154877, A165330, A165336.

A165337 Numbers that eventually reach the cycle 136-244 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

136, 163, 244, 316, 361, 424, 442, 613, 631, 1036, 1063, 1306, 1360, 1489, 1498, 1603, 1630, 1849, 1894, 1948, 1984, 2044, 2344, 2347, 2374, 2404, 2434, 2437, 2440, 2443, 2467, 2473, 2476, 2647, 2674, 2734, 2743, 2746, 2764, 3016, 3061, 3106, 3160, 3244
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 17 2009

Keywords

Comments

A165330(a(n)) = 136;
Subsequence of A165336.

Crossrefs

Cf. A154820, A154877, A165330, A165336, A165339.

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • R
    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))))

A346630 Smallest number which reaches the narcissistic number 153 after n steps when repeatedly summing the cubes of its digits.

Original entry on oeis.org

153, 135, 18, 3, 9, 12, 33, 114, 78, 126, 6, 117, 669, 177, 12558, 44499999999999999999
Offset: 0

Views

Author

Jörg Zurkirchen, Jul 25 2021

Keywords

Comments

All the terms a(n) as well as the intermediate results will be multiples of 3:
x^3 mod 3 = x mod 3 [0^3 = 0; 1^3 = 1; (-1)^3 = -1].
Therefore (sum of cubes of digits) mod 3 = (sum of digits) mod 3.
Because the only multiple of 3 in A046197 is 153, every number which is a multiple of 3 will end up at 153.
Some other terms (not dealt with here) may reach a cycle of length > 1:
Elizabeth Todd has shown that only numbers (1 mod 3) and (2 mod 3) may reach a cycle, and the only possible cycles are {55, 230, 130}, {136, 244}, {160, 217, 352}, {919, 1459}. That means that numbers (0 mod 3) never reach a cycle but just a single number, namely 153.
Shyam Sunder Gupta tested all the multiples of 3 less than 10^5. He found that they all reach 153, in accordance with the above statements.
The values a(n) for n>15 are really too big to be fully written out (and so are missing in the list), as Jon E. Schoenfield calculated for n=16 and n=17:
a(16) = 3.777999...999*10^61042524005486970; it has one 3, three 7's, and 61042524005486967 9's, so the sum of the cubes of its digits is 1*3^3 + 3*7^3 + 61042524005486967*9^3 = 44499999999999999999 = a(15).
a(17) consists of the digit string 45888 followed by a very, very long string of 9's. The number of 9's in that string is (a(16) - 1725)/729, which is a 61042524005486968-digit number consisting of the digit 5 followed by 753611407475147 copies of the 81-digit string 182441700960219478737997256515775034293552812071330589849108367626886145404663923 followed by a single instance of the 60-digit string 182441700960219478737997256515775034293552812071330589849106.

Examples

			a(3) = 3, for 3^3 = 27, 2^3 + 7^3 = 351, 3^3 + 5^3 + 1^3 = 153.
a(13) = 177, for 177 -> 687 -> 1071 -> 345 -> 216 -> 225 -> 141 -> 66 -> 432 -> 99 -> 1458 -> 702 -> 351 -> 153 (13 = longest chain for numbers up to 10^4).
The process ends because 153 = 1^3 + 5^3 + 3^3.

Links

Crossrefs

Cf. A055012 (sum of cubes of digits), A182111 (number of steps to a cycle), A165330 (cycle end), A046156.
Cf. A046197 (proving that {0, 1, 153, 370, 371, 407} are the only possible fixed points for all numbers of any size when repeatedly summing the cubes of its digits).
Cf. A346789 (concluding the number from the sum of the cubes of its digits).

Programs

  • Mathematica
    Table[k=0;While[Last[s=NestList[Total[IntegerDigits@#^3]&,k,n]]!=153||Count[s,153]!=1,k=k+3];k,{n,0,14}] (* Giorgos Kalogeropoulos, Jul 30 2021 *)
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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