A350300 Irregular triangle read by rows: The n-th row lists the "n-th power friends", numbers k such that digsum(digsum(k^n)^n) = k but digsum(k^n) is not k.

13, 16, 19, 28, 18, 27, 23, 29, 31, 34, 38, 44, 46, 47, 55, 56, 62, 65, 64, 73, 35, 80, 109, 127, 52, 70, 112, 118, 121, 127, 136, 181, 97, 108, 117, 130, 144, 153, 88, 144, 153, 160, 139, 152, 153, 154, 161, 173, 176, 178, 181, 184, 187, 189, 189, 198

Offset

2,1

Comments

Two numbers x and y with x < y are said to be "n-th power friends" if digsum(x^n)=y and digsum(y^n)=x. This sequence lists both x and y; A350301 lists just the x's and A350302 lists just the y's.

The name "n-th power friends" comes from "The Man Who Counted", where the n=2 case is discussed:

"The digits of the number 256 add up to 13. The square of 13 is 169. The digits of 169 add up to 16. As a result, the numbers 13 and 16 have a curious relation, which we could call a quadratic friendship" (p. 33).

There are finitely many such k for a particular n since digsum(digsum(k^n)^n) <= 9n log_10(9n log_10(k)). There are no such k for n=1 since either k is a single digit or else digsum(k) < k.

References

M. Tahan, The Man Who Counted: A Collection of Mathematical Adventures, W. W. Norton & Company, 1993.

Links

Table of n, a(n) for n=2..57.

Example

Triangle begins:
  13, 16;
  19, 28;
  18, 27;
  23, 29, 31, 34;
  ;
  38, 44, 46, 47, 55, 56, 62, 65;
  64, 73;
  35, 80;
  ;
  ;
  109, 127;
  52, 70, 112, 118, 121, 127, 136, 181;
  97, 108, 117, 130;
  144, 153;
  88, 144, 153, 160;
  ...
18 and 27 are in row n=4 since 18^4 = 104976 and 1 + 0 + 4 + 9 + 7 + 6 = 27, and 27^4 = 531441 and 5 + 3 + 1 + 4 + 4 + 1 = 18.

Prog

(Python 3)
from math import log
n = 1
while n <= 50:
    k = 2
    while 9*n*log(9*n*log(k, 10), 10) >= k:
        s1 = sum(int(d) for d in str(k**n))
        s2 = sum(int(d) for d in str(s1**n))
        if k != s1 and k == s2:
            print(k)
        k += 1
    n += 1

Crossrefs

Cf. A007953, A350301, A350302.

Sequence in context: A295327 A179202 A159975 * A110623 A163674 A280062

Adjacent sequences: A350297 A350298 A350299 * A350301 A350302 A350303

Keyword

nonn,base,tabf

Author

Daniel Carter, Dec 23 2021

Status

approved