A114260 - OEIS

%I #45 Feb 29 2024 06:25:52

%S 5702631489,7264103985,7602314895,7824061395,8105793624,8174035962,

%T 8304269175,8904623175,8923670541,9451360827,9785261403,9804753612,

%U 9846032571,57026314890,59730829461,60947591328,64017823995,65190218436,67024081935,70645192839,72641039850,74991208356

%N Numbers k such that the 4th power of k contains exactly 4 copies of each digit of k.

%C First 13 terms of the sequence are also pandigital, i.e., they contain all the 10 digits at least once. This is probably accidental, but quite curious.

%C If a(n) is in the sequence, then 10*a(n) is also in the sequence. So 10^k*a(n) is also in the sequence for positive integers k.  Thus this sequence differs from A365144. - _Ray Chandler_, Aug 23 2023

%C From _David A. Corneth_, Aug 30 2023: (START)

%C Not all terms are pandigital, for example 65190218436, 75932056341 and 83581076421 are not.

%C For any k ends in, say, 425742 (which has six digits) the last six digits of k^4 are fixed. So if k ends in 425742 then k^4 ends in 425742^2 mod 10^6 = 318096 and so it must have a 3, 1, 8, 0, 9 and 6 none of which are contained in 425742.

%C Therefore if k ends in 425742 then it must have at least 12 digits. In a search for terms <= 10^11 this eases the search, in a search for terms <= 10^12 this leaves only 6! * 5 / 6 = 600 cases to check instead of 10^6.

%C An additional idea one might use is that the number of digits of k^4 must be a multiple of 4. I.e. 10^(4*m + 3) <= k^4 <= 10^(4*m + 4) so 10^m * 10^0.75 < m < 10^(m + 1) (all strict inequalities as 10^0.75 (cf. A210522) is irrational) which tells us a bunch about the leading digits of k.

%C Checking pandigital numbers seperately might ease the search. That way if the union of some k and last q digits of k^4 is all decimal digits one could end the search there. This goes for example for 100426. If a term ends in 100426 then it has all decimal digits. (END)

%C All terms are divisible by 9. First decimal digit of a term is 5 or larger. - _Chai Wah Wu_, Feb 27 2024

%H Chai Wah Wu, <a href="/A114260/b114260.txt">Table of n, a(n) for n = 1..69</a> (terms 1..53 from David A. Corneth)

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_971.htm">Puzzle 971. The cube of N such that...</a>, The Prime Puzzles & Problems Connection.

%e 5702631489 is a term since its 4th power 1057550783692741389295697108242363408641 contains four 5's, four 7's, four 0's and so on.

%o (Python)

%o from itertools import count, islice

%o from sympy import integer_nthroot

%o def A114260_gen(): # generator of terms

%o     for l in count(1):

%o         a = integer_nthroot(10**(4*l-1),4)[0]

%o         if (a9:=a%9):

%o             a += 9-a9

%o         for b in range(a,10**l,9):

%o             if sorted(str(b)*4)==sorted(str(b**4)):

%o                 yield b

%o A114260_list = list(islice(A114260_gen(), 5)) # _Chai Wah Wu_, Feb 27 2024

%Y Cf. A114258, A114259, A114261, A210522, A365144.

%K nonn,base

%O 1,1

%A _Giovanni Resta_, Nov 18 2005

%E a(14) from _Ray Chandler_, Aug 24 2023

%E More terms from _David A. Corneth_, Aug 30 2023