login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A114260 Numbers k such that the 4th power of k contains exactly 4 copies of each digit of k. 8
5702631489, 7264103985, 7602314895, 7824061395, 8105793624, 8174035962, 8304269175, 8904623175, 8923670541, 9451360827, 9785261403, 9804753612, 9846032571, 57026314890, 59730829461, 60947591328, 64017823995, 65190218436, 67024081935, 70645192839, 72641039850, 74991208356 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
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
From David A. Corneth, Aug 30 2023: (START)
Not all terms are pandigital, for example 65190218436, 75932056341 and 83581076421 are not.
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.
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.
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.
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)
All terms are divisible by 9. First decimal digit of a term is 5 or larger. - Chai Wah Wu, Feb 27 2024
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..69 (terms 1..53 from David A. Corneth)
Carlos Rivera, Puzzle 971. The cube of N such that..., The Prime Puzzles & Problems Connection.
EXAMPLE
5702631489 is a term since its 4th power 1057550783692741389295697108242363408641 contains four 5's, four 7's, four 0's and so on.
PROG
(Python)
from itertools import count, islice
from sympy import integer_nthroot
def A114260_gen(): # generator of terms
for l in count(1):
a = integer_nthroot(10**(4*l-1), 4)[0]
if (a9:=a%9):
a += 9-a9
for b in range(a, 10**l, 9):
if sorted(str(b)*4)==sorted(str(b**4)):
yield b
A114260_list = list(islice(A114260_gen(), 5)) # Chai Wah Wu, Feb 27 2024
CROSSREFS
Sequence in context: A322687 A322697 A365144 * A344737 A344738 A113644
KEYWORD
nonn,base
AUTHOR
Giovanni Resta, Nov 18 2005
EXTENSIONS
a(14) from Ray Chandler, Aug 24 2023
More terms from David A. Corneth, Aug 30 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 06:04 EDT 2024. Contains 371906 sequences. (Running on oeis4.)