

A056810


Numbers whose fourth power is a palindrome.


3



0, 1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001
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OFFSET

1,3


COMMENTS

Suppose a number is of the form a=10...01 then a^2=10..020..01, so a^2 is always a palindrome. a^3=10..030..030..01, so a^3 is always a palindrome. Similarly we also have a^4=10..040..060..040..01, so a^4 is always a palindrome. However, a^5 is in general not a palindrome, for example 101^5=10510100501.  Dmitry Kamenetsky, Apr 17 2009
The sequence contains no term with digit sum 3.  Vladimir Shevelev, May 23 2011. Proof: There are four possibilities for n:
1) 1+10^k+10^m, 0<k<m, 2) 1+2*10^r, r>0, 3) 2+10^s, s>0, 4) 3*10^t, t>=0.
In the last two cases n^4 is trivially not a palindrome.
For r>=2, in the second case we have n^4 = (1 + 2*10^r)^4 = 1 + 8*10^r + 4*10^(2*r) + 2*10^(2*r + 1) + 2*10^(3*r) + 3*10^(3*r + 1) + 6*10^(4*r) + 10^(4*r + 1)
which cannot be a palindrome.
If r=1, we have 1+8*10+...9*10^4+10^5 which also is not a palindrome.
The proof for the first case is similar. QED  Vladimir Shevelev, Oct 24 2015
Does every term have the structure 100...0001? Referring to the Simmons (1972) paper, we can also ask, if n is a number whose cube is a palindrome in base 4, must the base4 expansion of n have the form 100...0001?  N. J. A. Sloane, Oct 22 2015


LINKS

Table of n, a(n) for n=1..13.
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 9398. [Annotated scanned copy]
G. J. Simmons, On palindromic squares of nonpalindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 1119. [Annotated scanned copy]


MATHEMATICA

palQ[n_] := Block[{}, Reverse[idn = IntegerDigits@ n] == idn]; k = 0; lst = {}; While[k < 1000000002, If[ palQ[k^4], AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Oct 23 2015 *)


PROG

(Python)
def ispal(n): s = str(n); return s == s[::1]
def afind(limit):
for k in range(limit+1):
if ispal(k**4): print(k, end=", ")
afind(10000001) # Michael S. Branicky, Sep 05 2021


CROSSREFS

Cf. A186080.
Sequence in context: A031997 A116098 A116129 * A000533 A147759 A147757
Adjacent sequences: A056807 A056808 A056809 * A056811 A056812 A056813


KEYWORD

nonn,base,more


AUTHOR

Robert G. Wilson v, Aug 21 2000


EXTENSIONS

a(11) from Robert G. Wilson v, Oct 23 2015
a(12)a(13) from Michael S. Branicky, Sep 05 2021


STATUS

approved



