A229629 Numbers k such that k is in the middle of decimal expansion of k^k.

1, 6, 888, 1808, 2138, 65246, 268105

Offset: 1

Author

Farideh Firoozbakht, Oct 04 2013

Keywords

• nonn
• base
• nice
• more

Comments

a(6) is greater than 50000.

a(8) > 600000. - Giovanni Resta, Oct 08 2013

Examples

			6 is in the sequence because 6^6 = 46656, which includes a 6 in the middle.
11 is not in the sequence, because even though the substring 11 appears twice in 11^11 = 285311670611, neither occurrence is precisely in the middle.
		

Crossrefs

Cf. A033147, A049329, A131495.

Programs

Mathematica

Do[a = IntegerDigits[n^n]; b = Length[a]; c = IntegerLength[n]; If[EvenQ[b - c] && FromDigits[Take[a, {(b - c)/2 + 1, (b + c)/2}]] == n, Print[n]], {n, 50000}]

PARI

is(n)=my(d=digits(n),D=digits(n^n)); if((#d+#D)%2, return(0)); for(i=1,#d, if(d[i]!=D[#D/2-#d/2+i], return(0))); 1 \\ Charles R Greathouse IV, Jul 30 2016

Python

from itertools import islice
def A229629(): # generator of terms
    n = 1
    while True:
        s, sn = str(n**n), str(n)
        l, ln = len(s), len(sn)
        if (ln-l) % 2 == 0 and s[l//2-ln//2:l//2+(ln+1)//2] == sn: yield n
        n += 1
A229629_list = list(islice(A229629(),5)) # Chai Wah Wu, Nov 21 2021

Extensions

a(6)-a(7) from Giovanni Resta, Oct 08 2013