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Numbers k such that k is in the middle of decimal expansion of k^k.
2

%I #26 Nov 21 2021 13:36:01

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

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

%C a(6) is greater than 50000.

%C a(8) > 600000. - _Giovanni Resta_, Oct 08 2013

%e 6 is in the sequence because 6^6 = 46656, which includes a 6 in the middle.

%e 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.

%t 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}]

%o (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

%o (Python)

%o from itertools import islice

%o def A229629(): # generator of terms

%o n = 1

%o while True:

%o s, sn = str(n**n), str(n)

%o l, ln = len(s), len(sn)

%o if (ln-l) % 2 == 0 and s[l//2-ln//2:l//2+(ln+1)//2] == sn: yield n

%o n += 1

%o A229629_list = list(islice(A229629(),5)) # _Chai Wah Wu_, Nov 21 2021

%Y Cf. A033147, A049329, A131495.

%K nonn,base,nice,more

%O 1,2

%A _Farideh Firoozbakht_, Oct 04 2013

%E a(6)-a(7) from _Giovanni Resta_, Oct 08 2013