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Nonpalindromic numbers that are products of repunits.
3

%I #33 Dec 16 2020 15:12:14

%S 161051,1490841,1625151,1771561,14921841,15043941,16266151,16399251,

%T 17876661,19487171,137009631,149231841,149352841,150574941,151807041,

%U 162676151,164140251,165483351,178927661,180391761,196643271,214358881,1370219631,1371330631,1492331841

%N Nonpalindromic numbers that are products of repunits.

%C The first term is A308365(19).

%C G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see link, page 98). According to this conjecture, these perfect powers are terms: {11^k, k>=4}, {111^k, k>=4}, {1111^k, k>=3}, {11111^k, k>=3}, ...

%H David A. Corneth, <a href="/A339676/b339676.txt">Table of n, a(n) for n = 1..10000</a>

%H Gustavus J. Simmons, <a href="https://oeis.org/A002778/a002778_2.pdf">Palindromic Powers</a>, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].

%e a(1) = 161051 = 11^5.

%e a(2) = 1490841 = 11^2 * 111^2.

%e a(3) = 1625151 = 11^4 * 111.

%e a(4) = 1771561 = 11^6.

%e a(5) = 14921841 = 11^2 * 111 * 1111.

%t vec[max_] := Module[{m = Floor @ Log10[9*max + 1], r, s = {1}, s1}, r = (10^Range[2, m] - 1)/9; Do[emax = Floor@Log[r[[k]], max]; s1 = r[[k]]^Range[0, emax]; s = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &], {k, 1, m - 1}]; s]; Select[vec[1.5*10^9], !PalindromeQ[#] &] (* _Amiram Eldar_, Dec 12 2020 *)

%Y Intersection of A308365 and A029742.

%Y Cf. A083278, A334131.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Dec 12 2020