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A287513 Numbers whose cyclic permutations are pairwise coprime.

%I A287513 #19 Aug 27 2025 12:06:54
%S A287513 1,2,3,4,5,6,7,8,9,10,13,14,16,17,19,23,25,29,31,32,34,35,37,38,41,43,
%T A287513 47,49,52,53,56,58,59,61,65,67,71,73,74,76,79,83,85,89,91,92,94,95,97,
%U A287513 98,112,113,115,116,118,119,121,125,127,131,133,134,136,137
%N A287513 Numbers whose cyclic permutations are pairwise coprime.
%C A287513 No term, except 10, contains a '0' digit.
%C A287513 No term contains two even digits.
%C A287513 No term > 9 is a multiple of 3.
%C A287513 No term contains two '5' digits.
%C A287513 This sequence contains A287198.
%C A287513 This sequence does not contain any term > 9 of A084433.
%C A287513 In the scatterplot of the first 10000 terms:
%C A287513 - the jump from a(7128) = 99998 to a(7129) = 111112 is due to the fact that there is no term > 10 starting with "10",
%C A287513 - the dotted lines, for example between a(2545) = 21131 and a(2772) = 29999, are due to the fact that there is no term starting with two even digits,
%C A287513 - these features can be seen at different scales (see scatterplots in Links section).
%H A287513 Rémy Sigrist, <a href="/A287513/b287513.txt">Table of n, a(n) for n = 1..10000</a>
%H A287513 Rémy Sigrist, <a href="/A287513/a287513.png">Scatterplot of the first 2000 terms</a>
%H A287513 Rémy Sigrist, <a href="/A287513/a287513_1.png">Scatterplot of the first 10000 terms</a>
%H A287513 Rémy Sigrist, <a href="/A287513/a287513_2.png">Scatterplot of the first 150000 terms</a>
%e A287513 The cyclic permutations of 5992 are:
%e A287513 - 5992 = 2^3 * 7 * 107
%e A287513 - 9925 = 5^2 * 397
%e A287513 - 9259 = 47 * 197
%e A287513 - 2599 = 23 * 113.
%e A287513 These values are pairwise coprime, hence 5992 appear in the sequence.
%e A287513 The cyclic permutations of 5776 are:
%e A287513 - 5776 = 2^4 * 19^2,
%e A287513 - 7765 = 5 * 1553,
%e A287513 - 7657 = 13 * 19 * 31,
%e A287513 - 6577 = 6577.
%e A287513 gcd(5776, 7657) = 19, hence 5776 does not appear in the sequence.
%t A287513 A287513Q[k_] := k < 10 || CoprimeQ @@ Map[FromDigits, NestList[RotateLeft, #, Length[#] - 1] & [IntegerDigits[k]]];
%t A287513 Select[Range[200], A287513Q] (* _Paolo Xausa_, Aug 27 2025 *)
%o A287513 (PARI) is(n) = my (p=n, l=#digits(n)); for (k=1, l-1, n = (n\10) + (n%10)*(10^(l-1)); if (gcd(n, p)>1, return (0)); p = lcm(n, p);); return (1)
%Y A287513 Cf. A084433, A287198, A387356.
%K A287513 nonn,base,changed
%O A287513 1,2
%A A287513 _Rémy Sigrist_, May 26 2017