A084433 -id:A084433 - cp's OEIS frontend

A287513 Numbers whose cyclic permutations are pairwise coprime.

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, 47, 49, 52, 53, 56, 58, 59, 61, 65, 67, 71, 73, 74, 76, 79, 83, 85, 89, 91, 92, 94, 95, 97, 98, 112, 113, 115, 116, 118, 119, 121, 125, 127, 131, 133, 134, 136, 137

Offset: 1

Rémy Sigrist, May 26 2017

No term, except 10, contains a '0' digit.
No term contains two even digits.
No term > 9 is a multiple of 3.
No term contains two '5' digits.
This sequence contains A287198.
This sequence does not contain any term > 9 of A084433.
In the scatterplot of the first 10000 terms:
- the jump from a(7128) = 99998 to a(7129) = 111112 is due to the fact that there is no term > 10 starting with "10",
- 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,
- these features can be seen at different scales (see scatterplots in Links section).

			The cyclic permutations of 5992 are:
- 5992 = 2^3 * 7 * 107
- 9925 = 5^2 * 397
- 9259 = 47 * 197
- 2599 = 23 * 113.
These values are pairwise coprime, hence 5992 appear in the sequence.
The cyclic permutations of 5776 are:
- 5776 = 2^4 * 19^2,
- 7765 = 5 * 1553,
- 7657 = 13 * 19 * 31,
- 6577 = 6577.
gcd(5776, 7657) = 19, hence 5776 does not appear in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 2000 terms
Rémy Sigrist, Scatterplot of the first 10000 terms
Rémy Sigrist, Scatterplot of the first 150000 terms

Cf. A084433, A287198, A387356.

A287513Q[k_] := k < 10 || CoprimeQ @@ Map[FromDigits, NestList[RotateLeft, #, Length[#] - 1] & [IntegerDigits[k]]];
Select[Range[200], A287513Q] (* Paolo Xausa, Aug 27 2025 *)

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)

A084434 Numbers whose digit permutations have GCD > 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 18, 20, 21, 22, 24, 26, 27, 28, 30, 33, 36, 39, 40, 42, 44, 45, 46, 48, 50, 51, 54, 55, 57, 60, 62, 63, 64, 66, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132

Offset: 1

Amarnath Murthy, Jun 02 2003

Numbers k such that there is a number d>1 which divides every number that can be obtained by permuting the digits of k. - N. J. A. Sloane, Aug 27 2020

Theorem. The sequence consists of: (1) A008585 (multiples of 3), (2) A014263 (numbers with all digits even), (3) A014181 (numbers with all digits equal), (4) numbers with all digits 5 or 0, (5) numbers with all digits 7 or 0, (6) numbers with 6k digits, all of which are 1 or 8, and (7) numbers with 6k digits, all of which are 2 or 9. - David Wasserman, May 07 2004

			72 is in the sequence because 72 and 27 are both divisible by 9.

Cf. A008585, A014181, A014263, A071249.

Subsequence of A084433 which contains for example 592 which is not in here.

Select[Range[0, 150], GCD @@ FromDigits /@ Permutations[IntegerDigits[#]] > 1 &]  (* Harvey P. Dale, Jan 12 2011 *)

More terms from David Wasserman, May 07 2004
Initial zero removed, Harvey P. Dale, Jan 14 2011
Entry revised by N. J. A. Sloane, Aug 27 2020

cp's OEIS Frontend

A287513 Numbers whose cyclic permutations are pairwise coprime.

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