A287198 -id:A287198 - 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)

A287478 Positive numbers m with the property that m is the least cyclic permutation of its digits with the same number of digits as m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

David A. Corneth, May 25 2017

First differs from A179239 at n = 83; a(83) = 120 and A179239(83) = 122. This sequence is a supersequence of A179239.
If there is a zero digit, then we do not consider the cyclic shift which begins with the zero digit.
The number 0 should also be in this list. The initial digit of any term must be its smallest nonzero digit. - M. F. Hasler, Oct 18 2019

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A179239, A287198.

Select[Range@ 120, Function[d, First@ Sort@ Map[FromDigits, DeleteCases[ NestList[RotateLeft, d, Length@ d - 1], ?(First@ # == 0 &)]] == #]@ IntegerDigits@ # &] (* _Michael De Vlieger, May 27 2017 *)

is(n) = my(d=digits(n), v=vector(#d), no=n, nn=n, l=List(n)); for(i=2,#d, no = nn\10; no = no+(nn-no*10)*10^(#d-1); if(#digits(no)==#d,listput(l, no)); nn=no); listsort(l); n==l[1]
is(n) = {my(d = digits(n), dd = concat(d, d)); for(i=2,#d, c=vector(#d, j, dd[i+j-1]); if(fromdigits(c) < n, if(c[1]!=0, return(0)))); 1}

is_A287478(n,D=digits(n))={!for(i=2,#D,((D[i]M. F. Hasler, Oct 18 2019

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A287513 Numbers whose cyclic permutations are pairwise coprime.

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A287478 Positive numbers m with the property that m is the least cyclic permutation of its digits with the same number of digits as m.

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Mathematica

PARI

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