A225184 -id:A225184 - cp's OEIS frontend

A225185 Primes p which do not have a primitive root that divides p+1.

7, 23, 31, 43, 47, 71, 73, 79, 103, 127, 151, 157, 167, 191, 193, 199, 223, 239, 241, 263, 271, 277, 283, 311, 313, 331, 337, 359, 367, 383, 397, 409, 431, 439, 457, 463, 479, 487, 503, 571, 577, 599, 607, 631, 647, 673, 691, 719, 727, 733, 739, 743, 751, 811, 823, 839, 863, 887, 911, 919, 967, 983, 991, 997

Offset: 1

N. J. A. Sloane, May 04 2013

nonn

			The primitive roots modulo 97 are 5, 7, 10, 13, 14, 15, 17, 21, 23, 26, 29, 37, 38, 39, ..., and 7 divides 98, so 97 is not a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Arto Lepistö, Francesco Pappalardi and Kalle Saari, Transposition Invariant Words, Theoret. Comput. Sci., Vol. 380, No. 3 (2007), pp. 377-387.

Cf. A060749, A225184 (complement), A001122.

[p: p in PrimesUpTo(1000) | not exists{r: r in [1..p-1] | IsPrimitive(r,p) and IsZero((p+1) mod r)}]; // Bruno Berselli, May 10 2013

q[n_] := PrimeQ[n] && AllTrue[PrimitiveRootList[n], ! Divisible[n + 1, #] &]; Select[Range[1000], q] (* Amiram Eldar, Oct 07 2021 *)
Select[Prime[Range[200]],NoneTrue[(#+1)/PrimitiveRootList[#],IntegerQ]&] (* Harvey P. Dale, Sep 08 2024 *)

forprime(p=2,1000, i=0;fordiv(p+1,X, if(znorder(Mod(X,p))==eulerphi(p), i=1)); if(i==0,print1(p", "))) \\ V. Raman, May 04 2012

More terms from V. Raman, May 04 2013

A318712 Numbers k for which a riffle shuffle permutation with two or more packets permutes all except the first and the last of the k cards.

Original entry on oeis.org

4, 6, 12, 14, 18, 20, 30, 38, 42, 54, 60, 62, 68, 84, 90, 98, 102, 108, 110, 114, 132, 138, 140, 150, 164, 174, 180, 182, 198, 212, 228, 230, 234, 252, 258, 270, 282, 294, 308, 318, 348, 350, 354, 374, 380, 390, 402, 420, 422, 434, 444, 450, 462, 468, 492, 500, 510, 522, 524, 542, 548, 558, 564

Offset: 1

Mauro Rigo, Sep 01 2018

nonn

If k cards are cut into m packets of s cards each, the r-th card of the deck can be identified with the pair (p,q), where p=(r-1)%s+1 ('%' is the Euclidean division) is the number of the packet and q=((r-1) mod s)+1 is the position in the packet; e.g., k=4, m=2, s=2: 1->(1,1), 2->(1,2), 3->(2,1), 4->(2,2).
The shuffle is performed by taking the card with the lowest q of each packet in sequence and stacking them on top of one another. It is a generalization of the riffle shuffle with two cards (cf. A217948) and it can be described as a permutation where (p,q) becomes (((q-1)*m+p-1)%s+1,(((q-1)*m+p-1) mod s)+1). For example, k=8, m=4, s=2, (1,1)->(1,1), (2,1)->(1,2), (3,1)->(2,1), (4,1)->(2,2), (1,2)->(3,1), (2,2)->(3,2), (3,2)->(4,1), (4,2)->(4,2); this permutation can be described using the index r as (3,2,5)(4,6,7).
The terms of the sequence are the values of k for which the permutation consists of only one orbit with k-2 passages (e.g., k=6, m=3, s=2 is (3,2,4,5)).
Sequence A217948 is a subsequence of this one as it takes into consideration only the m=2 case.
If the formula is true then Tiago Januario's conjecture on A217948 would be solved (see formula).

Mauro Rigo, Table of n, a(n) for n = 1..5000

Cf. A217948, A225184.

arr = []; for i = 4:1000 pdiv = 2:ceil(sqrt(i)); divisors = pdiv(rem(i, pdiv)==0); stop = 0; for j = divisors if ~stop ndiv = j; neldiv = i/j; a0 = 1; b0 = 2; a = a0; b = b0; a1 = 0; b1 = 0; operations = 0; while a1~=a0 || b1~=b0 b1 = mod((b-1)*ndiv+a-1, neldiv)+1; a1 = floor(((b-1)*ndiv+a-1)/neldiv)+1; operations = operations + 1; a = a1; b = b1; end if operations==i-2 arr = [arr, i]; stop = 1; end end end end

Apparently a(n) = A225184(n+1) + 1.

cp's OEIS Frontend

A225185 Primes p which do not have a primitive root that divides p+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions