A285784 Possible nonprime residues of k > p# modulo p# for some primorial p# in A002110.

1, 121, 143, 169, 187, 209, 221, 247, 289, 299, 323, 361, 377, 391, 403, 437, 481, 493, 527, 529, 533, 551, 559, 589, 611, 629, 667, 689, 697, 703, 713, 731, 767, 779, 793, 799, 817, 841, 851, 871, 893, 899, 901, 923, 943, 949, 961, 989, 1003, 1007, 1027, 1037, 1073, 1079, 1081

Offset: 1

Author

Jamie Morken, Apr 26 2017

Keywords

• nonn

Comments

Original name: Union of nonprimes p_n# < k < p_(n+1)# and gcd(k, p_n#) = 1, with p_n# = A002110(n).

From Michael De Vlieger, May 24 2017: (Start)

Let p_n# = A002110(n). This sequence includes nonprime p_n# < k < p_(n+1)# but does not repeat terms that have already appeared in the sequence (mainly 1 for p_n# with n > 1).

If regarded as a number triangle T(n,k), row length <= A048863(n). (End)

Relevant for sieving primes with a wheel of circumference p#: For the 2*3*5 wheel, the only relevant nonprime residue is 1, while for a 2*3*5*7 wheel, there are 5 more nonprime residues {121, 143, 169, 187, 209}. - M. F. Hasler, Mar 25 2019

Examples

			Primorial(2) = 2*3 = 6 has two totatives (1 and 5), primorial(3) = 2*3*5 = 30 has eight totatives (1,7,11,13,17,19,23,29), etc.
Taking the union of these lists and removing the primes gives the sequence.
		

Links

• Jamie Morken, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002110, A048863.

Programs

Mathematica

MapIndexed[Select[Range @@ #1, Function[k, And[If[First@ #2 == 1, ! PrimeQ@ k, CompositeQ@ k > 1], CoprimeQ[Last@ #1, k]]]] &, Partition[FoldList[#1 #2 &, 1, Prime@ Range@ 5], 2, 1]] // Flatten (* Michael De Vlieger, May 24 2017 *)

PARI

select( n->!isprime(n), setunion((S(p,M)=Set(primes([1,p*M])%M))(11,210), S(13,2310))) \\ M. F. Hasler, Mar 25 2019

Extensions

Edited by N. J. A. Sloane, May 01 2017

Incorrect definition replaced and more terms added by M. F. Hasler, Mar 25 2019