A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

1, 1, 5, 1, 7, 11, 13, 17, 19, 23, 29, 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209

Offset

1,3

Comments

Values in row n of a(n) are those of row n of A286942 complement those of row n of A279864.

From Michael De Vlieger, May 18 2017: (Start)

Numbers t < p_n# such that gcd(t, p_n#) = 0, where p_n# = A002110(n).

Numbers in the reduced residue system of A002110(n).

A005867(n) = number of terms of a(n) in row n; local minimum of Euler's totient function.

A048862(n) = number of primes in row n of a(n).

A048863(n) = number of nonprimes in row n of a(n).

Since 1 is coprime to all n, it delimits the rows of a(n).

The prime A000040(n+1) is the second term in row n since it is the smallest prime coprime to A002110(n) by definition of primorial.

The smallest composite in row n is A001248(n+1) = A000040(n+1)^2.

The Kummer numbers A057588(n) = A002110(n) - 1 are the last terms of rows n, since (n - 1) is less than and coprime to all positive n. (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..6299 (rows 1 <= n <= 6 flattened).

Mathoverflow, Lower bound for Euler's totient for almost all integers. - Michael De Vlieger, May 18 2017

Eric Weisstein's World of Mathematics, Totative. - Michael De Vlieger, May 18 2017

Example

The triangle starts
1;
1, 5;
1, 7, 11, 13, 17, 19, 23, 29;
1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209;

Mathematica

Table[Function[P, Select[Range@ P, CoprimeQ[#, P] &]]@ Product[Prime@ i, {i, n}], {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)

Prog

(PARI) row(n) = my(P=factorback(primes(n))); select(x->(gcd(x, P) == 1), [1..P]); \\ Michel Marcus, Jun 02 2020

Crossrefs

Cf. A002110, A005867, A048862, A057588, A279864, A286941, A286942, A309497, A038110, A058250, A329815.

Cf. A335334 (row sums).

Sequence in context: A342921 A342417 A233091 * A332459 A051854 A006569

Adjacent sequences: A286938 A286939 A286940 * A286942 A286943 A286944

Keyword

nonn,tabf

Author

Jamie Morken and Michael De Vlieger, May 16 2017

Extensions

More terms from Michael De Vlieger, May 18 2017

Status

approved