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Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).
8

%I #46 Oct 30 2024 20:47:09

%S 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,

%T 61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,

%U 149,151,157,163,167,169,173,179,181,187,191,193,197,199,209

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

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

%C From _Michael De Vlieger_, May 18 2017: (Start)

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

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

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

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

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

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

%C 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.

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

%C 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)

%H Michael De Vlieger, <a href="/A286941/b286941.txt">Table of n, a(n) for n = 1..6299</a> (rows 1 <= n <= 6 flattened).

%H Mathoverflow, <a href="https://mathoverflow.net/questions/140104/lower-bound-for-eulers-totient-for-almost-all-integers">Lower bound for Euler's totient for almost all integers</a>. - _Michael De Vlieger_, May 18 2017

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Totative.html">Totative</a>. - _Michael De Vlieger_, May 18 2017

%e The triangle starts

%e 1;

%e 1, 5;

%e 1, 7, 11, 13, 17, 19, 23, 29;

%e 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;

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

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

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

%Y Cf. A285784 (nonprimes that appear), A335334 (row sums).

%K nonn,tabf

%O 1,3

%A _Jamie Morken_ and _Michael De Vlieger_, May 16 2017

%E More terms from _Michael De Vlieger_, May 18 2017