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

%I A286941 #47 Feb 16 2025 08:33:45
%S A286941 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 A286941 61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,
%U A286941 149,151,157,163,167,169,173,179,181,187,191,193,197,199,209
%N A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).
%C A286941 Values in row n of a(n) are those of row n of A286942 complement those of row n of A279864.
%C A286941 From _Michael De Vlieger_, May 18 2017: (Start)
%C A286941 Numbers t < p_n# such that gcd(t, p_n#) = 0, where p_n# = A002110(n).
%C A286941 Numbers in the reduced residue system of A002110(n).
%C A286941 A005867(n) = number of terms of a(n) in row n; local minimum of Euler's totient function.
%C A286941 A048862(n) = number of primes in row n of a(n).
%C A286941 A048863(n) = number of nonprimes in row n of a(n).
%C A286941 Since 1 is coprime to all n, it delimits the rows of a(n).
%C A286941 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 A286941 The smallest composite in row n is A001248(n+1) = A000040(n+1)^2.
%C A286941 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 A286941 Michael De Vlieger, <a href="/A286941/b286941.txt">Table of n, a(n) for n = 1..6299</a> (rows 1 <= n <= 6 flattened).
%H A286941 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 A286941 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Totative.html">Totative</a>. - _Michael De Vlieger_, May 18 2017
%e A286941 The triangle starts
%e A286941 1;
%e A286941 1, 5;
%e A286941 1, 7, 11, 13, 17, 19, 23, 29;
%e A286941 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 A286941 Table[Function[P, Select[Range@ P, CoprimeQ[#, P] &]]@ Product[Prime@ i, {i, n}], {n, 4}] // Flatten (* _Michael De Vlieger_, May 18 2017 *)
%o A286941 (PARI) row(n) = my(P=factorback(primes(n))); select(x->(gcd(x, P) == 1), [1..P]); \\ _Michel Marcus_, Jun 02 2020
%Y A286941 Cf. A002110, A005867, A048862, A057588, A279864, A286941, A286942, A309497, A038110, A058250, A329815.
%Y A286941 Cf. A285784 (nonprimes that appear), A335334 (row sums).
%K A286941 nonn,tabf
%O A286941 1,3
%A A286941 _Jamie Morken_ and _Michael De Vlieger_, May 16 2017
%E A286941 More terms from _Michael De Vlieger_, May 18 2017