A279864 -id:A279864 - cp's OEIS frontend

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

Jamie Morken and Michael De Vlieger, May 16 2017

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)

			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;

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

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

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

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

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

More terms from Michael De Vlieger, May 18 2017

A286942 Irregular triangle read by rows: numbers 1 <= k <= (A002110(n) - 1) where gcd(k, A002110(n - 1)) = 1.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173

Offset: 1

Jamie Morken and Michael De Vlieger, May 16 2017

From Michael De Vlieger, May 18 2017: (Start)
Row n of a(n) is the list of numbers 1 <= k <= A002110(n) that are coprime to A002110(n-1).
A286941(n) and A279864(n) are subsets of a(n) such that the terms of the rows of each sequence combined and sorted comprise all the terms of a(n).
Row lengths = A005867(n) + A005867(n-1): {2, 3, 10, 56, 528, 6240, 97920, ...}.
1 is coprime to all n thus delimits the rows of a(n).
The smallest prime q in row n of a(n) is gpf(primorial(n)) = A006530(A002110(n)) = prime(n) by definition of primorial.
The smallest composite x in row n of a(n) is q^2 = A001248(n).
The Kummer number A057588(n) = A002110(n) - 1 is the largest term in row n of a(n). (End)

			The triangle starts:
1, 2;
1, 3, 5;
1, 5, 7, 11, 13, 17, 19, 23, 25, 29
Example1:
To find row n of the irregular triangle A286942, take a running sum for each value in the irregular triangle row n-1 of A286941 with A002110(n-1) b-1 times, where b is the largest prime factor in A002110(n).
For example to find row 3 of A286942: Take a running sum for both 1 and 5 in row n-1 of A286941 with A002110(3-1)=6, 5-1=4 times, where b is the largest prime factor 5 in A002110(3).
Result:
1 5
7 11
13 17
19 23
25 29
Equal to row 3 of A286942: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
Example2:
To find row n of the irregular triangle A279864, multiply each value in row n-1 of A286941 with the largest prime factor b in A002110(n).
Example for n=3: b=5.
1*5=5
5*5=25
Example3:
To find row n of the irregular triangle A286941, remove the values that are in row n of the irregular triangle A279864 from the values that are in row n of the irregular triangle A286942.
For n=3.
A286942 row n = 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
A279864 row n = 5, 25.
Removing values 5, 25 from the values in A286942 row n gives row n of A286941: 1, 7, 11, 13, 17, 19, 23, 29.

Michael De Vlieger, Table of n, a(n) for n = 1..6839 (rows 1 <= n <= 6).
Eric Weisstein's World of Mathematics, Relatively Prime - _Michael De Vlieger_, May 18 2017
Seqfan, Formula for the sequence.

Cf. A002110, A005867, A279864, A286941, A286942.

Table[Select[Range@ #2, Function[k, CoprimeQ[k, #1]]] & @@ Map[Times @@ # &, {Most@ #, #}] &@ Prime@ Range@ n, {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)

a(n) = union(A286941(n), A279864(n)) where n consists of all terms in row n of each sequence. - Michael De Vlieger, May 18 2017

More terms from Michael De Vlieger, May 18 2017

A335284 Numbers k > 1 such that, if p is the least prime dividing k, k is less than or equal to the product of all prime numbers up to (and including) p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203

Offset: 1

Javier Múgica, May 30 2020

nonn

The sequence A279864 contains the same terms as this one in different order, namely, sorted according to their least prime factor.
A number k > 1 belongs to this sequence if k <= A002110(A055396(k)) = A034386(A020639(k)). This condition approaches log(k) <= p as k -> infinity, p being the least prime factor of k.
All prime numbers belong to this sequence. Squares of prime numbers are included starting at 5^2; cubes are included starting at 11^3, and so on. That is, for all m there exists a p(m) such that all m-th powers of prime numbers from p(m)^m onwards belong to the sequence.
For large N the number of integers 1 < k <= N which belong to this sequence is ~ e^(-gamma)*N/log(log(N)), where gamma is Euler's constant: A001620.
Let p = p_r denote the r-th prime number and P_r = A034386(p) (the product of primes <= p). This sequence contains 1*2*4*...*(p_(r-1)-1) = A005867(r-1) elements whose least prime factor is p. These are distributed symmetrically about P_r/2, the first ones being p and, for p >= 5, p^2, and the last one being P_r-p.

			The least prime factor of 77 is 7, and 77 < 2*3*5*7 = 210, therefore 77 belongs to the sequence.

Javier Múgica, Table of n, a(n) for n = 1..19720 (terms <= 100000)

Cf. A034386, A020639.
A279864 contains the same terms as this sequence in a different order.
Contains A308966. Both sequences agree in their first 38 terms.

isok(k) = if (k>1, my(p=vecmin(factor(k)[,1])); k <= prod(j=1, primepi(p), prime(j))); \\ Michel Marcus, May 31 2020

Asymptotic expression for a(n): e^(gamma)*n*(log(log(n))+O(1)), where gamma is Euler's constant: A001620.

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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).

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A286942 Irregular triangle read by rows: numbers 1 <= k <= (A002110(n) - 1) where gcd(k, A002110(n - 1)) = 1.

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A335284 Numbers k > 1 such that, if p is the least prime dividing k, k is less than or equal to the product of all prime numbers up to (and including) p.

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