[seqfan] Re: list of all non-prime coprimes of the full set of primorial numbers

[Michael De Vlieger]

Mr. Morken,

 

Irregular triangle T(n,k) of non-prime totatives k of primorials p_n# =
A002110(n).

 

Let primorial p_n# = A002110(n) and non-prime 1 <= k <= p_n# such that
gcd(k, p_n#) = 1.

 

Mathematica:

Table[Function[P, Select[Range@ P, And[! PrimeQ@ #, CoprimeQ[#, P]] &]]@
Product[Prime@ i, {i, n}], {n, 5}] // Flatten

 

Triangle begins:

1, 

1, 

1, 

1, 121, 143, 169, 187, 209, 

1, 169, 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, 1121, 1139, 1147, 1157, 1159, 1189,
1207, 1219, 1241, 1247, 1261, 1271, 1273, 1313, 1333, 1339, 1343, 1349,
1357, 1363, 1369, 1387, 1391, 1403, 1411, 1417, 1457, 1469, 1501, 1513,
1517, 1537, 1541, 1577, 1591, 1633, 1643, 1649, 1651, 1679, 1681, 1691,
1703, 1711, 1717, 1739, 1751, 1763, 1769, 1781, 1807, 1817, 1819, 1829,
1843, 1849, 1853, 1891, 1909, 1919, 1921, 1927, 1937, 1943, 1957, 1961,
1963, 2021, 2033, 2041, 2047, 2059, 2071, 2077, 2117, 2119, 2147, 2159,
2171, 2173, 2183, 2197, 2201, 2209, 2227, 2231, 2249, 2257, 2263, 2279, 2291

.

 

Some thoughts:

At first I considered not having 1 included in a(n) because 1 is coprime to
all numbers, but it does serve to hold the place of primorials p_n# with n <
4 and generally delimit the data. 

All the terms outside of 1 are composite (by definition of the sequence),
the product of at least two not necessarily distinct primes q > p_n (q that
are, by definition of "primorial", coprime to p_n#). 

The terms k > 1 of each row are products of the primes q < sqrt(p_n#) that
involve at least 2 such factors q. 

The first composite k in the sequence is 11^2; this occurs when the
primorial function overtakes the square function on primes, i.e., p_n# >
q_1^2 (i.e., sqrt(p_n#) > q_1), where q_1 = nextprime(p_n). 

The first k of p_n# repeated in row (n + 1) is 13^2, for n = 4, since
sqrt(p_4#) is greater than not only q = 11, but 13 as well, which is
nextprime(p_(n+1)). 

For n >= 4 there must be a number of squares q^2, and for n >= 5 there must
be at least one repeated term that is also in row (n - 1). As n increases
from 5, the largest squares q^2 will persist increasingly longer, as the
range p_n < q < sqrt(p_n#) widens. 

 

Note : the sequence of non-prime numbers 1 <= k <= n coprime to n does not
appear to be in OEIS, nor is composite numbers 1 <= k <= n coprime to n.
These would be (in Mathematica):

 

Table[Select[Range@ n, And[! PrimeQ@ #, CoprimeQ[#, n]] &], {n,  12}] //
Flatten

 

and

 

Table[Select[Range@ n, And[CompositeQ@ #, CoprimeQ[#, n]] &], {n,  12}] //
Flatten

 

Respectively (maybe need only one of these, probably preferably the former,
since it does include 1 and thus populates all the rows n.)

The former would be the complement of A112484: Array where n-th row contains
the primes < n and coprime to n, in Mathematica:

 

Table[Select[Prime@ Range@ PrimePi@ n, CoprimeQ[#, n] &], {n, 12}] //
Flatten

 

I have just proposed A285788: Irregular triangle T(n,k): non-prime 1 <= k <=
n such that n and k are coprime.

Thus a(n) = A285788(A002110(n)).

 

Best regards,

 

Michael De Vlieger