A369252 Arithmetic derivative applied to the numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

27, 39, 51, 55, 75, 71, 87, 75, 91, 111, 103, 123, 95, 119, 147, 131, 119, 151, 183, 151, 135, 195, 167, 155, 231, 147, 199, 191, 187, 255, 167, 267, 211, 291, 195, 215, 247, 191, 263, 215, 327, 251, 247, 363, 203, 375, 311, 271, 255, 239, 411, 231, 311, 343, 299, 231, 435, 359, 331, 447, 311, 263, 391, 483, 263

Offset: 1

Author

Antti Karttunen, Jan 22 2024

Keywords

• nonn

Comments

The table showing the possible modulo 3 combinations for p, q, r and the sum ((p*q) + (p*r) + (q*r)):

| p | q | r | sum ((p*q) + (p*r) + (q*r)) (mod 3)

--+------+------+------+----------------------------------------

| 0 | 0 | 0 | 0, p=q=r=3, sum is 27.

--+------+------+------+----------------------------------------

| 0 | 0 | +/-1 | 0, p=q=3, r > 3.

--+------+------+------+----------------------------------------

| 0 | +1 | +1 | +1

--+------+------+------+----------------------------------------

| 0 | -1 | -1 | +1

--+------+------+------+----------------------------------------

| 0 | -1 | +1 | -1

--+------+------+------+----------------------------------------

| 0 | +1 | -1 | -1

--+------+------+------+----------------------------------------

| +1 | +1 | +1 | 0

--+------+------+------+----------------------------------------

| -1 | -1 | -1 | 0

--+------+------+------+----------------------------------------

| -1 | +1 | +1 | -1, regardless of the order, thus x3.

--+------+------+------+----------------------------------------

| +1 | -1 | -1 | -1, regardless of the order, thus x3.

--+------+------+------+----------------------------------------

Notably a(n) is a multiple of 3 only when A046316(n) is either a multiple of 9, or all primes p, q and r are either == +1 (mod 3) or all are == -1 (mod 3), and the case a(n) == +1 (mod 3) is only possible when A046316(n) is a multiple of 3, but not of 9, and furthermore, it is required that r == q (mod 3). See how these combinations affects sequences like A369241, A369245, A369450, A369451, A369452.

For n=1..9 the number of terms of the form 3k, 3k+1 and 3k+2 in range [1..10^n-1] are:

6, 2, 1,

39, 22, 38,

291, 209, 499,

2527, 1884, 5588,

23527, 17020, 59452,

227297, 156240, 616462,

2232681, 1453030, 6314288,

22119496, 13661893, 64218610,

220098425, 129624002, 650277572.

It seems that 3k+2 terms are slowly gaining at the expense of 3k+1 terms when n grows, while the density of the multiples of 3 might converge towards a limit.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A369251 (same sequence sorted into ascending order, with duplicates removed).

Cf. A369464 (numbers that do not occur in this sequence).

Cf. A003415, A046316, A369054, A369058, A369248.

Cf. also the trisections of A369055: A369460, A369461, A369462 and their partial sums A369450, A369451, A369452, also A369241, A369245.

Only terms of A004767 occur here.

Formula

a(n) = A003415(A046316(n)).