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Talk:Coprimality
(Not so...) pairwise coprime infinite sequences
I removed the following text (which I previously added, wrongfully) since these sequences definitions DO NOT guarantee that they are pairwise coprime infinite sequences (although it seems that their terms are pairwise coprime with a high probability). They are NOT pairwise coprime infinite sequences!
The Euclid numbers (A006862) form yet another example, since the th term is the product of the first primes (Cf. primorial), plus 1. Again, some terms are composite. Another example involving multiplication is that of the Kummer numbers (A057588) for which the th term is the product of the first primes (Cf. primorial), minus 1.
— Daniel Forgues 19:14, 23 April 2012 (UTC)
Haha! Check
- Hisanori Mishima, PI Pn + 1 (n = 1 to 110)
for the prime factorization of the Euclid numbers (A006862), a(7) is not coprime with a(17)
a(7) = 510511 = 19 * 97 * 277
a(17) = 1922760350154212639071 = 277 * 3467 * 105229 * 19026377261
— Daniel Forgues 19:14, 23 April 2012 (UTC)
Check
- Hisanori Mishima, PI Pn - 1 (n = 1 to 110)
for the prime factorization of the Kummer numbers (A057588), a(35) is not coprime with a(44)
a(35) = 1492182350939279320058875736615841068547583863326864530409 = 673 * 448045542064369 * 4948626474214096948642213863754187837657
a(44) = 198962376391690981640415251545285153602734402721821058212203976095413910572269 = 673 * 65473937 * 566471804985844321 * 7970932666248247010325264452352519508898124959389
— Daniel Forgues 19:22, 23 April 2012 (UTC)
Check
- Hisanori Mishima, n! + 1 (n = 1 to 100)
for the prime factorization of the n! + 1 numbers (A038507), a(16) is not coprime with a(18)
a(16) = 20922789888001 = 17 * 61 * 137 * 139 * 1059511
a(18) = 6402373705728001 = 19 * 23 * 29 * 61 * 67 * 123610951
and check
- Hisanori Mishima, n! - 1 (n = 1 to 100)
for the prime factorization of the n! - 1 numbers (A033312), a(15) is not coprime with a(29)
a(15) = 1307674367999 = 17 * 31 * 31 * 53 * 1510259
a(29) = 8841761993739701954543615999999 = 31 * 59 * 311 * 26156201 * 594278556271609021
The same could be said for (Hisanori Mishima, WIFC (World Integer Factorization Center))
- A049650 Compositorial + 1 (a(10) = 17281= 11 * 1571; a(12) = 207361= 7 * 11 * 2693)
- A060880 Compositorial − 1 (a(14) = 2903039= 17 * 170767; a(22) = 115880067071999= 17 * 6816474533647)
and
- A?????? Compositorial + Next Composite (obviously, pairwise noncoprime with very high probability!)
- A?????? Compositorial − Next Composite (obviously, pairwise noncoprime with very high probability!)
and so on...
But I didn't succeed in finding noncoprime pairs for those two sequences
- A060881 Primorial + Next Prime (pairwise coprime with very high probability!)
- A060882 Primorial − Next Prime (pairwise coprime with very high probability!)
by looking at
- Hisanori Mishima, PI Pn + NextPrime (n = 1 to 100)
- Hisanori Mishima, PI Pn - NextPrime (n = 1 to 100)
although I suspect that the strong law of small numbers might apply here...
— Daniel Forgues 19:40, 23 April 2012 (UTC)