A267540 Primes p such that p (mod 3) = p (mod 5).
2, 17, 31, 47, 61, 107, 137, 151, 167, 181, 197, 211, 227, 241, 257, 271, 317, 331, 347, 421, 467, 541, 557, 571, 587, 601, 617, 631, 647, 661, 677, 691, 751, 797, 811, 827, 857, 887, 947, 977, 991, 1021, 1051, 1097, 1171, 1187, 1201, 1217, 1231, 1277, 1291
Offset: 1
Programs
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Magma
[p: p in PrimesUpTo(2000) | p mod 3 eq p mod 5]; // Vincenzo Librandi, Jan 17 2016
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Maple
select(isprime, [seq(seq(15*i+j, j= 1..2), i=0..10000)]); # Robert Israel, Jan 17 2016
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Mathematica
Select[ Prime[ Range[10000]], (Mod[#,3] == Mod[#,5]) &] (* Or *) Select[ Prime[ Range[10000]], 0 < Mod[#,15] < 3 &]
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PARI
lista(nn) = forprime(p=2, nn, if(p%3 == p%5, print1(p, ", "))); \\ Altug Alkan, Jan 17 2016
Formula
a(n) = 1/2*((-1)^n*(3*(-1)^n*(10n+81)-1)) with (1
G.f.: (x*(-14x^6-32x^5+16x^4+30x^3-x+14)+17)/((x-1)^2*(x+1)) generates a(2)...a(16), (0<=x<15).
G.f.: (x*(x*(30x*(-2x^4-x^3+x+2)-301)+14)+317)/((x-1)^2*(x+1)) generates a(17)...a(32), (0<=x<16).
Extensions
More terms from Vincenzo Librandi, Jan 17 2016
A267781 Expansion of (x*(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x-1)^2*(x+1)).
17, 31, 47, 61, 107, 137, 151, 167, 181, 197, 211, 227, 241, 257, 271, 287, 301, 317, 331, 347, 361, 377, 391, 407, 421, 437, 451, 467, 481, 497, 511, 527, 541, 557, 571, 587, 601, 617, 631, 647, 661, 677, 691, 707, 721, 737, 751, 767, 781, 797, 811, 827, 841, 857, 871, 887, 901, 917, 931, 947, 961, 977, 991, 1007
Offset: 1
Comments
The terms which are primes are the same (p mod 3 = p mod 5) as in A267540 (starting from a(2)=17, their correspondence is verified up to 150000047).
Primes here frequently also have regular intervals and occur mostly in short blocks (consisting of 2-4 primes) rather than singletons, but some blocks can be much longer (e.g., a(1)..a(15) and a(33)..a(43)).
Programs
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Mathematica
CoefficientList[ Series[(x*(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x - 1)^2*(x + 1)), {x, 0, 63}], x] (* Michael De Vlieger, Jan 21 2016 *)(* Or *) Flatten @Prepend[ Table[(30*n - (-1)^n + 123)/2, {n, 5, 1000}],{17,31,47,61,107}](* Efficient. Mikk Heidemaa, Jan 21 2016 *) -
PARI
Vec((x*(-14*x^6-32*x^5+16*x^4+30*x^3-x+14)+17)/((x-1)^2*(x+1)) + O(x^80)) \\ Michel Marcus, Jan 20 2016
Formula
a(n) = (30*n - (-1)^n + 123)/2 for n > 4. - Colin Barker, Jan 21 2016
Comments