A123907 a(n) = T(p(n)) - p(T(n)) = Commutator[triangular numbers, primes] at n.

1, 1, 2, -1, 19, 18, 46, 39, 79, 178, 179, 306, 394, 375, 469, 662, 887, 872, 1127, 1265, 1248, 1553, 1703, 2018, 2600, 2780, 2763, 2987, 2958, 3134, 4587, 4849, 5380, 5373, 6518, 6503, 7100, 7725, 8089, 8750, 9431, 9452, 10859, 10892, 11260, 11219, 13275, 15485, 15947, 15908, 16358, 17257, 17222, 19189

Offset: 1

Author

Jonathan Vos Post, Oct 28 2006

Keywords

• easy
• sign
• less

Comments

Asymptotically p(T(n)) ~ (n^2 + n)*(log n) and T(p(n)) ~ (1/2)(n log n)^2, hence asymptotically a(n) ~ (1/2)(n log n)^2 - (n^2 + n)*(log n) = O((n^2)(log n)^2). a(4) = -1 should be the only negative value.

Examples

			a(1) = T(p(1)) - p(T(1)) = T(2) - p(1) = 3 - 2 = 1.
a(2) = T(p(2)) - p(T(2)) = T(2) - p(1) = 6 - 5 = 1.
a(3) = T(p(3)) - p(T(3)) = T(2) - p(1) = 15 - 13 = 1.
a(4) = T(p(4)) - p(T(4)) = T(2) - p(1) = 28 - 29 = -1.
a(5) = T(p(5)) - p(T(5)) = T(2) - p(1) = 66 - 47 = 19.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000217, A011756, A034953.

Programs

Magma

P:=NthPrime; B:=Binomial; [B(P(n)+1,2) - P(B(n+1,2)): n in [1..60]]; // G. C. Greubel, Aug 06 2019

Maple

A000040 := proc(n) ithprime(n) ; end; A000217 := proc(n) n*(n+1)/2 ; end; A123907 := proc(n) A000217(A000040(n))-A000040(A000217(n)) ; end ; for n from 1 to 80 do printf("%d,",A123907(n)) ; end; # R. J. Mathar, Jan 13 2007

Mathematica

With[{B=Binomial, P=Prime}, Table[B[P[n]+1, 2] -P[B[n+1, 2]], {n, 60}]] (* G. C. Greubel, Aug 06 2019 *)

PARI

vector(60, n, p=prime; b=binomial; b(p(n)+1,2) - p(b(n+1,2)) ) \\ G. C. Greubel, Aug 06 2019

Sage

p=nth_prime; b=binomial; [b(p(n)+1,2) - p(b(n+1,2)) for n in (1..60)] # G. C. Greubel, Aug 06 2019

Formula

a(n) = T(p(n)) - p(T(n)) where T(i) = i*(i+1)/2, p(i) = prime(i).

a(n) = A000217(A000040(n)) - A000040(A000217(n)).

a(n) = p(n)*(p(n)+1)/2 - p(n*(n+1)/2) where p(i) = prime(i).

a(n) = A034953(n) - A011756(n).

Extensions

More terms from R. J. Mathar, Jan 13 2007