This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A123907 #17 Sep 17 2024 08:45:24
%S A123907 1,1,2,-1,19,18,46,39,79,178,179,306,394,375,469,662,887,872,1127,
%T A123907 1265,1248,1553,1703,2018,2600,2780,2763,2987,2958,3134,4587,4849,
%U A123907 5380,5373,6518,6503,7100,7725,8089,8750,9431,9452,10859,10892,11260,11219,13275,15485,15947,15908,16358,17257,17222,19189
%N A123907 a(n) = T(p(n)) - p(T(n)) = Commutator[triangular numbers, primes] at n.
%C A123907 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.
%H A123907 G. C. Greubel, <a href="/A123907/b123907.txt">Table of n, a(n) for n = 1..1000</a>
%F A123907 a(n) = T(p(n)) - p(T(n)) where T(i) = i*(i+1)/2, p(i) = prime(i).
%F A123907 a(n) = A000217(A000040(n)) - A000040(A000217(n)).
%F A123907 a(n) = p(n)*(p(n)+1)/2 - p(n*(n+1)/2) where p(i) = prime(i).
%F A123907 a(n) = A034953(n) - A011756(n).
%e A123907 a(1) = T(p(1)) - p(T(1)) = T(2) - p(1) = 3 - 2 = 1.
%e A123907 a(2) = T(p(2)) - p(T(2)) = T(2) - p(1) = 6 - 5 = 1.
%e A123907 a(3) = T(p(3)) - p(T(3)) = T(2) - p(1) = 15 - 13 = 1.
%e A123907 a(4) = T(p(4)) - p(T(4)) = T(2) - p(1) = 28 - 29 = -1.
%e A123907 a(5) = T(p(5)) - p(T(5)) = T(2) - p(1) = 66 - 47 = 19.
%p A123907 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
%t A123907 With[{B=Binomial, P=Prime}, Table[B[P[n]+1, 2] -P[B[n+1, 2]], {n, 60}]] (* _G. C. Greubel_, Aug 06 2019 *)
%o A123907 (PARI) vector(60, n, p=prime; b=binomial; b(p(n)+1,2) - p(b(n+1,2)) ) \\ _G. C. Greubel_, Aug 06 2019
%o A123907 (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
%o A123907 (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
%Y A123907 Cf. A000040, A000217, A011756, A034953.
%K A123907 easy,sign,less
%O A123907 1,3
%A A123907 _Jonathan Vos Post_, Oct 28 2006
%E A123907 More terms from _R. J. Mathar_, Jan 13 2007