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The Programme inserts symbolics in the Sequence Aplet to reproduce the sequence

http://oeis.org/A025581

in U1 & its partial sums

http://oeis.org/A121924

in U2. The sub-programme SEQSET produces an agreeable environment in the Sequence Aplet for the necessary symbolics.

Note: I considered programming this for the Prime but was deterred by the upper limit of 40,000 for indices in the Prime Sequence App - the 38G produces exactly correct results up to 2,000,000 & beyond (I don't know the upper limit for exactly correct) & accordingly decided not to bother.

RUN SEQSET:
RECURSE(U,COMB(1+ROUND(√(2*N),0),2)-N,0,1)►U1(N):
CHECK 1:
RECURSE(U,COMB(U3(N),3)+(N-COMB(U3(N),2))*(U3(N)^2+3*U3(N)-2*N-2)/4,0,1)►U2(N):
CHECK 2:
RECURSE(U,ROUND(√(2*N+.25),0),0,0)►U3(N):

SEQSET

SELECT Sequence:
UNCHECK 0:
0►NumFont:
0►Simult:
2►Angle:
1►InvCross:
1►NumStep:
1►Format:
1►NumCol:
1►NumStart:
6►NumRow:
RECURSE(U,0,0,0)►U1(N):
Ans►U2(N):
Ans►U3(N):
Ans►U4(N):
Ans►U5(N):
Ans►U6(N):
Ans►U7(N):
Ans►U8(N):
Ans►U9(N):
Ans►U0(N):