A381150 a(0) = 1, a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + (sum of prior prime terms or whose negatives are prime) - (sum of prior composite terms or whose negatives are composite).

1, 2, 3, 8, 5, 7, 16, 9, -7, -30, -23, -39, -16, 23, 85, 62, -23, -131, -370, -239, -347, -802, -455, 347, 1496, 1149, -347, -2190, -1843, 347, 2884, 2537, -347, -3578, -3231, 347, 4272, 3925, -347, -4966, -4619, 347, 5660, 5313, -347, -6354, -6007, -11667, -5660

Offset: 0

Author

James C. McMahon, Feb 15 2025

Keywords

• sign

Examples

			For n=5, a(5) = 5 + (2 + 3 + 5) - 8 = 7.
For n=9, a(9) = -7 + (2 + 3 + 5 + 7 -7) - (8 + 16 + 9) = -7 + 10 - 33 = -30
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10003 (a(n) for n = 1..1000 from James C. McMahon)
• Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^10, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).
• Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^16, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).

Crossrefs

Cf. A000040, A002808, A101135, A119746, A131093, A381149.

Programs

Maple

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+(t->
     `if`(isprime(abs(t)), t, `if`(abs(t)>1, -t, 0)))(a(n)))
    end:
a:= proc(n) option remember; `if`(n<3, n+1, a(n-1)+b(n-1)) end:
seq(a(n), n=0..48);  # Alois P. Heinz, Feb 15 2025

Mathematica

Nest[Append[#,#[[-1]]+Total[Select[#,PrimeQ]]-Total[Select[#,CompositeQ]]]&,{1,2,3},46]