A381150 Contribution to the OEIS

   

Recently proposed for the On-Line Encyclopedia of Integer Sequences (OEIS) is the sequence:

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).

   The sequence starts: 2, 3, 8, 5, 7, 16, 9, -7, -30, -23, -39, -16, 23, 85, 62, -23, -131... and continues to cycle through negative and positive integers. Although the sequence is not periodic, it does repeat terms. For example, the  term 347 (or its negative) occurs nine times in the first 48 terms. The 418th (or its negative) occurs 71 times in a span of 211 terms.

    My colleague, Michael De Vlieger, produced the graphic showing the sequence's progression shown above.

    If approved, the sequence will become A381150. Currently, it is still in draft form: https://oeis.org/A381150.

    Update: The sequence was published 2/25/2025.

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).

0

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

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

0,2

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).

EXAMPLE

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

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]

CROSSREFS

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

Sequence in context: A284047 A265343 A347193 * A268392 A154760 A231817

Adjacent sequences: A381147 A381148 A381149 * A381152 A381153 A381154

KEYWORD

sign,new

AUTHOR

James C. McMahon, Feb 15 2025

STATUS

approved