A381149 Contribution to the OEIS

 A381149

a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + sum of prior prime terms.

+40
0

2, 3, 8, 13, 31, 80, 129, 178, 227, 503, 1282, 2061, 2840, 3619, 4398, 5177, 5956, 6735, 7514, 8293, 17365, 26437, 61946, 97455, 132964, 168473, 203982, 239491, 275000, 310509, 346018, 381527, 798563, 1215599, 1632635, 2049671, 2466707, 5350450, 8234193, 11117936

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

OFFSET

1,1

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

EXAMPLE

For n=5, a(5) = a(4) + sum of prior primes = 13 + (2 + 3 + 13) = 31, so that 13 is counted twice.

MATHEMATICA

Nest[Append[#, #[[-1]]+Total[Select[#, PrimeQ]]]&, {2, 3}, 38]

PROG

(Python)

from sympy import isprime

from itertools import count, islice

def agen(): # generator of terms

yield from [2, 3]

primesum, an = 5, 3

while True:

an += primesum

if isprime(an): primesum += an

yield an

print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 19 2025

CROSSREFS

Cf. A101135, A119746, A131093, A381150.

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Feb 15 2025

STATUS

approved

This sequence is similar to my other sequence submitted at the same time, A381150. However, I think A381150 is more interesting because of the terms that repeat and also how it cycles between positive and negative terms. Link to prior post: https://jamesmacmath.blogspot.com/2025/02/a-sequence-with-competing-sums-of-prior.html

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

Using Excel Solver to Find the Solution to a Geometry Puzzle

The Monday Puzzle in the Guardian by Alex Bellos had a good problem to illustrate the use of Excel Solver. The link to the article is: https://www.theguardian.com/science/2025/feb/17/did-you-solve-it-the-simple-geometry-problem-that-fools-almost-everyone.

Restated, the problem is to divide an 11 x 11 square into five rectangles so that the ten side lengths are the whole numbers from 1 to 10? 

Below is the 11 x 11 square with five rectangles and the ten sides labeled A through J. 

To use Excel Solver to find a solution, ten cells of the spreadsheet will be designated as the decision variables. In the attached spreadsheet, these are simply the cells A1 through J1. Solver will change their values until a solution is found. 

In rows 3 through 8 are 6 constraints:

A+D=11

C+F=11

E+H=11

G+B=11

H+J+D=11

B+I+F=11

Also in Solver, additional constraints are entered:

All values, A through J, are => 1 and <= 10.

All values, A through J, are integers.

All values, A through J, are different.

With these constraints entered, click the Solve button, and one of the solutions is displayed:

A=6, B=3, C=10, D=5, E=9, F=1, G=8, H=2, I=7, J=4

eleven by eleven square into 5 rectangles.xlsx

More about Solver: https://jamesmacmath.blogspot.com/2021/12/optimization-solutions-using-microsoft.html