∩ prime sums ± i

[allie dicelli]

hellos!

below is a small commonlisp generator for three index sequences.


let (s_n) be the sum of the first (n) primes.

* **a**: indices (n) where (s_n + n) is prime
* **b**: indices (n) where (s_n - n) is prime
* **c**: indices where both conditions hold (the intersection)


(defun %generate-sequences (&key (predicate #'cl-primality::primep) (upto 1000))
  "prime indices where the prime sum at index plus-or-minus index is also prime"
  (flet ((make-seq () (make-array 0 :element-type 'integer :fill-pointer 0 :adjustable t)))
    (loop with primes = (loop with primes-array = (make-array 1 :element-type 'integer
                                                             :initial-element 2
                                                             :fill-pointer 1
                                                             :adjustable t)
                              while (< (length primes-array) upto)
                              for i from 3 by 2
                              when (funcall predicate i)
                              do (vector-push-extend i primes-array)
                              finally (return primes-array))
          with seq-a = (make-seq)
          with seq-b = (make-seq)
          with seq-c = (make-seq)
          for prime across primes
          for index from 1
          for sum = prime then (+ sum prime)
          for candidate-a = (funcall predicate (+ sum index))
          for candidate-b = (funcall predicate (- sum index))
          do (when (and candidate-a candidate-b)
               (vector-push-extend index seq-c))
             (when candidate-a
               (vector-push-extend index seq-a))
             (when candidate-b
               (vector-push-extend index seq-b))
          finally (return (list seq-c seq-a seq-b)))))

i suspect **a** and **b** already appear in oeis under other descriptions, but the intersection **c**, as well as the complements and gap structure in (a \cup b), may be of independent interest.

comments welcome; pointers to existing entries appreciated.

best,
zoë trout
librecell

[allie dicelli]

addendum: these “sum-prime twins” (sequence c) are the intersection of A046155 and A046156.

zoë trout

librecell

On Jan 1, 2026, at 11:52, allie dicelli <allied...@gmail.com> wrote:

hellos!

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[Hugo Pfoertner]

None of the three index sequences **a**, **b**, **c** currently exist in the database. The reference to A046155 and A046156 must be an error. Neither appears to be related to the described task. I suggest you submit the sequences **a**, ...