A354369 Successive pairs of terms (a, b) such that (a + b) is a prime number and at least one of a and b is a prime number. This is the lexicographically earliest infinite sequence of distinct terms > 0 with this property.

1, 2, 3, 4, 5, 6, 7, 10, 8, 11, 12, 17, 13, 16, 14, 23, 18, 19, 20, 41, 22, 31, 24, 29, 26, 47, 28, 43, 30, 37, 32, 71, 34, 67, 36, 53, 38, 59, 40, 61, 42, 89, 44, 83, 46, 103, 48, 79, 50, 101, 52, 97, 54, 73, 56, 107, 58, 109, 60, 113, 62, 131, 64, 127, 66, 157, 68, 173, 70, 163, 72, 139, 74, 137

Offset

1,2

Comments

The terms 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, ... will never appear in the sequence; they form A071904, the "Odd composite numbers".

Links

Table of n, a(n) for n=1..74.

Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..10^4, showing records in red, local minima in blue, composite powers of 2 in magenta, and fixed points in gold.

Example

The earliest pairs with their prime sum: (1, 2) = 3, (3, 4) = 7, (5, 6) = 11, (7, 10) = 17, (8, 11) = 19, (12, 17) = 29, (13, 16) = 29, (14, 23) = 37, etc.

Mathematica

nn = 120; c[_] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; If[EvenQ[i], While[Nand[c[k] == 0, PrimeQ[# + k]] &[a[i - 1]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[Or[c[u] > 0, And[OddQ[u], CompositeQ[u]]], u++]], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 24 2022 *)

Prog

(Python)
from sympy import isprime
from itertools import islice
def agen(): # generator of terms
    i, j, v, aset = 1, 2, 3, set()
    while True:
        aset.update((i, j)); yield from (i, j)
        i = v
        while i in aset or (i%2 == 1 and not isprime(i)):
            i += 1
        j, p = v, isprime(i)
        while j==i or j in aset or not (isprime(i+j) and (p or isprime(j))):
            j += 1
        while v in aset: v += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Jun 24 2022

Crossrefs

Cf. A354367, A354368, A354370 (same idea), A071904.

Sequence in context: A072794 A181820 A371249 * A199426 A119257 A266645

Adjacent sequences: A354366 A354367 A354368 * A354370 A354371 A354372

Keyword

nonn

Author

Eric Angelini and Carole Dubois, May 24 2022

Status

approved