A354370 -id:A354370 - cp's OEIS frontend

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

Eric Angelini and Carole Dubois, May 24 2022

nonn

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".

			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.

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.

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

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 *)

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

A354367 Successive pairs of terms (a, b) such that (a + b) is a square 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.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 6, 19, 8, 17, 10, 71, 11, 14, 12, 13, 15, 181, 18, 31, 20, 29, 21, 43, 22, 59, 23, 26, 24, 97, 27, 37, 28, 53, 30, 139, 32, 89, 33, 67, 34, 47, 35, 109, 38, 83, 39, 61, 40, 41, 42, 79, 44, 317, 45, 151, 46, 179, 48, 73, 50, 239, 51, 349, 52, 173, 54, 307, 55, 269, 56, 113, 57, 199, 58

Offset: 1

Eric Angelini and Carole Dubois, May 24 2022

nonn

This is not a permutation of the integers > 0 as no square > 4 will appear. Two prime terms can form a pair (2 and 7 for instance) but at least one term must be prime [the pair (1, 3) is ok].

			The earliest pairs with their square sum: (1, 3) = 4, (2, 7) = 9, (4, 5) = 9, (6, 19) = 25, (8, 17) = 25, (10, 71) = 81, (11, 14) = 25, (12, 13) = 25, etc.

Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..10^4, showing records in red, numbers entering late in blue, highlighting primes in green, fixed points in gold, and composite prime powers in magenta.

Cf. A354368, A354369, A354370 (same idea).

nn = 10^4; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; If[EvenQ[i], While[Nand[c[k] == 0, AnyTrue[{#, k}, PrimeQ], IntegerQ@ Sqrt[# + k]] &[a[i - 1]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[Or[c[u] > 0, And[IntegerQ@ Sqrt@ u, u > 4]], u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 24 2022 *)

A354368 Successive pairs of terms (a, b) such that (a + b) is a square 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.

Original entry on oeis.org

2, 7, 3, 13, 5, 11, 17, 19, 23, 41, 29, 71, 31, 113, 37, 107, 43, 101, 47, 53, 59, 137, 61, 83, 67, 257, 73, 251, 79, 821, 89, 167, 97, 227, 103, 797, 109, 467, 127, 197, 131, 193, 139, 761, 149, 751, 151, 173, 157, 419, 163, 1601, 179, 397, 181, 719, 191, 293, 199, 701, 211, 1553, 223, 353, 229, 347

Offset: 1

Eric Angelini and Carole Dubois, May 24 2022

nonn

This sequence is a permutation of the prime numbers.

			The earliest pairs with their square sum: (2, 7) = 9, (3, 13) = 16, (5, 11) = 16, (17, 19) = 36, (23, 41) = 64, (29, 71) = 100, (31, 113) = 144, (37, 107) = 144, etc.

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

Cf. A354367, A354369, A354370 (same idea).

Clear[c, a]; nn = 66; c[] = 0; a[1] = 2; c[2] = 1; u = 3; Do[k = u; If[EvenQ@ i, While[Nand[c[k] == 0, IntegerQ@ Sqrt[# + k]] &[a[i - 1]], k = NextPrime[k]]]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u = NextPrime[u]]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 24 2022 *)

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A354367 Successive pairs of terms (a, b) such that (a + b) is a square 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A354368 Successive pairs of terms (a, b) such that (a + b) is a square 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica