A354367 -id:A354367 - 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

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 *)

A354370 Successive pairs of terms (i, j) such that (i + j) is a prime number and at least i is a prime number. This is the lexicographically earliest infinite sequence of distinct terms > 1 with this property.

Original entry on oeis.org

2, 3, 5, 6, 7, 4, 11, 8, 13, 10, 17, 12, 19, 18, 23, 14, 29, 24, 31, 16, 37, 22, 41, 20, 43, 28, 47, 26, 53, 30, 59, 38, 61, 36, 67, 34, 71, 32, 73, 40, 79, 48, 83, 44, 89, 42, 97, 52, 101, 50, 103, 46, 107, 56, 109, 54, 113, 60, 127, 64, 131, 62, 137, 74, 139, 58, 149, 78, 151, 72, 157, 66, 163, 70

Offset: 1

Eric Angelini and Carole Dubois, May 24 2022

nonn

The terms 1, 9, 15, 21, 25, 27, 33, 35, 39, 45, ... will never appear in the sequence; they form A014076, the "Odd nonprimes". Two prime terms can form a pair (2 and 3 for instance) but the first term must always be a prime [the pair (5, 6) is ok].

			The earliest pairs with their prime sum: (2, 3) = 5, (5, 6) = 11, (7, 4) = 11, (11, 8) = 19, (13, 10) = 23, (17, 12) = 29, (19, 18) = 37, (23, 14) = 37, etc.

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

Cf. A354367, A354368, A354369 (same idea), A014076.

nn = 120; c[] = 0; a[1] = 2; c[2] = 1; u = 3; Do[If[EvenQ[i], k = u; While[Nand[c[k] == 0, PrimeQ[# + k]] &[a[i - 1]], k++], k = u; While[Nand[c[k] == 0, PrimeQ[k]], 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 = 2, 3, 4, set()
    while True:
        aset.update((i, j)); yield from (i, j)
        i = j = v
        while i in aset or not isprime(i): i += 1
        while j == i or j in aset or not isprime(i+j): j += 1
        while v in aset: v += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Jun 24 2022

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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.

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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.

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Comments

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Crossrefs

Programs

Mathematica

A354370 Successive pairs of terms (i, j) such that (i + j) is a prime number and at least i is a prime number. This is the lexicographically earliest infinite sequence of distinct terms > 1 with this property.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python