A354369 -id:A354369 - cp's OEIS frontend

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.

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 *)

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

A354770 Numbers k such that d(k)/log(k) sets a new record, where d(k) is the number-of-divisors function A000005(k).

Original entry on oeis.org

2, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640

Offset: 1

N. J. A. Sloane, Jun 22 2022

nonn

A related sequence, not yet in the OEIS, is "Numbers k such that log(d(k))/log(k) > log(d(m))/log(m) for all m > k". It begins 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, ..., and up to this point it agrees with A236021 (except that it doesn't include 1). Does it continue to agree with A236021?

			The values of d(k)/log(k) for k = 2, 3, ... are 2.885390082, 1.820478453, 2.164042562, 1.242669869, 2.232442506, 1.027796685, 1.923593388, 1.365358840, 1.737177928, 0.8340647828, ... and reach record highs at k = 2 (2.885390082...), k = 60 (2.930872040...), and so on.

David desJardins, Posting to Math Fun Mailing List, Jun 22 2022.

Cf. A000005, A066523, A236020, A236021, A354768, A354369.

s = {}; rm = 0; Do[If[(r = DivisorSigma[0, n]/Log[n]) > rm, rm = r; AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Jun 22 2022 *)

More terms from Amiram Eldar, Jun 22 2022

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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.

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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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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.

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A354770 Numbers k such that d(k)/log(k) sets a new record, where d(k) is the number-of-divisors function A000005(k).

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