A127484 - OEIS

%I #12 Sep 08 2022 08:45:29

%S 1,2,3,8,13,14,22,23,24,34,38,64,98,99,133,147,153,178,232,253,254,

%T 297,328,343,344,367,407,498,573,574,582,587,624,638,639,653,668,679,

%U 702,703,759,772,793,797,849,874,944,958,1023,1058,1067,1087,1203,1212,1322

%N Numbers k such that A127483(k) = A127483(k+1) - 1.

%C A127483(n) = {1,2,3,4,8,9,13,14,15,17,22,23,24,25,30,32,34,35,38,39,42,45,50,...} are the numbers n such that A100705(n) = n^3 + (n+1)^2 is prime.

%C Corresponding primes of the form n^3 + (n+1)^2 are listed in A100662(n) = {5, 17, 43, 89, 593, 829, 2393, 2969, 3631, 5237, ...}.

%C Note that there are many consecutive twins, triples and quadruplets in A127483(n). For example: (1,2,3,4), {8,9}, {13,14,15}, {22,23,24,25}, {34,35}, {38,39}, {64,65}, {98,99,100}. Twins in A127483(k) start with numbers k = a(n). Triplets in A127483(k) start with k = {1,2,13,22,23,98,253,343,573,638,702,...} = A127485, or numbers n such that a(k) = a(k+1) - 1 = a(k+2) - 2. Quadruplets in A127483(k) start with k = {1,22,13077,14267,16092,16267,16282,36387,47012,51912,54662,...} = A127486.

%H Amiram Eldar, <a href="/A127484/b127484.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Range[3000],PrimeQ[ #^3+(#+1)^2]&&PrimeQ[(#+1)^3+(#+2)^2]&]

%o (Magma) [k:k in [1..1350]|IsPrime(k^3+(k+1)^2) and IsPrime((k+1)^3+(k+2)^2)]; // _Marius A. Burtea_, Jan 20 2020

%Y Cf. A100705, A100662, A127483, A127485, A127486.

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Jan 16 2007