A028873 -id:A028873 - cp's OEIS frontend

A162860 Numbers k such that k^2+4*k+1 is prime.

2, 6, 8, 12, 18, 24, 30, 32, 38, 42, 44, 56, 62, 66, 78, 84, 86, 90, 96, 108, 110, 116, 122, 126, 134, 138, 140, 144, 162, 170, 188, 192, 200, 204, 206, 216, 218, 248, 252, 264, 266, 294, 296, 302, 308, 318, 320, 324, 326, 330, 338, 348, 354, 360, 368, 414, 416

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2009

			a(1) = k = 2 is in the sequence because 2^2+4*2+1=13 = A028874(1) is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028874.

f[a_]:=a^2+4*a+1; lst={};Do[If[PrimeQ[f[n]],AppendTo[lst,n]],{n,6!}]; lst
Select[Range[500],PrimeQ[#^2+4#+1]&] (* Harvey P. Dale, May 28 2012 *)

a(n) = A028873(n)-2. - R. J. Mathar, Aug 12 2009

A309726 Numbers k such that k^2 - 12 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 25, 29, 35, 41, 49, 53, 59, 61, 79, 85, 91, 95, 97, 103, 107, 113, 119, 121, 137, 139, 145, 149, 163, 169, 173, 179, 181, 185, 191, 205, 209, 227, 233, 235, 245, 251

Offset: 1

Daniel Starodubtsev, Aug 14 2019

nonn

All terms are odd and not divisible by 3.

			11 is in the sequence because 11^2 - 12 = 109, which is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A028876, A028879, A028882, A028885, A186815, A090696, A296507.

Cf. A056927.

Select[Range[5,301,2],PrimeQ[#^2-12]&] (* Harvey P. Dale, Dec 23 2019 *)

select(n->isprime(n^2-12), [1..1000]) \\ Andrew Howroyd, Aug 14 2019

If A056927(k) = 12, then k is a term. - A.H.M. Smeets, Aug 15 2019

A329103 Numbers k such that both k^2 - 3 and 2^k - 3 are primes.

Original entry on oeis.org

4, 10, 14, 20, 266, 452, 694

Offset: 1

Alex Ratushnyak, Nov 04 2019

a(8) > 2086750, using A050414. - Michael S. Branicky, Feb 16 2024

Intersection of A050414 and A028873.

Cf. A000040, A329102.

isok(k) = isprime(k^2 - 3) && isprime(2^k - 3); \\ Michel Marcus, Jul 02 2021

A330438 Numbers k such that k^2-2 and k^3-2 are prime.

Original entry on oeis.org

9, 15, 19, 27, 37, 121, 135, 145, 211, 217, 259, 265, 267, 279, 355, 357, 387, 391, 435, 489, 525, 561, 615, 621, 727, 951, 987, 1029, 1119, 1141, 1177, 1251, 1287, 1357, 1435, 1491, 1561, 1617, 1717, 1785, 1819, 1839, 1875, 1909, 1989, 2001, 2077, 2107, 2211

Offset: 1

K. D. Bajpai, Dec 14 2019

nonn

Intersection of A028870 and A038599.

			a(1) = 9: 9^2 - 2 = 79; 9^3 - 2 = 727; both results are prime.
a(2) = 15: 15^2 - 2 = 223; 15^3 - 2 = 3373; both results are prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A028870, A028873, A038599, A038600, A108701.

[n : n in [1 .. 100] | IsPrime (n^2 - 2) and IsPrime (n^3 - 2)];

filter:= k -> isprime(k^2-2) and isprime(k^3-2):
select(filter, [$2..10000]); # Robert Israel, Dec 24 2019

Select[Range[10000], PrimeQ[#^3 - 2] && PrimeQ[#^2 - 2] &]

A348425 Squares whose second arithmetic derivative is a square.

Original entry on oeis.org

0, 1, 4, 49, 529, 2209, 6241, 27889, 28561, 35344, 49729, 128881, 192721, 250000, 431649, 528529, 703921, 1181569, 1495729, 1610361, 1868689, 3411409, 4870849, 5755201, 9138529, 11390625, 12250000, 13830961, 13845841, 15737089, 22648081, 25391521, 31618129

Offset: 1

Marius A. Burtea, Oct 18 2021

nonn

For prime numbers of the form p = k^2 - 2 (A028871) the number m = p^2 is a term because m'' = (p^2)'' = (2*p*p')' = (2*p)'= p + 2*p' = p + 2 = k^2.
If m is a term in A028873 then p = m^2 - 3 is prime and k = p^4 is a term. Indeed: k' = 4*p^3 and k'' = 4*p^3 + 12*p^2 = 4*p^2*(p + 3) = 4*p^2*m^2.
If m is a term in A201787  then p = 5*m^2 - 6 is prime and k = p^6 is a term. Indeed: k' = 6*p^5 and k'' = 5*p^5 + 30*p^4 = 5*p^4*(p + 6) = 25*p^4*m^2.

			4'' = 4' = 4 so 4 is a term.
49'' = 14' = 9 so 49 is a term.

Cf. A000290, A003415, A028871, A028873, A068346, A201787, A266890, A192192, A328244.

f:=func; [n:n in [s*s:s in [0.. 5623]] | IsSquare(Floor(f(Floor(f(n)))))];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6000]^2, IntegerQ @ Sqrt[d[d[#]]] &] (* Amiram Eldar, Oct 18 2021 *)

ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415
lista(nn) = {for (n=0, nn, if (issquare(ad(ad(n^2))), print1(n^2, ", ")); ); } \\ Michel Marcus, Oct 30 2021

cp's OEIS Frontend

A162860 Numbers k such that k^2+4*k+1 is prime.

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A309726 Numbers k such that k^2 - 12 is prime.

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A329103 Numbers k such that both k^2 - 3 and 2^k - 3 are primes.

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A330438 Numbers k such that k^2-2 and k^3-2 are prime.

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Magma

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A348425 Squares whose second arithmetic derivative is a square.

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Magma

Mathematica

PARI