A199367 -id:A199367 - cp's OEIS frontend

A115104 Numbers n such that 4*n^3 + 1 is prime.

1, 3, 4, 7, 9, 10, 19, 25, 34, 37, 39, 42, 49, 54, 55, 72, 73, 78, 85, 87, 93, 94, 102, 108, 109, 118, 138, 142, 147, 157, 160, 165, 168, 175, 192, 195, 202, 210, 214, 220, 228, 232, 243, 247, 249, 250, 252, 253, 258, 267, 273, 274, 279, 289, 297

Offset: 1

Parthasarathy Nambi, Mar 02 2006

nonn

For any n in this sequence, 3*(4*n^3 + 1) has the same nonzero digits as its prime factors in base 2n. - Ely Golden, Dec 12 2016

			If n=94 then (4*n^3 + 1) = 3322337 (prime).

Ely Golden, Table of n, a(n) for n = 1..1000

Cf. A001912. See A199307 for the actual primes.

See also A199364, A199365, A199366, A199367, A199368, A199369.

Select[Range[300], PrimeQ[4*#^3 + 1] &] (* Stefan Steinerberger, Mar 04 2006 *)

is(n)=isprime(4*n^3+1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Stefan Steinerberger, Mar 04 2006

A199366 Numbers k such that 4*k^3-1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 12, 15, 18, 30, 32, 45, 48, 51, 63, 66, 87, 90, 98, 101, 113, 116, 122, 125, 132, 150, 153, 155, 156, 161, 170, 171, 173, 183, 195, 198, 203, 213, 233, 237, 243, 246, 260, 266, 282, 288, 291, 297, 300, 302, 305, 308, 321, 335, 341, 342, 347, 366, 371, 377, 381, 386, 393, 398, 401, 402, 407, 408, 411, 423, 425, 426, 437, 443, 452, 455, 456

Offset: 1

N. J. A. Sloane, Nov 05 2011

See comment in A199307.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A199307, A199367.

[n: n in [0..500] | IsPrime(4*n^3-1)]; // Vincenzo Librandi, Jan 07 2013

Select[Range[0, 1000], PrimeQ[4#^3 - 1]&] (* Vincenzo Librandi, Jan 07 2013 *)

is(n)=isprime(4*n^3-1) \\ Charles R Greathouse IV, May 22 2017

A352330 Squares whose arithmetic derivative (A003415) is a cube.

Original entry on oeis.org

0, 1, 11664, 20736, 2313441, 2985984, 9150625, 28005264, 236421376, 655360000, 1871773696, 3340840000, 4294967296, 10435031104, 10485760000, 11716114081, 33556377856, 50054665441, 80706559921, 156531800881, 203928109056, 258439040161, 282429536481, 414998793616

Offset: 1

Marius A. Burtea, Mar 13 2022

nonn

For p prime number of the form p = 4*m^3 - 1 (A199367) the number k = 2^8*p^4 is a term. Indeed, k' = (2^8*p^4)' = 8*2^7*p^4 + 2^8*4*p^3 = 2^9*p^3*(2*(p + 1)) = 2^9*p^3*(2*(p + 1)) = 2^9*p^3*2*4*m^3 = (2^3*p*8*m)^3 so k is a term.
The sequence is infinite because numbers of the form m = 2^(2^(6*k + 5)), k >= 0, are terms. Indeed: m' = 2^(6*k + 5)*2^(2^(6*k + 5) - 1) = 2^(6*k + 4 + 2^(6*k + 5)) = 2^(6*k + 3 + 2^(6*k + 5) + 1), and the exponent 6*k + 3 + 2^(6*k + 5) + 1 is divisible by 3.
If p is a prime number then the numbers of the form m = p^(64^k), k >= 1 are terms.

			11664 = 108^2 and 11664' = 46656 = 36^3 so 11664 is a term.
20736 = 144^2 and 20376' = 110592 = 48^3 so 20736 is a term.

Cf. A003415, A199367, A266890.

f:=func; [p:p in [s*s:s in [0.. 450000]]| IsPower(Floor(f(p)),3)];

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6.5*10^5]^2, IntegerQ@Surd[d[#], 3] &] (* Amiram Eldar, Mar 13 2022 *)

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A115104 Numbers n such that 4*n^3 + 1 is prime.

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A199366 Numbers k such that 4*k^3-1 is prime.

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A352330 Squares whose arithmetic derivative (A003415) is a cube.

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