A084380 -id:A084380 - cp's OEIS frontend

A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

0, 1, 3, 5, 29, 71, 83, 113, 173, 263, 311, 419, 431, 491, 503, 509, 683, 701, 761, 773, 839, 911, 953, 1031, 1091, 1103, 1151, 1193, 1259, 1283, 1373, 1451, 1523, 1583, 1601, 1733, 1823, 1889, 1931, 2099, 2153, 2213, 2273, 2339, 2351, 2441, 2531, 2543

Offset: 1

Jonathan Vos Post, Aug 16 2007

The corresponding near-cube primes are A132282. Analog of near-square primes. After a(1) = 0, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.

			a(1) = 0 because 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1 because 1^3 + 2 = 5 is prime and 1 is noncomposite.
a(3) = 3 because 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5 because 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29 because 29^3 + 2 = 24391 is prime.
45 is not in the sequence because, although 45^3 + 2 = 91127 is prime, 45 is not prime.
63 is not in the sequence because, although 63^3 + 2 = 250049 is prime, 63 is not prime.
65 is not in the sequence because, although 65^3 + 2 = 274627 is prime, 65 is not prime.
a(6) = 71 because 71^3 + 2 = 357913 is prime.
a(7) = 83 because 83^3 + 2 = 571789 is prime.
a(8) = 113 because 113^3 + 2 = 1442899 is prime.
123 is not in the sequence because, although 123^3 + 2 = 1860869 is prime, 123 is not prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132282.

{p in A000040 such that A067200(p) = A084380(p) = p^3 + 2 is in A000040}.

Union of {0,1} and A048637. - R. J. Mathar, Oct 18 2007

More terms from R. J. Mathar, Oct 18 2007

A300480 Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t+m)/2)exp(-t)dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 2, 1, 2, 2, 0, 2, 3, 3, 3, 2, 4, 8, 10, 18, 2, 5, 15, 29, 47, 95, 2, 6, 24, 66, 130, 256, 592, 2, 7, 35, 127, 327, 697, 1610, 4277, 2, 8, 48, 218, 722, 1838, 4376, 11628, 35010, 2, 9, 63, 345, 1423, 4459, 11770, 31607, 95167, 320589, 2, 10, 80, 514, 2562, 9820, 30248, 85634, 258690

Offset: 0

Max Alekseyev, Mar 06 2018

a(m,n) is a polynomial in m of degree n.

For any integers m>=0, n>=0, 2 * Integral_{t=-m..m} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-m/2..m/2} T_n(z)*exp(-2*z)*dz = A300481(m,n)*exp(m) - a(m,n)*exp(-m).

			Array starts with:
m=0: 2,  1,   0,    3,    18,     95,     592, ...
m=1: 2,  2,   3,   10,    47,    256,    1610, ...
m=2: 2,  3,   8,   29,   130,    697,    4376, ...
m=3: 2,  4,  15,   66,   327,   1838,   11770, ...
m=4: 2,  5,  24,  127,   722,   4459,   30248, ...
...

Values for m<=0 are given in A300481.
Rows: A300482 (m=0), A300483 (m=1), A300484 (m=2), A300485 (m=-1), A102761 (m=-2).
Columns: A007395 (n=0), A000027 (n=1), A005563 (n=2), A084380 (n=3).
Cf. A000179 (almost row m=-2), A127672, A156995.

{ A300480(m,n) = if(n==0,return(2)); subst( serlaplace( 2*polchebyshev(n,1,(x+m)/2)), x, 1); }

a(m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} m^j/j!.

a(m,n) = Sum_{i=0..n} A127672(n,i) * A080955(m,i) = Sum_{i=0..n} A127672(n,i) * A089258(i,m).

A185065 a(n) = n*(n^3 + 2).

Original entry on oeis.org

0, 3, 20, 87, 264, 635, 1308, 2415, 4112, 6579, 10020, 14663, 20760, 28587, 38444, 50655, 65568, 83555, 105012, 130359, 160040, 194523, 234300, 279887, 331824, 390675, 457028, 531495, 614712, 707339, 810060, 923583, 1048640

Offset: 0

Vincenzo Librandi, Mar 31 2011

Numbers a(n) such that a(n)^3 = x^3*(x-2). The values of x are in A084380.

A058895(n)^3 + A068601(n)^3 + A033562(n)^3 = a(n)^3, for n > 0. - Vincenzo Librandi, Mar 13 2012

			20^3 = 10^3*(10-2); 87^3 = 29^3*(29-2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A084380, A058895, A068601, A033562.

Table[n(n^3+2),{n,0,40}]  (* Harvey P. Dale, Apr 11 2011 *)

a(n)=n*(n^3+2) \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(3 + 5*x + 17*x^2 - x^3)/(1-x)^5. - Bruno Berselli, Mar 31 2011

A132282 Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.

Original entry on oeis.org

2, 3, 29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793

Offset: 1

Jonathan Vos Post, Aug 16 2007

The corresponding near-cube prime indices q are A132281. Analog of near-square primes. After a(1) = 2, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.

			a(1) = 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1^3 + 2 = 3 is prime and 1 is noncomposite.
a(3) = 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29^3 + 2 = 24391 is prime and 29 is prime.
45^3 + 2 = 91127 is prime, but not in this sequence because 45 is not prime.
63^3 + 2 = 250049 is prime, but not in this sequence because 63 is not prime.
a(6) = 71^3 + 2 = 357913 is prime.
a(7) = 83^3 + 2 = 571789 is prime.
a(8) = 113^3 + 2 = 1442899 is prime.

Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Cf. A000040, A048636, A056899, A059100, A067200, A067201, A084380, A132281.

Join[{2, 5}, Select[Prime[Range[200]]^3 + 2, PrimeQ[ # ] &]] (* Stefan Steinerberger, Aug 17 2007 *)

v=[2,3]; forprime(p=3, 1e4, if(isprime(t=p^3+2), v=concat(v, t))); t \\ Charles R Greathouse IV, Feb 14 2011

a(n) = A132281(n)^3 + 2. {p in A000040 such that for some q = 0, 1, or q in A000040, we have p = A067200(q) = A084380(q) = q^3 + 2 is in A000040}.

a(n) = A048636(n-2) for n >= 3. - Georg Fischer, Nov 03 2018

More terms from Stefan Steinerberger, Aug 17 2007

a(2) corrected by Charles R Greathouse IV, Feb 14 2011

A034324 a(n) = (n-1)(n-2)(n-3) + n.

Original entry on oeis.org

1, 2, 3, 10, 29, 66, 127, 218, 345, 514, 731, 1002, 1333, 1730, 2199, 2746, 3377, 4098, 4915, 5834, 6861, 8002, 9263, 10650, 12169, 13826, 15627, 17578, 19685, 21954, 24391, 27002, 29793, 32770, 35939, 39306, 42877, 46658, 50655, 54874, 59321

Offset: 1

Laurence Michaels (guardian(AT)ntplx.net)

(n*a(n+1)^3+1)/(n^3+1) is the smallest integer of the form (n*k^3+1)/(n^3+1). - Benoit Cloitre, May 02 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A084378, A084381.

List([1..50], n-> (n-2)^3 +2); # G. C. Greubel, Aug 23 2019

[(n-2)^3 +2: n in [1..50]]; // G. C. Greubel, Aug 23 2019

seq( (n-2)^3 +2, n=1..50); # G. C. Greubel, Aug 23 2019

Table[(n-3)(n-2)(n-1)+n, {n,50}] (* or *) Table[n^3+2, {n,-1,50}]  (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
CoefficientList[Series[(1 -2x +x^2 +6x^3)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Feb 24 2014 *)
LinearRecurrence[{4,-6,4,-1},{1,2,3,10},50] (* Harvey P. Dale, Aug 06 2018 *)

a(n)=(n-3)*(n-2)*(n-1)+n \\ Charles R Greathouse IV, Jul 02 2016

[(n-2)^3 +2 for n in (1..50)] # G. C. Greubel, Aug 23 2019

a(n) = (n-2)^3 + 2 = A084380(n-2). - Philippe Deléham, Feb 23 2014
a(n+1) = A002061(n)*(n-2) + 3. - Philippe Deléham, Feb 23 2014
G.f.: x*(1-2*x+x^2+6*x^3)/(1-x)^4. - Philippe Deléham, Feb 23 2014
E.g.f.: 6 + (x^3-3*x^2+7*x-6)*exp(x). - Nikolaos Pantelidis, Feb 06 2023

Extended and corrected by Erich Friedman

A274077 a(n) = n^3 + 4.

Original entry on oeis.org

4, 5, 12, 31, 68, 129, 220, 347, 516, 733, 1004, 1335, 1732, 2201, 2748, 3379, 4100, 4917, 5836, 6863, 8004, 9265, 10652, 12171, 13828, 15629, 17580, 19687, 21956, 24393, 27004, 29795, 32772, 35941, 39308, 42879, 46660, 50657, 54876, 59323, 64004, 68925

Offset: 0

Vincenzo Librandi, Jun 09 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Sequences of the type n^3+k: A000578 (k=0), A001093 (k=1), A084380 (k=2), A084378 (k=3), this sequence (k=4), A084381 (k=5), A084382 (k=6), A084377 (k=7).

Magma
```
[n^3+4: n in [0..50]];
```

seq(n^3+4, n=0..100); # Robert Israel, Jun 09 2016

Table[n^3 + 4, {n, 0, 60}]
Range[0,50]^3+4 (* or *) LinearRecurrence[{4,-6,4,-1},{4,5,12,31},50] (* Harvey P. Dale, Jul 01 2017 *)

a(n) = n^3 + 4 \\ Felix Fröhlich, Jun 09 2016

O.g.f.: (4 - 11*x + 16*x^2 - 3*x^3)/(1 - x)^4.
E.g.f.: (x^3 + 3*x^2 + x + 4)*exp(x). - Robert Israel, Jun 09 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

A259189 Semiprimes of the form n^3 + 2.

Original entry on oeis.org

10, 218, 514, 731, 1333, 2199, 2746, 3377, 4915, 5834, 6861, 8002, 9263, 12169, 15627, 29793, 35939, 42877, 54874, 59321, 68923, 117651, 125002, 132653, 148879, 185195, 205381, 314434, 405226, 421877, 474554, 531443, 592706, 658505, 704971

Offset: 1

Morris Neene, Jun 20 2015

nonn

Intersection of A001358 and A084380. - Michel Marcus, Jun 20 2015
Since there are no squares of the form n^3 + 2, all semiprimes in this sequence are products of distinct primes.
No term in A040034 divides any term in this sequence.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A084380 (n^3+2), A144953 (primes of same form).

Cf. A237040 (similar sequence with n^3+1).

IsSP:=func;[r:n in [1..1000]|IsSP(r) where r is 2+n^3];

Select[Range[100]^3 + 2, PrimeOmega[#] == 2 &] (* Alonso del Arte, Jun 20 2015 *)

is(n)=bigomega(n^3 + 2)==2 \\ Anders Hellström, Sep 07 2015

use ntheory ":all"; my @sp = grep { scalar(factor($))==2 } map { $**3+2 } 1..100; say "@sp"; # Dana Jacobsen, Sep 07 2015

cp's OEIS Frontend

A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

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A300480 Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t+m)/2)*exp(-t)*dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

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A185065 a(n) = n*(n^3 + 2).

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A132282 Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.

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A034324 a(n) = (n-1)*(n-2)*(n-3) + n.

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A274077 a(n) = n^3 + 4.

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A259189 Semiprimes of the form n^3 + 2.

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A300480 Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t+m)/2)exp(-t)dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

A034324 a(n) = (n-1)(n-2)(n-3) + n.