A127918 -id:A127918 - cp's OEIS frontend

A117762 a(1) = 6; for n>1, a(n) = prime(n)*(prime(n)^2 - 1)/2.

6, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608, 1342740, 1653900

Offset: 1

Roger L. Bagula, Apr 14 2006

a(n) is the order of the matrix group PSL(2,prime(n)). - corrected by Tom Edgar, Sep 28 2015

Blyth and Robertson, Essential Student Algebra, Volume 5: Groups,Chapman and Hall, New York, page 14

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A084920, A127918.

[6] cat [NthPrime(n)*(NthPrime(n)^2-1)/2: n in [2..40]]; // Vincenzo Librandi, Sep 29 2015

a[n_]= If[n==1, 6, Prime[n]*(Prime[n]^2 -1)/2];
Table[a[n], {n,40}]
Join[{6}, Table[Prime[n] (Prime[n]^2 - 1)/2, {n, 2, 40}]] (* Vincenzo Librandi, Sep 29 2015 *)

a(n) = prime(n)*(prime(n)^2-1)/2;
vector(40, n, a(n+1)) \\ Altug Alkan, Sep 28 2015

def A117762(n): return nth_prime(n)*(nth_prime(n)^2-1)/2 + 3*int(n==1)
[A117762(n) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = A127918(n), n>1.

a(n) = A000040(n)*A084921(n). - R. J. Mathar, Jan 29 2024

A127919 1/3 of product of three numbers: the n-th prime, the previous number and the following number.

Original entry on oeis.org

2, 8, 40, 112, 440, 728, 1632, 2280, 4048, 8120, 9920, 16872, 22960, 26488, 34592, 49608, 68440, 75640, 100232, 119280, 129648, 164320, 190568, 234960, 304192, 343400, 364208, 408312, 431640, 480928, 682752, 749320, 857072, 895160, 1102600

Offset: 1

Artur Jasinski, Feb 06 2007

Number of irreducible monic cubic polynomials over GF(prime(n)). - Robert Israel, Jan 06 2015

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A036689, A034953, A127917, A127918, A127920.

[(p^3 - p) div 3: p in PrimesUpTo(150)]; // Vincenzo Librandi, Jan 08 2015

seq((ithprime(n)^3 - ithprime(n))/3, n=1..100); # Robert Israel, Jan 06 2015

Table[(Prime[n] + 1) Prime[n] (Prime[n] - 1)/3, {n, 100}]

forprime(p=2,1e3,print1(2*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = (prime(n)^3 - prime(n))/3. - Wesley Ivan Hurt, Oct 15 2023

A127920 1/6 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

1, 4, 20, 56, 220, 364, 816, 1140, 2024, 4060, 4960, 8436, 11480, 13244, 17296, 24804, 34220, 37820, 50116, 59640, 64824, 82160, 95284, 117480, 152096, 171700, 182104, 204156, 215820, 240464, 341376, 374660, 428536, 447580, 551300, 573800, 644956

Offset: 1

Artur Jasinski, Feb 06 2007

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

Cf. A036689, A034953, A127917, A127918, A127919.

[(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/6: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/6, {n, 1, 100}]
((#-1)#(#+1))/6&/@Prime[Range[40]] (* Harvey P. Dale, Dec 23 2019 *)

forprime(p=2,1e3,print1(binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

from sympy import prime
print([(prime(n) - 1)*prime(n)*(prime(n) + 1)//6 for n in range(1, 101)]) # Indranil Ghosh, Apr 09 2017

a(n) = A127918(n)/3. - Michel Marcus, Apr 09 2017

A127922 1/24 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

1, 5, 14, 55, 91, 204, 285, 506, 1015, 1240, 2109, 2870, 3311, 4324, 6201, 8555, 9455, 12529, 14910, 16206, 20540, 23821, 29370, 38024, 42925, 45526, 51039, 53955, 60116, 85344, 93665, 107134, 111895, 137825, 143450, 161239, 180441, 194054

Offset: 2

Artur Jasinski, Feb 06 2007

nonn

The product of (n-1), n, and (n+1) = n^3 - n. - Harvey P. Dale, Jan 17 2011
For n > 2, a(n) = A001318(n-2) * A007310(n-1), if A007310(n-1) is prime. Also a(n) is a subsequence of A000330. - Richard R. Forberg, Dec 25 2013
If p is an odd prime it can always be the side length of a leg of a primitive Pythagorean triangle. However it constrains the other leg to have a side length of (p^2-1)/2 and the hypotenuse to have a side length of (p^2+1)/2. The resulting triangle has an area equal to (p-1)*p*(p+1)/4. a(n) is 1/6 the area of such triangles. - Frank M Jackson, Dec 06 2017

G. C. Greubel, Table of n, a(n) for n = 2..1000
César Aguilera, Two Prime Number Objects and The Velucchi Numbers, hal-02909691 [math.NT], 2020.

Cf. A000040, A000330, A011842.

Cf. A036689, A034953, A127917, A127918, A127919, A127920, A127921, A011842.

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/24, {n, 1, 100}] (#^3-#)/ 24&/@ Prime[Range[2,40]] (* Harvey P. Dale, Jan 17 2011 *)
((#-1)#(#+1))/24&/@Prime[Range[2,40]] (* Harvey P. Dale, Jan 20 2023 *)

for(n=2,25, print1((prime(n)+1)*prime(n)*(prime(n)-1)/24, ", ")) \\ G. C. Greubel, Jun 19 2017

a(n) = A011842(A000040(n) + 1) = A000330((A000040(n) - 1) / 2).

A138416 a(n) = (p^3 - p^2)/2, where p = prime(n).

Original entry on oeis.org

2, 9, 50, 147, 605, 1014, 2312, 3249, 5819, 11774, 14415, 24642, 33620, 38829, 50807, 73034, 100949, 111630, 148137, 176435, 191844, 243399, 282449, 348524, 451632, 510050, 541059, 606797, 641574, 715064, 1016127, 1115465, 1276292, 1333149

Offset: 1

Artur Jasinski, Mar 19 2008

Differences (p^k - p^m)/q with k > m:
    expression     OEIS sequence
  --------------   -------------
   p^2 - p            A036689
  (p^2 - p)/2         A008837
   p^3 - p            A127917
  (p^3 - p)/2         A127918
  (p^3 - p)/3         A127919
  (p^3 - p)/6         A127920
   p^3 - p^2          A135177
  (p^3 - p^2)/2    this sequence
   p^4 - p            A138401
  (p^4 - p)/2         A138417
   p^4 - p^2          A138402
  (p^4 - p^2)/2       A138418
  (p^4 - p^2)/3       A138419
  (p^4 - p^2)/4       A138420
  (p^4 - p^2)/6       A138421
  (p^4 - p^2)/12      A138422
   p^4 - p^3          A138403
  (p^4 - p^3)/2       A138423
   p^5 - p            A138404
  (p^5 - p)/2         A138424
  (p^5 - p)/3         A138425
  (p^5 - p)/5         A138426
  (p^5 - p)/6         A138427
  (p^5 - p)/10        A138428
  (p^5 - p)/15        A138429
  (p^5 - p)/30        A138430
   p^5 - p^2          A138405
  (p^5 - p^2)/2       A138431
   p^5 - p^3          A138406
  (p^5 - p^3)/2       A138432
  (p^5 - p^3)/3       A138433
  (p^5 - p^3)/4       A138434
  (p^5 - p^3)/6       A138435
  (p^5 - p^3)/8       A138436
  (p^5 - p^3)/12      A138437
  (p^5 - p^3)/24      A138438
   p^5 - p^4          A138407
  (p^5 - p^4)/2       A138439
   p^6 - p            A138408
  (p^6 - p)/2         A138440
   p^6 - p^2          A138409
  (p^6 - p^2)/2       A138441
  (p^6 - p^2)/3       A138442
  (p^6 - p^2)/4       A138443
  (p^6 - p^2)/5       A138444
  (p^6 - p^2)/6       A138445
  (p^6 - p^2)/10      A138446
  (p^6 - p^2)/12      A138447
  (p^6 - p^2)/15      A138448
  (p^6 - p^2)/20      A122220
  (p^6 - p^2)/30      A138450
  (p^6 - p^2)/60      A138451
   p^6 - p^3          A138410
  (p^6 - p^3)/2       A138452
   p^6 - p^4          A138411
  (p^6 - p^4)/2       A138453
  (p^6 - p^4)/3       A138454
  (p^6 - p^4)/4       A138455
  (p^6 - p^4)/6       A138456
  (p^6 - p^4)/8       A138457
  (p^6 - p^4)/12      A138458
  (p^6 - p^4)/24      A138459
   p^6 - p^5          A138412
  (p^6 - p^5)/2       A138460
.
We can prove that for n>1, a(n) is the remainder of the Euclidean division of Sum_{k=0..p-1} k^p by p^3 where p = prime(n). - Pierre Vandaële, Nov 30 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..168

[(p^3-p^2)/2: p in PrimesUpTo(1000)]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, (p^3 - p^2)/2], {n, 1, 50}]; a
(#^3-#^2)/2&/@Prime[Range[50]] (* Harvey P. Dale, Nov 01 2020 *)

forprime(p=2,1e3,print1((p^3-p^2)/2", ")) \\ Charles R Greathouse IV, Jun 16 2011

Definition corrected by T. D. Noe, Aug 25 2008

A127921 1/12 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

2, 10, 28, 110, 182, 408, 570, 1012, 2030, 2480, 4218, 5740, 6622, 8648, 12402, 17110, 18910, 25058, 29820, 32412, 41080, 47642, 58740, 76048, 85850, 91052, 102078, 107910, 120232, 170688, 187330, 214268, 223790, 275650, 286900, 322478, 360882, 388108, 431462

Offset: 2

Artur Jasinski, Feb 06 2007

Summation of products of partitions into two parts of prime(n): a(6) = (1*12) + (2*11) + (3*10) + (4*9) + (5*8) + (6*7) = 182. - César Aguilera, Feb 20 2018

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A000040, A036689, A034953, A127917, A127918, A127919, A127920.

[(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/12: n in [2..50]]; // G. C. Greubel, Apr 30 2018

a:= n-> (p->p*(p^2-1)/12)(ithprime(n)):
seq(a(n), n=2..40);  # Alois P. Heinz, Mar 08 2022

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/12, {n, 2, 100}]
((#-1)#(#+1))/12&/@Prime[Range[2,40]] (* Harvey P. Dale, Mar 08 2022 *)

a(n,p=prime(n))=binomial(p+1,3)/2 \\ Charles R Greathouse IV, Feb 28 2018

a(n) ~ (n log n)^3/12. - Charles R Greathouse IV, Feb 28 2018

A138459 a(n) = ((n-th prime)^6-(n-th prime)^4)/12.

Original entry on oeis.org

4, 54, 1250, 9604, 146410, 399854, 2004504, 3909630, 12313004, 49509670, 73881680, 213654354, 395606540, 526495354, 897861304, 1846372554, 3514034690, 4292210710, 7536519254, 10672906020, 12608819004, 20254042120, 27241076254

Offset: 1

Artur Jasinski, Mar 22 2008

Differences (p^k-p^m)/q such that k > m:
p^2-p is given in A036689
(p^2-p)/2 is given in A008837
p^3-p is given in A127917
(p^3-p)/2 is given in A127918
(p^3-p)/3 is given in A127919
(p^3-p)/6 is given in A127920
p^3-p^2 is given in A135177
(p^3-p^2)/2 is given in A138416
p^4-p is given in A138401
(p^4-p)/2 is given in A138417
p^4-p^2 is given in A138402
(p^4-p^2)/2 is given in A138418
(p^4-p^2)/3 is given in A138419
(p^4-p^2)/4 is given in A138420
(p^4-p^2)/6 is given in A138421
(p^4-p^2)/12 is given in A138422
p^4-p^3 is given in A138403
(p^4-p^3)/2 is given in A138423
p^5-p is given in A138404
(p^5-p)/2 is given in A138424
(p^5-p)/3 is given in A138425
(p^5-p)/5 is given in A138426
(p^5-p)/6 is given in A138427
(p^5-p)/10 is given in A138428
(p^5-p)/15 is given in A138429
(p^5-p)/30 is given in A138430
p^5-p^2 is given in A138405
(p^5-p^2)/2 is given in A138431
p^5-p^3 is given in A138406
(p^5-p^3)/2 is given in A138432
(p^5-p^3)/3 is given in A138433
(p^5-p^3)/4 is given in A138434
(p^5-p^3)/6 is given in A138435
(p^5-p^3)/8 is given in A138436
(p^5-p^3)/12 is given in A138437
(p^5-p^3)/24 is given in A138438
p^5-p^4 is given in A138407
(p^5-p^4)/2 is given in A138439
p^6-p is given in A138408
(p^6-p)/2 is given in A138440
p^6-p^2 is given in A138409
(p^6-p^2)/2 is given in A138441
(p^6-p^2)/3 is given in A138442
(p^6-p^2)/4 is given in A138443
(p^6-p^2)/5 is given in A138444
(p^6-p^2)/6 is given in A138445
(p^6-p^2)/10 is given in A138446
(p^6-p^2)/12 is given in A138447
(p^6-p^2)/15 is given in A138448
(p^6-p^2)/20 is given in A122220
(p^6-p^2)/30 is given in A138450
(p^6-p^2)/60 is given in A138451
p^6-p^3 is given in A138410
(p^6-p^3)/2 is given in A138452
p^6-p^4 is given in A138411
(p^6-p^4)/2 is given in A138453
(p^6-p^4)/3 is given in A138454
(p^6-p^4)/4 is given in A138455
(p^6-p^4)/6 is given in A138456
(p^6-p^4)/8 is given in A138457
(p^6-p^4)/12 is given in A138458
(p^6-p^4)/24 is given in A138459
p^6-p^5 is given in A138412
(p^6-p^5)/2 is given in A138460

a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^4)/12], {n, 1, 24}]; a

forprime(p=2,1e3,print1((p^6-p^4)/12", ")) \\ Charles R Greathouse IV, Jul 15 2011

A140392 Triples of height (a prime p), base length x and side length y=z of isosceles triangles.

Original entry on oeis.org

3, 8, 5, 5, 24, 13, 7, 48, 25, 11, 120, 61, 13, 168, 85, 17, 288, 145, 19, 360, 181, 23, 528, 265, 29, 840, 421, 31, 960, 481, 37, 1368, 685, 41, 1680, 841, 43, 1848, 925, 47, 2208, 1105, 53, 2808, 1405, 59, 3480, 1741, 61, 3720, 1861, 67, 4488, 2245, 71, 5040, 2521

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

Or: consecutive triples of p=A000040(j), x=2*A084920(j), y=z= A066885(j), j>=2.

The area of the triangles is half the product of height and base length, p*x/2=A127918(j).

			Contains (p,x,y) = (3,8,5), (5,24,13), (7,48,25), (11,120,61), ...

Edited and extended by R. J. Mathar, Jun 17 2008

cp's OEIS Frontend

A117762 a(1) = 6; for n>1, a(n) = prime(n)*(prime(n)^2 - 1)/2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A127919 1/3 of product of three numbers: the n-th prime, the previous number and the following number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A127920 1/6 of product of three numbers: n-th prime, previous and following number.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A127922 1/24 of product of three numbers: n-th prime, previous and following number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A138416 a(n) = (p^3 - p^2)/2, where p = prime(n).

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Extensions

A127921 1/12 of product of three numbers: n-th prime, previous and following number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A138459 a(n) = ((n-th prime)^6-(n-th prime)^4)/12.

Views

Author

Keywords

Comments