A127917 -id:A127917 - cp's OEIS frontend

A127918 Half of product of three numbers: n-th prime, previous and following number.

3, 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

Offset: 1

Artur Jasinski, Feb 06 2007

Apart from the first term, the same as A117762. - R. J. Mathar, Jun 14 2008
Except the first term, a(n) is the area of the integer-sided isosceles triangle ABC with AB=AC such that the altitude AH is of prime(n) length.
The couples (a(n), altitude) are (12,3), (60,5), (168,7), (660,11), (1092,13), ... and the sequence of the ratio a(n)/prime(n) is {4, 12, 24, 60, 84, 144, 180, ...} - see A084921. - Michel Lagneau, Oct 23 2013
a(n) is also equal to the number of reducible quadratic polynomials in the field of size prime(n). - James East, Apr 26 2024

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

Cf. A036689, A034953, A127917, A127920.

[(NthPrime(n)+1)*NthPrime(n)*(NthPrime(n)-1)/2: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/2, {n, 1, 100}]

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

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).

A069487 Areas of Pythagorean triangles (A069482, A069484, A069486).

Original entry on oeis.org

30, 240, 840, 5544, 6864, 26520, 23256, 73416, 208104, 107880, 467976, 473304, 296184, 727560, 1494600, 2101344, 863760, 3138816, 2625864, 1492704, 5259504, 4248936, 7623384, 12845904, 7759224, 4244424

Offset: 1

Reinhard Zumkeller, Mar 29 2002

nonn

			prime(2)^3 * prime(1) - prime(1)^3 * prime(2) = 3^3 * 2 - 2^3 * 3 = 54 - 24 = 30 that is the area of the Pythagorean triangle (5, 12, 13), so a(1) = 30. - _Bernard Schott_, Sep 23 2019

Marius A. Burtea, Table of n, a(n) for n = 1..5000
César Aguilera, Two Prime Number Objects and The Velucchi Numbers, hal-02909691 [math.NT], 2020.

Cf. A069482, A069484, A069486.

Cf. A000040, A030078, A127917.

[NthPrime(n+1)^3*NthPrime(n)-NthPrime(n+1)*(NthPrime(n)^3):n in [1..26]]; // Marius A. Burtea, Sep 19 2019

a(n) = A030078(n+1)*A000040(n) - A000040(n+1)*A030078(n).
a(n) = A000040(n+1)^3*A000040(n) - A000040(n+1)*A000040(n)^3.
a(n) = A000040(n)*A127917(n+1) - A127917(n)*A000040(n+1). - César Aguilera, Sep 18 2019

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

A369632 Decimal expansion of Sum_{primes p} 1/(p*(p^2 - 1)).

Original entry on oeis.org

2, 2, 1, 4, 6, 3, 3, 7, 1, 3, 9, 2, 7, 9, 5, 9, 4, 3, 4, 2, 4, 6, 3, 6, 4, 3, 5, 8, 8, 4, 5, 9, 8, 8, 1, 7, 4, 8, 7, 2, 4, 0, 9, 5, 8, 3, 0, 4, 5, 5, 7, 7, 9, 6, 0, 8, 0, 3, 8, 8, 7, 3, 3, 2, 9, 7, 1, 4, 3, 4, 3, 0, 8, 4, 8, 1, 6, 2, 7, 2, 6, 7, 5, 6, 0, 4, 7, 7, 7, 5, 6, 5, 5, 0, 4, 2, 8, 5, 7, 6, 0, 3, 8, 7, 9

Offset: 0

R. J. Mathar, Jan 28 2024

			0.22146337139279594342463643588459881748724095830455...

Index to constants which are prime zeta sums {1,1,1}

Cf. A127917, A136141, A179119.

RealDigits[NSum[PrimeZetaP[2*k + 1], {k, 1, Infinity}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Jan 28 2024 *)

sumeulerrat(1/(p*(p^2-1))) \\ Amiram Eldar, Jan 28 2024

Equals Sum_{i>=1} 1/A127917(i) = (A136141 - A179119)/2.

Equals Sum_{k>=1} P(2*k+1), where P(s) is the prime zeta function. - Amiram Eldar, Jan 28 2024

More terms from Amiram Eldar, Jan 28 2024

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

A244509 Order of GL_2(p), the general linear group over F_p, where p runs through the primes.

Original entry on oeis.org

6, 48, 480, 2016, 13200, 26208, 78336, 123120, 267168, 682080, 892800, 1822176, 2755200, 3337488, 4773696, 7738848, 11908560, 13615200, 19845936, 25048800, 28003968, 38450880, 46879728, 62029440, 87607296, 103020000, 111447648, 129843216, 139851360

Offset: 1

John McGee, Nov 15 2014

nonn

			For n=3 (p=5) we have a(3) = 4*5*(25-1) = 480.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abbey Bourdon, Ozlem Ejder, Yuan Liu, Frances Odumodu, Bianca Viray, On the level of modular curves that give rise to sporadic j-invariants, arXiv:1808.04520 [math.NT], 2018. See Table 7.2 (an extract of current sequence).

Cf. A127917 (order of SL_2(p)), A047927.

[(NthPrime(n)-1)*NthPrime(n)*(NthPrime(n)^2-1): n in [1..100]]; // Vincenzo Librandi, Aug 15 2018

gl2psz[p_] := (p - 1) p (p^2 - 1); sqg = gl2psz/@Prime@Range[m]
Table[(Prime[n] - 1) Prime[n] (Prime[n]^2 - 1), {n, 30}] (* Vincenzo Librandi, Aug 15 2018 *)

a(n) = { my(p=prime(n)); (p-1)*p*(p^2-1) } \\ Joerg Arndt, Nov 23 2014

a(n) = (p-1)*p*(p^2-1) where p = prime(n).
a(n) = A127917(n)*(prime(n)-1).
Subsequence of A047927. - Michel Marcus, Nov 25 2014
Sum 1/a(n) = A382584. - R. J. Mathar, Mar 31 2025

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

cp's OEIS Frontend

A127918 Half of product of three numbers: n-th prime, previous and following number.

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

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

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

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A069487 Areas of Pythagorean triangles (A069482, A069484, A069486).

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

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A369632 Decimal expansion of Sum_{primes p} 1/(p*(p^2 - 1)).

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