A138426 -id:A138426 - cp's OEIS frontend

A138420 a(n) = ((prime(n))^4-(prime(n))^2)/4.

3, 18, 150, 588, 3630, 7098, 20808, 32490, 69828, 176610, 230640, 468198, 706020, 854238, 1219368, 1971918, 3028470, 3460530, 5036658, 6351660, 7098228, 9735960, 11862858, 15683580, 22129968, 26012550, 28135068, 32767038, 35286570

Offset: 1

Artur Jasinski, Mar 19 2008

Number of monic irreducible polynomials of degree 4 over GF(prime(n)). - Robert Israel, Jan 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A008837, A127919, A138420, A138426.

[(NthPrime((n))^4 - NthPrime((n))^2)/4: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

seq(1/4*(ithprime(i)^4 - ithprime(i)^2), i=1..100); # Robert Israel, Jan 07 2015

a = {}; Do[p = Prime[n]; AppendTo[a, (p^4 - p^2)/4], {n, 1, 50}]; a
(#^4-#^2)/4&/@Prime[Range[30]] (* Harvey P. Dale, Aug 01 2025 *)

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

a(n) = A138402(n)/4. - R. J. Mathar, Oct 15 2017

Name edited by Robert Israel, Jan 07 2015

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

A138430 a(n) = (prime(n)^5 - prime(n))/30.

Original entry on oeis.org

1, 8, 104, 560, 5368, 12376, 47328, 82536, 214544, 683704, 954304, 2311464, 3861872, 4900280, 7644832, 13939848, 23830808, 28153208, 45004168, 60140976, 69102384, 102568544, 131301352, 186135312, 286244672, 350336680, 386424688, 467517240, 512874648

Offset: 1

Artur Jasinski, Mar 19 2008

a(n) = Sum_{i+j=p} (i^2)*(j^2) where p = prime(n). - Michel Lagneau, May 21 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040, A138404, A138426.

[(NthPrime((n))^5 - NthPrime((n)))/30: n in [1..30]]; // Vincenzo Librandi, Jun 18 2011

Table[p = Prime[n]; (p^5 - p)/30, {n, 50}]

forprime(p=2,1e3,print1((p^5-p)/30", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A138404(n)/30. - R. J. Mathar, Oct 15 2017

a(n) = A138426(n)/6. - Bernard Schott, Feb 28 2023

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

A253655 Number of monic irreducible polynomials of degree 6 over GF(prime(n)).

Original entry on oeis.org

9, 116, 2580, 19544, 295020, 804076, 4022064, 7839780, 24670536, 99133020, 147912160, 427612404, 791672280, 1053546956, 1796518224, 3694034916, 7030054140, 8586690620, 15076346164, 21349986840, 25222305336, 40514492720, 54489965796, 82830096360, 138828513824, 176919851700

Offset: 1

Robert Israel, Jan 07 2015

nonn

			For n=1 the a(1) = 9 irreducible monic polynomials of degree 6 over GF(2) are
x^6+x^5+1, x^6+x^3+1, x^6+x^5+x^4+x^2+1, x^6+x^5+x^3+x^2+1, x^6+x+1, x^6+x^5+x^4+x+1, x^6+x^4+x^3+x+1, x^6+x^5+x^2+x+1, x^6+x^4+x^2+x+1.

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

Cf. A008837, A127919, A138420, A138426.

[(p^6 - p^3 - p^2 + p) div 6: p in PrimesUpTo(110)]; // Vincenzo Librandi, Jan 08 2015

f:= p-> (p^6 - p^3 - p^2 + p)/6:
seq(f(ithprime(i)), i=1..100); # Robert Israel, Jan 07 2015

Table[(Prime[n]^6 - Prime[n]^3 - Prime[n]^2 + Prime[n]) / 6, {n, 1, 30}] (* Vincenzo Librandi, Jan 08 2015 *)

a(n) = (p^6 - p^3 - p^2 + p)/6, where p = prime(n).

cp's OEIS Frontend

A138420 a(n) = ((prime(n))^4-(prime(n))^2)/4.

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

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A138430 a(n) = (prime(n)^5 - prime(n))/30.

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

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A253655 Number of monic irreducible polynomials of degree 6 over GF(prime(n)).

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Programs

Magma

Maple

Mathematica

Formula