| Displaying 1-8 of 8 results found.
page
1
1/6 of product of three numbers: n-th prime, previous and following number.
+10
7
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
MATHEMATICA
Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/6, {n, 1, 100}]
PROG
(Python)
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 (Magma) [(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/6: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017
a(n) = ((prime(n))^5-prime(n))/5.
+10
6
6, 48, 624, 3360, 32208, 74256, 283968, 495216, 1287264, 4102224, 5725824, 13868784, 23171232, 29401680, 45868992, 83639088, 142984848, 168919248, 270025008, 360845856, 414614304, 615411264, 787808112, 1116811872
COMMENTS
Number of monic irreducible polynomials of degree 5 over GF(prime(n)). - Robert Israel, Jan 07 2015
MAPLE
seq((ithprime(i)^5-ithprime(i))/5, i = 1 .. 50); # Robert Israel, Jan 07 2015
MATHEMATICA
a = {}; Do[p = Prime[n]; AppendTo[a, (p^5 - p)/5], {n, 1, 50}]; a
PROG
(Magma) [(NthPrime((n))^5 - NthPrime((n)))/5: n in [1..30] ]; // Vincenzo Librandi, Jun 18 2011
1/24 of product of three numbers: n-th prime, previous and following number.
+10
5
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
COMMENTS
The product of (n-1), n, and (n+1) = n^3 - n. - Harvey P. Dale, Jan 17 2011 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
MATHEMATICA
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 *)
PROG
(PARI) for(n=2, 25, print1((prime(n)+1)*prime(n)*(prime(n)-1)/24, ", ")) \\ G. C. Greubel, Jun 19 2017
a(n) = ((prime(n))^4-(prime(n))^2)/4.
+10
5
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
COMMENTS
Number of monic irreducible polynomials of degree 4 over GF(prime(n)). - Robert Israel, Jan 07 2015
MAPLE
seq(1/4*(ithprime(i)^4 - ithprime(i)^2), i=1..100); # Robert Israel, Jan 07 2015
MATHEMATICA
a = {}; Do[p = Prime[n]; AppendTo[a, (p^4 - p^2)/4], {n, 1, 50}]; a
PROG
(Magma) [(NthPrime((n))^4 - NthPrime((n))^2)/4: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011
a(n) = (p^3 - p^2)/2, where p = prime(n).
+10
3
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
COMMENTS
Differences (p^k - p^m)/q with k > m:
.
expression OEIS sequence
-------------- -------------
(p^3 - p^2)/2 this sequence
MATHEMATICA
a = {}; Do[p = Prime[n]; AppendTo[a, (p^3 - p^2)/2], {n, 1, 50}]; a
EXTENSIONS
Definition corrected by T. D. Noe, Aug 25 2008
1/12 of product of three numbers: n-th prime, previous and following number.
+10
2
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
COMMENTS
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
MAPLE
a:= n-> (p->p*(p^2-1)/12)(ithprime(n)):
MATHEMATICA
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 *)
PROG
(Magma) [(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/12: n in [2..50]]; // G. C. Greubel, Apr 30 2018
a(n) = ((n-th prime)^6-(n-th prime)^4)/12.
+10
2
4, 54, 1250, 9604, 146410, 399854, 2004504, 3909630, 12313004, 49509670, 73881680, 213654354, 395606540, 526495354, 897861304, 1846372554, 3514034690, 4292210710, 7536519254, 10672906020, 12608819004, 20254042120, 27241076254
COMMENTS
Differences (p^k-p^m)/q such that k > m:
MATHEMATICA
a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^4)/12], {n, 1, 24}]; a
Number of monic irreducible polynomials of degree 6 over GF(prime(n)).
+10
1
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
FORMULA
a(n) = (p^6 - p^3 - p^2 + p)/6, where p = prime(n).
EXAMPLE
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.
MAPLE
f:= p-> (p^6 - p^3 - p^2 + p)/6:
MATHEMATICA
Table[(Prime[n]^6 - Prime[n]^3 - Prime[n]^2 + Prime[n]) / 6, {n, 1, 30}] (* Vincenzo Librandi, Jan 08 2015 *)
PROG
(Magma) [(p^6 - p^3 - p^2 + p) div 6: p in PrimesUpTo(110)]; // Vincenzo Librandi, Jan 08 2015
Search completed in 0.007 seconds
|