cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A133071 a(n) = n^5 - n^3 + n^2.

Original entry on oeis.org

0, 1, 28, 225, 976, 3025, 7596, 16513, 32320, 58401, 99100, 159841, 247248, 369265, 535276, 756225, 1044736, 1415233, 1884060, 2469601, 3192400, 4075281, 5143468, 6424705, 7949376, 9750625, 11864476, 14329953, 17189200, 20487601, 24273900, 28600321, 33522688, 39100545, 45397276
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2007

Keywords

Comments

Exponents are prime numbers in decreasing order.

Examples

			a(7)=16513 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807-343+49=16513.

Links

Crossrefs

Cf. A000290, A000578, A000584, A011379, A045991, A100019.
Cf. A133070, A133072, A133073.

Programs

  • Magma
    [n^5-n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    Table[n^5 - n^3 + n^2, {n,0,50}] (* G. C. Greubel, Oct 20 2017 *)
  • PARI
    for(n=0,50, print1(n^5 - n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

Formula

a(n) = n^5 - n^3 + n^2.
G.f.: x*(1 + 22*x + 72*x^2 + 26*x^3 - x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133072 a(n) = n^5 + n^3 - n^2.

Original entry on oeis.org

0, 1, 36, 261, 1072, 3225, 7956, 17101, 33216, 59697, 100900, 162261, 250416, 373321, 540372, 762525, 1052416, 1424481, 1895076, 2482597, 3207600, 4092921, 5163796, 6447981, 7975872, 9780625, 11898276, 14367861, 17231536, 20534697, 24326100, 28657981, 33586176, 39170241, 45473572
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2007

Keywords

Comments

Exponents are the prime numbers in decreasing order.

Examples

			a(7)=17101 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807+343-49=17101.

Links

Crossrefs

Cf. A000290, A000578, A000584, A011379, A045991, A100019.
Cf. A133070, A133071, A133073.

Programs

  • Magma
    [n^5+n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    Table[n^5 + n^3 - n^2, {n, 0, 50}] (* G. C. Greubel, Oct 20 2017 *)
  • PARI
    for(n=0,50, print1(n^5 + n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

Formula

a(n) = n^5 + n^3 - n^2.
G.f.: x*(1 + 30*x + 60*x^2 + 26*x^3 + 3*x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133073 a(n) = n^5 + n^3 + n^2.

Original entry on oeis.org

0, 3, 44, 279, 1104, 3275, 8028, 17199, 33344, 59859, 101100, 162503, 250704, 373659, 540764, 762975, 1052928, 1425059, 1895724, 2483319, 3208400, 4093803, 5164764, 6449039, 7977024, 9781875, 11899628, 14369319, 17233104, 20536379, 24327900, 28659903, 33588224, 39172419, 45475884
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2007

Keywords

Comments

Exponents are prime numbers in decreasing order.

Examples

			a(7) = 17199 because 7^5 = 16807, 7^3 = 343, 7^2 = 49 and we can write 16807 + 343 + 49 = 17199.

Links

Crossrefs

Cf. A000290, A000578, A000584, A011379, A045991, A100019.
Cf. A133070, A133071, A133072.

Programs

  • Magma
    [n^5+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    Total[#^{5,3,2}]&/@Range[0,40]  (* Harvey P. Dale, Jan 18 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,3,44,279,1104,3275},35] (* James C. McMahon, Mar 10 2025 *)
  • PARI
    for(n=0,50, print1(n^5 + n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

Formula

G.f.: x*(3 + 26*x + 60*x^2 + 30*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007
a(n) = n^2*(n^3 + n + 1). - Wesley Ivan Hurt, Mar 02 2023

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133061 a(n) = 5*p^5 - 3*p^3 - 2*p^2, where p = prime(n).

Original entry on oeis.org

128, 1116, 15200, 82908, 801020, 1849536, 7083968, 12359196, 32144156, 102480896, 143054460, 346565088, 579070880, 734799996, 1146409148, 2090525216, 3573998396, 4222293120, 6749714268, 9020062940, 10364180256, 15383790396, 19693474076, 27918166496, 42933944448, 52547391200
Offset: 1

Views

Author

Omar E. Pol, Nov 05 2007

Keywords

Examples

			a(4)=82908 because the 4th prime is 7, 5*7^5=84035, 3*7^3=1029, 2*7^2=98 and we can write 84035-1029-98=82908.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A133070. Prime numbers: A000040.

Programs

  • Magma
    [5*p^5-3*p^3-2*p^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010
  • Maple
    a:= n-> (p-> (5*p^3-3*p-2)*p^2)(ithprime(n)):
seq(a(n), n=1..26);  # Alois P. Heinz, Sep 23 2024
  • Mathematica
    Table[(Prime[n])^2*(5*Prime[n]^3 - 3*Prime[n] - 2), {n, 1, 50}] (* G. C. Greubel, Oct 09 2017 *)
  • PARI
    for(n=1,25, print1(5*prime(n)^5 - 3*prime(n)^3 - 2*prime(n)^2, ", ")) \\ G. C. Greubel, Oct 09 2017

Formula

a(n) = 5*(p(n))^5 - 3*(p(n))^3 - 2*(p(n))^2, where p(n)=A000040(n).

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A134630 a(n) = 5*n^5 - 3*n^3 - 2*n^2.

Original entry on oeis.org

0, 0, 128, 1116, 4896, 15200, 38160, 82908, 162176, 292896, 496800, 801020, 1238688, 1849536, 2680496, 3786300, 5230080, 7083968, 9429696, 12359196, 15975200, 20391840, 25735248, 32144156, 39770496, 48780000, 59352800, 71684028, 85984416, 102480896, 121417200, 143054460, 167671808, 195566976, 227056896
Offset: 0

Views

Author

Omar E. Pol, Nov 04 2007

Keywords

Comments

Coefficients and exponents are the first three prime numbers in decreasing order.

Examples

			a(4)=4896 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120-192-32=4896.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A100019, A133070.

Programs

  • Magma
    [5*n^5-3*n^3 -2*n^2: n in [0..50]]; // Vincenzo Librandi, Dec 14 2010
  • Maple
    A134630:=n->5*n^5 - 3*n^3 - 2*n^2; seq(A134630(n), n=0..50); # Wesley Ivan Hurt, May 21 2014
  • Mathematica
    CoefficientList[Series[4 x^2 (32 + 87 x + 30 x^2 + x^3)/(-1 + x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, May 21 2014 *)
Table[5n^5-3n^3-2n^2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,0,128,1116,4896,15200},40] (* Harvey P. Dale, Jun 01 2014 *)

Formula

a(n) = 5*n^5 - 3*n^3 - 2*n^2.
G.f.: 4*x^2*(32+87*x+30*x^2+x^3)/(-1+x)^6. - R. J. Mathar, Nov 14 2007
a(0)=0, a(1)=0, a(2)=128, a(3)=1116, a(4)=4896, a(5)=15200, a(n)= 6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jun 01 2014

Extensions

More terms from Vincenzo Librandi, Dec 14 2010
Showing 1-5 of 5 results.

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