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.

A036660 Product of prime p with sum of next p consecutive primes.

Original entry on oeis.org

16, 69, 335, 1001, 3883, 6851, 15385, 22553, 40273, 80765, 101897, 173567, 239563, 283327, 373509, 538321, 746999, 841007, 1119905, 1344811, 1483725, 1887231, 2211369, 2738619, 3540597, 4025961, 4319511, 4877167, 5210309
Offset: 1

Views

Author

G. L. Honaker, Jr., Dec 15 1998

Keywords

Examples

			p=2 -> 2*(3+5)=16.
p=7 -> 7*(11+13+17+19+23+29+31)=1001.

Crossrefs

Cf. A036659.

Programs

  • PARI
    a(n) = prime(n)*sum(i=1, prime(n), prime(n+i)); \\ Michel Marcus, Jan 06 2014

Extensions

Offset corrected to 1 by Michel Marcus, Jan 06 2014

A037050 Numbers n such that product of n with sum of next n consecutive integers is palindromic.

Original entry on oeis.org

1, 121, 187, 14014
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Comments

Also, number n such that n^2*(3*n+1)/2 is palindromic.
No additional terms below 10^10. - Jens Voß, Feb 20 2009
a(5) > 3.7*10^12, if it exists. - Giovanni Resta, Aug 26 2019

Examples

			187 * (188+189+...+373+374) = 187 * 52547 = palindrome 9826289.

Crossrefs

Cf. A036659, A037051.

Programs

  • Maple
    A036659 := proc(n) n^2*(3*n+1)/2 ; end: isA002113 := proc(n) local b10,i ; b10 := convert(n,base,10) ; for i from 1 to nops(b10)/2 do if op(i,b10) <> op(-i,b10) then RETURN(false) ; fi ; od ; RETURN(true) ; end: for n from 1 to 1000000 do c := A036659(n) : if isA002113(c) then print(n) ; fi ; od : # R. J. Mathar, Jun 26 2007

A037051 Palindromic product of some n with sum of next n consecutive integers.

Original entry on oeis.org

2, 2664662, 9826289, 4128458548214
Offset: 1

Views

Author

Patrick De Geest, Jan 04 1999

Keywords

Comments

Values of n see A037050.
a(5) > 7.5*10^37, if it exists. - Giovanni Resta, Aug 26 2019

Examples

			2664662 = 121 * (122 + 123 + ... + 241 + 242) = 121 * 22022.

Crossrefs

Cf. A036659, A037050.

A087887 a(n) = 18n^3 + 6n^2.

Original entry on oeis.org

0, 24, 168, 540, 1248, 2400, 4104, 6468, 9600, 13608, 18600, 24684, 31968, 40560, 50568, 62100, 75264, 90168, 106920, 125628, 146400, 169344, 194568, 222180, 252288, 285000, 320424, 358668, 399840, 444048, 491400, 542004, 595968, 653400, 714408, 779100, 847584
Offset: 0

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Oct 13 2003

Keywords

Comments

Another parametric representation of the solutions of the Diophantine equation x^2 - y^2 = z^3 is (x,y,z) = (15n^3, 3n^3, 6n^2), thus getting a(n) = 18n^3 + 6n^2.

Crossrefs

Cf. A036659, A085409, A085482.

Programs

  • Mathematica
    a[n_] := 18*n^3 + 6*n^2; Array[a, 50, 0] (* Amiram Eldar, Jan 10 2023 *)

Formula

O.g.f.: 12x(2+6x+x^2)/(-1+x)^4. a(n) = 12*A036659(n). - R. J. Mathar, Apr 07 2008
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/36 + sqrt(3)*Pi/12 + 3*log(3)/4 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/72 - sqrt(3)*Pi/6 - log(2) + 3/2. (End)

Extensions

More terms from Ray Chandler, Nov 06 2003

A294315 a(n) = 3*n^3 + n^2.

Original entry on oeis.org

0, 4, 28, 90, 208, 400, 684, 1078, 1600, 2268, 3100, 4114, 5328, 6760, 8428, 10350, 12544, 15028, 17820, 20938, 24400, 28224, 32428, 37030, 42048, 47500, 53404, 59778, 66640, 74008, 81900, 90334, 99328, 108900, 119068, 129850, 141264, 153328, 166060, 179478
Offset: 0

Views

Author

Jason Morgan, Oct 28 2017

Keywords

Comments

All terms are even.

Examples

			a(3)=90 because 3*3^3 + 3^2 = 3*27 + 9 = 90.

Links

Crossrefs

Cf. A117642, A000290, A036659.

Programs

  • GAP
    A294315:=List([0..10^4],n -> 3 *n^3 + n^2 ); # Muniru A Asiru, Dec 11 2017
  • Mathematica
    Array[3 #^3 + #^2 &, 40, 0] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 28, 90}, 40] (* or *)
CoefficientList[Series[2 x (2 + 6 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Dec 12 2017 *)
  • PARI
    a(n) = 3*n^3 + n^2;
  • PARI
    concat(0, Vec(2*x*(2 + 6*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 11 2017

Formula

a(n) = 3*n^3 + n^2.
a(n) = A117642(n) + A000290(n).
a(n) = 2*A036659(n).
From Colin Barker, Dec 11 2017: (Start)
G.f.: 2*x*(2 + 6*x + x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + sqrt(3)*Pi/2 + 9*log(3)/2 - 9.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - sqrt(3)*Pi - 6*log(2) + 9. (End)
Showing 1-5 of 5 results.

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