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.

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A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

Keywords

Comments

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

Examples

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

Links

Crossrefs

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

Programs

  • Mathematica
    lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Extensions

Edited by R. J. Mathar, Aug 22 2009
Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A063497 Number of atoms in first n shells of type I hyperfullerene.

Original entry on oeis.org

0, 60, 300, 840, 1800, 3300, 5460, 8400, 12240, 17100, 23100, 30360, 39000, 49140, 60900, 74400, 89760, 107100, 126540, 148200, 172200, 198660, 227700, 259440, 294000, 331500, 372060, 415800, 462840, 513300, 567300, 624960, 686400, 751740, 821100, 894600
Offset: 0

Views

Author

N. J. A. Sloane, Aug 01 2001

Keywords

Comments

The third derivative of the sum of 5th powers: A000539. - Gregory R. Bryant, Jun 14 2013

References

  • T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (14).

Links

Crossrefs

Cf. A000539.

Programs

  • Magma
    [20*n^3 + 30*n^2 + 10*n : n in [0..50]]; // Wesley Ivan Hurt, May 03 2021
  • Mathematica
    Table[20n^3 + 30n^2 + 10n, {n, 0, 50}] (* David Nacin, Mar 01 2012 *)
  • PARI
    a(n) = { 20*n^3 + 30*n^2 + 10*n } \\ Harry J. Smith, Aug 23 2009

Formula

a(n) = 20*n^3 + 30*n^2 + 10*n.
G.f.: 60*x * (1 + x)/(1 - x)^4. - Colin Barker, Mar 01 2012

A081175 Numbers of the form Sum_{i=1..k} i^j, j >= 1, k >= 1.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 14, 15, 17, 21, 28, 30, 33, 36, 45, 55, 65, 66, 78, 91, 98, 100, 105, 120, 129, 136, 140, 153, 171, 190, 204, 210, 225, 231, 253, 257, 276, 285, 300, 325, 351, 354, 378, 385, 406, 435, 441, 465, 496, 506, 513, 528, 561, 595, 630, 650, 666, 703
Offset: 1

Views

Author

N. J. A. Sloane, Apr 18 2003

Keywords

Comments

Union of sums of k-th powers, for k >= 1.

Examples

			30 is in the set because 30 = 1^2 + 2^2 + 3^2 + 4^2 (j=2, k=4).

Links

Crossrefs

Cf. A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.
For primes in this sequence see A164307.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
A:=select(`<=`,{1, seq(seq(sum(i^k,i=1..m), m=2..floor((N*(k+1))^(1/(k+1)))),k = 1 ..ilog2(N-1))},N):
sort(convert(A,list)); # Robert Israel, Jan 26 2015
  • Mathematica
    Take[ Union[ Flatten[ Table[ Sum[ i^j, {i, 1, n}], {j, 1, 9}, {n, 1, 40}]]], 60]

Extensions

Corrected and extended by Robert G. Wilson v, May 08 2003

A132336 Sum of the integers from 1 to n, excluding perfect fifth powers.

Original entry on oeis.org

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 495, 528, 562, 597, 633, 670, 708, 747, 787, 828, 870, 913, 957, 1002, 1048, 1095, 1143, 1192, 1242, 1293, 1345, 1398, 1452
Offset: 1

Views

Author

Cino Hilliard, Nov 07 2007

Keywords

Examples

			a(1)=0+1, excluding 0 and 1, so a(1)=0.
a(2)=0+1+2, excluding 0 and 1, so a(2)=2.
a(3)=0+1+2+3, excluding 0 and 1, so a(3)=2+3=5.

Links

Crossrefs

Cf. A000217, A000539, A178487.
Different from A000096.
Cf. A132337.

Programs

  • Maple
    A000217 := proc(n) n*(n+1)/2 ; end proc:
A000539 := proc(n) (2*n^6+6*n^5+5*n^4-n^2)/12 ; end proc:
A132336 := proc(n) r := floor(n^(1/5)) ; A000217(n)-A000539(r); end proc: seq(A132336(n),n=1..40) ;
  • PARI
    g5(n)=for(x=1, n, r=floor(x^(1/5)); sum5=(2*r^6+6*r^5+5*r^4-r^2)/12; sn=x* (x+1)/2; print1(sn-sum5, ", "))
  • PARI
    a(n) = my(r=sqrtnint(n,5)); n*(n+1)/2 - (2*r^6+6*r^5+5*r^4-r^2)/12; \\ Ruud H.G. van Tol, Nov 02 2023
  • Python
    from sympy import integer_nthroot
def A132336(n): return n*(n+1)-(m:=integer_nthroot(n,5)[0])**2*(m**2*(m*(m+3<<1)+5)-1)//6>>1 # Chai Wah Wu, Jun 06 2025

Formula

a(n) = A000217(n) - A000539(r) where r = floor(n^(1/5)).
a(n) = n(n+1)/2 - (2r^6 + 6r^5 + 5r^4 - r^2)/12.
a(n) = A000217(n) - A000539(r) where r= A178487(n). - R. J. Mathar, Oct 12 2010

Extensions

Edited by the Assoc. Editors of the OEIS, Oct 12 2010. Thanks to Daniel Mondot for pointing out that the sequence needed editing.

A135095 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^2 if n is even.

Original entry on oeis.org

1, 5, 248, 264, 3389, 3425, 20232, 20296, 79345, 79445, 240496, 240640, 611933, 612129, 1371504, 1371760, 2791617, 2791941, 5268040, 5268440, 9352541, 9353025, 15789368, 15789944, 25555569, 25556245, 39905152, 39905936, 60417085
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/24)*(3 + 2*n + 5*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 2*n^6 - 3*(-1)^n*(1 + n* (-2 - 7*n + 5*n^3 + 2*n^4))): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 5; s = 2; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
Table[(1/24)*(3 + 2*n + 5*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 2*n^6 - 3*(-1)^n*(1 + n* (-2 - 7*n + 5*n^3 + 2*n^4))), {n,1,50}] (* G. C. Greubel, Sep 23 2016 *)
  • PARI
    for(n=1,50, print1((1/24)*(3 + 2*n + 5*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 2*n^6 - 3*(-1)^n*(1 + n* (-2 - 7*n + 5*n^3 + 2*n^4))), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(x^8 - 4*x^7 + 236*x^6 + 12*x^5 + 1446*x^4 - 12*x^3 + 236*x^2 + 4*x + 1)*(x^2 + 1)/( (1+x)^6 * (x-1)^7 ). - R. J. Mathar, Feb 22 2009
E.g.f.: (1/24)*( (-3 - 6*x - 17*x^2 + 240*x^3 - 75*x^4 + 6*x^5)*exp(x) + (3 + 24*x + 204*x^2 + 364*x^3 + 195*x^4 + 36*x^5 + 2*x^6)*exp(x) ). - G. C. Greubel, Sep 23 2016

A135099 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^3 if n is even.

Original entry on oeis.org

1, 9, 252, 316, 3441, 3657, 20464, 20976, 80025, 81025, 242076, 243804, 615097, 617841, 1377216, 1381312, 2801169, 2807001, 5283100, 5291100, 9375201, 9385849, 15822192, 15836016, 25601641, 25619217, 39968124, 39990076, 60501225
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/48)*(9*(1-(-1)^n) +4*n^2*(n+1)^2*(n^2 +n+1) -6*(-1)^n*n^2*(n + 2)*(2*n^2 +n-4)): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 5; s = 3; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a
Table[(1/48)*(9*(1 - (-1)^n) + 4*n^2*(n + 1)^2*(n^2 + n + 1) - 6*(-1)^n*n^2*(n + 2)*(2*n^2 + n - 4)), {n, 1, 50}] (* G. C. Greubel, Sep 23 2016 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^5,a+(n+1)^3]}; NestList[nxt,{1,1},30][[All,2]] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,9,252,316,3441,3657,20464,20976,80025,81025,242076,243804,615097},30] (* Harvey P. Dale, Oct 02 2022 *)
  • PARI
    for(n=1,50, print1((1/48)*(9*(1-(-1)^n) +4*n^2*(n+1)^2*(n^2 +n +1) -6*(-1)^n*n^2*(n+2)*(2*n^2 +n-4)), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(1 + 8*x + 237*x^2 + 16*x^3 + 1682*x^4 - 48*x^5 + 1682*x^6 + 16*x^7 + 237*x^8 + 8*x^9 + x^ 10)/((1+x)^6 * (x-1)^7). - R. J. Mathar, Feb 22 2009
E.g.f.: (1/48)*( (-9 - 18*x - 306*x^2 + 468*x^3 - 150*x^4 + 12*x^5)*exp(-x) + (9 + 48*x + 456*x^2 + 768*x^3 + 396*x^4 + 72*x^5 + 4*x^6)*exp(x) ). - G. C. Greubel, Sep 23 2016

A135214 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^4 if n is even.

Original entry on oeis.org

1, 17, 260, 516, 3641, 4937, 21744, 25840, 84889, 94889, 255940, 276676, 647969, 686385, 1445760, 1511296, 2931153, 3036129, 5512228, 5672228, 9756329, 9990585, 16426928, 16758704, 26524329, 26981305, 41330212, 41944868, 62456017
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6))) // G. C. Greubel, Jul 04 2018
  • Mathematica
    a = {}; r = 5; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a
LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {1, 17, 260, 516, 3641, 4937, 21744, 25840, 84889, 94889, 255940, 276676, 647969}, 50] (* G. C. Greubel, Oct 04 2016 *)
  • PARI
    x='x+O('x^50); Vec(x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6)) \\ G. C. Greubel, Jul 04 2018

Formula

From R. J. Mathar, May 17 2008: (Start)
O.g.f.: x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6).
a(2*n-1) = n*(-8 + 80*n^2 + 48*n^4 + 80*n^5 + 35*n - 220*n^3)/15.
a(2*n) = n*(-8 + 80*n^2 + 48*n^4 + 80*n^5 + 35*n + 20*n^3)/15 . (End)

A135301 a(1)=1, a(n)=a(n-1)+n^0 if n odd, a(n)=a(n-1)+ n^2 if n is even.

Original entry on oeis.org

1, 5, 6, 22, 23, 59, 60, 124, 125, 225, 226, 370, 371, 567, 568, 824, 825, 1149, 1150, 1550, 1551, 2035, 2036, 2612, 2613, 3289, 3290, 4074, 4075, 4975, 4976, 6000, 6001, 7157, 7158, 8454, 8455, 9899, 9900, 11500, 11501, 13265, 13266, 15202, 15203
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 0; s = 2; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+1,a+(n+1)^2]}; Transpose[NestList[nxt,{1,1},50]][[2]] (* or *) LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,5,6,22,23,59,60},50] (* Harvey P. Dale, Jul 16 2014 *)

Formula

O.g.f.: x*(x^4+4*x^3-2*x^2+4*x+1)/((-1+x)^4*(1+x)^3) . a(2n-1) = 4*n^3/3-2*n^2+5*n/3, a(2n) = 4*n^3/3+2*n^2+5*n/3. - R. J. Mathar, May 17 2008
a(1)=1, a(2)=5, a(3)=6, a(4)=22, a(5)=23, a(6)=59, a(7)=60, a(n)=a(n-1)+ 3*a(n-2)- 3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a (n-7). - Harvey P. Dale, Jul 16 2014
a(n) = ( (2*n+1)*(n^2+n+3)+3*(n^2+n-1)*(-1)^n )/12. - Luce ETIENNE, Jul 26 2014

A135332 a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^3 if n is even.

Original entry on oeis.org

1, 9, 10, 74, 75, 291, 292, 804, 805, 1805, 1806, 3534, 3535, 6279, 6280, 10376, 10377, 16209, 16210, 24210, 24211, 34859, 34860, 48684, 48685, 66261, 66262, 88214, 88215, 115215, 115216, 147984, 147985, 187289, 187290, 233946, 233947, 288819
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,9,10,74,75,291,292,804,805},40] (* Harvey P. Dale, Nov 28 2014 *)

Formula

From R. J. Mathar, Feb 22 2009: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
G.f.: x*(1 + 8*x - 3*x^2 + 32*x^3 + 3*x^4 +8*x^5 -x^6)/((1+x)^4*(1-x)^5). (End)

A140142 a(1)=1, a(n)=a(n-1)+n^0 if n odd, a(n)=a(n-1)+ n^4 if n is even.

Original entry on oeis.org

1, 17, 18, 274, 275, 1571, 1572, 5668, 5669, 15669, 15670, 36406, 36407, 74823, 74824, 140360, 140361, 245337, 245338, 405338, 405339, 639595, 639596, 971372, 971373, 1428349, 1428350, 2043006, 2043007, 2853007, 2853008, 3901584, 3901585
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Maple
    a:= n-> (Matrix([[275,274,18,17, 1,0,0,-1,-17, -18,-274]]). Matrix(11, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,5,-5,-10,10,10, -10,-5,5,1,-1][i] else 0 fi)^n)[1,6]: seq(a(n), n=1..33); # Alois P. Heinz, Aug 06 2008
  • Mathematica
    a = {}; r = 0; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
nxt[{n_,a_}]:={n+1,If[OddQ[n+1],a+1,a+(n+1)^4]}; Transpose[ NestList[ nxt,{1,1},40]][[2]] (* Harvey P. Dale, Dec 24 2012 *)

Formula

O.g.f.: x*(x^8+16*x^7-4*x^6+176*x^5+6*x^4+176*x^3-4*x^2+16*x+1)/((-1+x)^6*(1+x)^5) - R. J. Mathar, May 17 2008
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