A000583 -id:A000583 - cp's OEIS frontend

A050468 a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.

1, 16, 80, 256, 626, 1280, 2400, 4096, 6481, 10016, 14640, 20480, 28562, 38400, 50080, 65536, 83522, 103696, 130320, 160256, 192000, 234240, 279840, 327680, 391251, 456992, 524960, 614400, 707282, 801280, 923520, 1048576, 1171200

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative because it is the Dirichlet convolution of A000583 = n^4 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005
Called E'_4(n) by Hardy.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = x + 16*x^2 + 80*x^3 + 256*x^4 + 626*x^5 + 1280*x^6 + 2400*x^7 + 4096*x^8 + ...

Emil Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 120.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135, section 9.3. MR0106147 (21 #4881)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Cf. A000122, A000583, A000700, A010054, A101455, A121373, A175571, A204342, A204372.

Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, this sequence, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

A := Basis( ModularForms( Gamma1(4), 5), 34); A[2] + 16*A[3]; /* Michael Somos, May 03 2015 */

edashed[r_,n_] := Plus@@(Select[Divisors[n], Mod[n/#,4] == 1 &]^r) - Plus@@(Select[Divisors[n], Mod[n/#,4] == 3 &]^r); edashed[4,#] &/@Range[33] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] (EllipticTheta[ 2, 0, x]^8 + 4 EllipticTheta[ 2, 0, x^2]^8) / 256, {x, 0, 2 n}]; (* Michael Somos, Jan 11 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(4*e+4) - s[p]^(e+1))/(p^4 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * (-1)^((n/d - 1)/2) * d^4))}; /* Michael Somos, Sep 12 2005 */

{a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, n\d)))}; /* Michael Somos, Jan 14 2012 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^4 * (eta(x + A)^4 + 20 * x * eta(x^4 + A)^8 / eta(x + A)^4), n))}; /* Michael Somos, Jan 14 2012 */

a(2*n + 1) = A204342(n). a(2*n) = 16 * a(n).
G.f.: Sum_{n>=1} n^4*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Michael Somos, Jan 14 2012: (Start)
Expansion of eta(q^2)^2 * eta(q^4)^4 * (eta(q)^4 + 20 * eta(q^4)^8 / eta(q)^4) in powers of q.
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4), a(p^e) = ((p^4)^(e+1) - (-1)^(e+1)) / (p^4 + 1) if p == 3 (mod 4). (End)
From Michael Somos, Jan 15 2012: (Start)
Expansion of theta_3(q^2) * (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.
Expansion of x * phi(x)^2 * (psi(x)^8 + 4 * x * psi(x^2)^8) in powers of x where phi(), psi() are Ramanujan theta functions. (End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^5 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204372. - Michael Somos, May 03 2015
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(4*e+4) - A101455(p)^(e+1))/(p^4 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/1536 (A175571). (End)
a(n) = Sum_{d|n} (n/d)^4*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A262941 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is an even square or twice a square.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 2, 3, 1, 3, 3, 6, 3, 4, 4, 4, 4, 3, 4, 2, 3, 3, 4, 3, 2, 5, 3, 4, 3, 6, 5, 6, 4, 2, 3, 2, 4, 4, 4, 5, 3, 3, 1, 3, 5, 6, 6, 4, 3, 3, 4, 1, 5, 4, 3, 4, 3, 4, 3, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 2, 4, 6, 3, 4, 6, 4, 5, 2, 7, 7, 4, 3, 3, 5, 4, 5, 6, 6, 5, 2, 6, 4, 8

Offset: 1

Zhi-Wei Sun, Oct 04 2015

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer n can be written as x^4 + 2^k*y^2 + z*(z+1)/2, where k is 1 or 2, and x,y,z are integers with z > 0.
This has been verified for n up to 2*10^6. We also guess that a(n) = 1 only for n = 1, 2, 13, 16, 50, 59, 239, 493, 1156, 1492, 1984, 3332.
See also A262944, A262945, A262954, A262955, A262956 for similar conjectures.

			a(1)    = 1 since    1 = 0^4 +    0^2 +  1*2/2  with  0 even.
a(2)    = 1 since    2 = 1^4 +    0^2 +  1*2/2  with  0 even.
a(13)   = 1 since   13 = 1^4 + 2* 1^2 +  4*5/2.
a(16)   = 1 since   16 = 1^4 +    0^2 +  5*6/2  with  0 even.
a(50)   = 1 since   50 = 1^4 +    2^2 +  9*10/2 with  2 even.
a(59)   = 1 since   59 = 0^4 +    2^2 + 10*11/2 with  2 even.
a(239)  = 1 since  239 = 0^4 + 2* 2^2 + 21*22/2 with  2 even.
a(493)  = 1 since  493 = 2^4 +   18^2 + 17*18/2 with 18 even.
a(1156) = 1 since 1156 = 1^4 + 2*24^2 +  2*3/2  with 24 even.
a(1492) = 1 since 1492 = 2^4 + 2* 7^2 + 52*53/2.
a(1984) = 1 since 1984 = 5^4 +   18^2 + 45*46/2 with 18 even.
a(3332) = 1 since 3332 = 5^4 +   52^2 +  2*3/2  with 52 even.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Zhi-Wei Sun, On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and aT_x+by^2+f(z), arXiv:1502.03056 [math.NT], 2015.

Cf. A000217, A000290, A000583, A254885, A262813, A262827, A262944, A262945, A262954, A262955, A262956.

SQ[n_]:=IntegerQ[Sqrt[n/2]]||IntegerQ[Sqrt[n/4]]
Do[r=0;Do[If[SQ[n-x^4-y(y+1)/2],r=r+1],{x,0,n^(1/4)},{y,1,(Sqrt[8(n-x^4)+1]-1)/2}];Print[n," ",r];Continue,{n,1,100}]

A270559 Number of ordered ways to write n as x^4 + x^3 + y^2 + z*(z+1)/2, where x, y and z are integers with x nonzero, y nonnegative and z positive.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 1, 3, 4, 2, 5, 2, 3, 4, 2, 3, 4, 5, 1, 4, 3, 3, 4, 3, 4, 5, 5, 3, 6, 5, 3, 3, 6, 2, 4, 6, 3, 9, 4, 2, 3, 4, 3, 7, 6, 3, 6, 2, 4, 2, 6, 5, 7, 6, 4, 5, 3, 6, 4, 11, 1, 5, 9, 3, 6, 5, 3, 8, 8

Offset: 1

Zhi-Wei Sun, Mar 18 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0. In other words, for each n = 1,2,3,... there are integers x and y such that n-(x^4+x^3+y^2) is a positive triangular number.
(ii) a(n) = 1 only for n = 1, 2, 8, 20, 62, 97, 296, 1493, 4283, 4346, 5433.
In contrast, the author conjectured in A262813 that any positive integer can be expressed as the sum of a nonnegative cube, a square and a positive triangular number.

			a(1) = 1 since 1 = (-1)^4 + (-1)^3 + 0^2 + 1*2/2.
a(2) = 1 since 2 = (-1)^4 + (-1)^3 + 1^2 + 1*2/2.
a(8) = 1 since 8 = 1^4 + 1^3 + 0^2 + 3*4/2.
a(20) = 1 since 20 = (-2)^4 + (-2)^3 + 3^2 + 2*3/2.
a(62) = 1 since 62 = (-2)^4 + (-2)^3 + 3^2 + 9*10/2.
a(97) = 1 since 97 = 1^4 + 1^3 + 2^2 + 13*14/2.
a(296) = 1 since 296 = (-4)^4 + (-4)^3 + 7^2 + 10*11/2.
a(1493) = 1 since 1493 = (-2)^4 + (-2)^3 + 0^2 + 54*55/2.
a(4283) = 1 since 4283 = (-6)^4 + (-6)^3 + 50^2 + 37*38/2.
a(4346) = 1 since 4346 = (-3)^4 + (-3)^3 + 49^2 + 61*62/2.
a(5433) = 1 since 5433 = (-8)^4 + (-8)^3 + 14^2 + 57*58/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000290, A000578, A000583, A262813, A262815, A262816, A262827, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270516, A270533.

TQ[n_]:=TQ[n]=n>0&&IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[x!=0&&TQ[n-y^2-x^4-x^3],r=r+1],{y,0,Sqrt[n]},{x,-1-Floor[(n-y^2)^(1/4)],(n-y^2)^(1/4)}];Print[n," ",r];Continue,{n,1,10000}]

A357633 Half-alternating sum of the partition having Heinz number n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 1, 4, 4, 5, 2, 6, 5, 5, 0, 7, 3, 8, 3, 6, 6, 9, 1, 6, 7, 2, 4, 10, 4, 11, 1, 7, 8, 7, 2, 12, 9, 8, 2, 13, 5, 14, 5, 3, 10, 15, 2, 8, 5, 9, 6, 16, 1, 8, 3, 10, 11, 17, 3, 18, 12, 4, 2, 9, 6, 19, 7, 11, 6, 20, 3, 21, 13, 4, 8, 9, 7, 22, 3, 0

Offset: 1

Gus Wiseman, Oct 09 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 + 3 - 3 - 2 = 2.

The original alternating sum is A316524, reverse A344616.
The version for standard compositions is A357622, non-reverse A357621.
The skew-alternating form is A357634, non-reverse A357630.
Positions of zeros are A000583, non-reverse A357631.
The reverse version is A357629.
These partitions are counted by A357637, skew A357638.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357623-A357626, A357632, A357636, A357640.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[halfats[Reverse[primeMS[n]]],{n,30}]

A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1

Offset: 2

Matthew Vandermast, Jun 03 2012

Length of row n equals A001221(n).
The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences.  Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
Table lists exponents in the order in which they appear in the prime factorization of a member of A025487.  This ordering is common in database comments (e.g., A008966).
Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

			First rows of table read:
  1;
  1;
  2;
  1;
  1,1;
  1;
  3;
  2;
  1,1;
  1;
  2,1;
  ...
The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.

Jason Kimberley, Table of i, a(i) for i = 2..24301 (n = 2..10000)

Cf. A025487, A001221 (row lengths), A001222 (row sums). A118914 gives the nondecreasing version. A124010 lists exponents in n's prime factorization in natural order, with A124010(1) = 0.
A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688.  Other sequences determined by the exponents in the prime factorization of n include:
Multiplicative: A000005, A007425, A008683, A008836, A034444, A037445, A181819.
Additive: A001221, A001222, A056169.
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012

apply( {A212171_row(n)=vecsort(factor(n)[,2]~,,4)}, [1..40])\\ M. F. Hasler, Apr 19 2022

Row n of A118914, reversed.

Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.

A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 49, 63, 64, 70, 81, 90, 100, 108, 121, 144, 154, 165, 169, 192, 196, 210, 225, 256, 273, 286, 289, 300, 324, 325, 360, 361, 400, 441, 442, 462, 480, 484, 525, 529, 550, 561, 576, 588, 595, 625, 646, 676, 700, 729, 741, 750, 784

Offset: 1

Gus Wiseman, Oct 09 2022

nonn

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   25: {3,3}
   30: {1,2,3}
   36: {1,1,2,2}
   49: {4,4}
   63: {2,2,4}
   64: {1,1,1,1,1,1}
   70: {1,3,4}
   81: {2,2,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}

The version for original alternating sum is A000290.
The half-alternating form is A000583, non-reverse A357631.
The version for standard compositions is A357628, non-reverse A357627.
The non-reverse version is A357632.
Positions of zeros in A357634, non-reverse A357630.
These partitions are counted by A357640, half A357639.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even A357642.
Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[1000],skats[Reverse[primeMS[#]]]==0&]

A079901 Triangle of powers, T(n,k) = n^k, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 27, 1, 4, 16, 64, 256, 1, 5, 25, 125, 625, 3125, 1, 6, 36, 216, 1296, 7776, 46656, 1, 7, 49, 343, 2401, 16807, 117649, 823543, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721

Offset: 0

Reinhard Zumkeller, Feb 21 2003

Matrix inverse equals the triangle R where R(n,k) = A107045(n,k)/A107046(n,k) are coefficients with exponential-like properties. - Paul D. Hanna, May 22 2005

			Triangle begins:
  1;
  1,1;
  1,2,4;
  1,3,9,27;
  1,4,16,64,256;
  1,5,25,125,625,3125;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A107045/A107046 (matrix inverse).

Cf. A002262, A000290, A000578, A000583, A000272, A000169, A000312.

a079901 n k = a079901_tabl !! n !! k
a079901_row n = a079901_tabl !! n
a079901_tabl = zipWith (map . (^)) [0..] a002262_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{1},Flatten[Table[n^k,{n,9},{k,0,n}]]] (* Harvey P. Dale, Feb 08 2013 *)

row(n) = vector(n+1, k, n^(k-1)); \\ Amiram Eldar, May 09 2025

T(n,k) = if k=0 then 1 else T(n,k-1)*n.
T(n,0) = 1; T(n,1) = n for n>0; T(n,2) = A000290(n) for n > 1; T(n,3) = A000578(n) for n > 2; T(n,4) = A000583(n) for n>3.
T(n,n-2) = A000272(n) for n>2; T(n,n-1) = A000169(n) for n>1; T(n,n) = A000312(n).

A000190 Number of solutions to x^4 == 0 (mod n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 16, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3

Offset: 1

N. J. A. Sloane

Shadow transform of fourth powers A000583. - Michel Marcus, Jun 06 2013

T. D. Noe, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.

Cf. A000583.

Array[ Function[ n, Count[ Array[ PowerMod[ #, 4, n ]&, n, 0 ], 0 ] ], 100 ]
f[p_, e_] := p^Floor[3*e/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],f[i,1]^(3*f[i,2]\4)) \\ Charles R Greathouse IV, Jun 07 2013

Multiplicative with a(p^e) = p^[3e/4]. - David W. Wilson, Aug 01 2001

Dirichlet g.f.: zeta(4*s-3) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) + 1/p^(3*s-2)). - Amiram Eldar, Dec 18 2023

A002593 a(n) = n^2(2n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.

Original entry on oeis.org

0, 1, 28, 153, 496, 1225, 2556, 4753, 8128, 13041, 19900, 29161, 41328, 56953, 76636, 101025, 130816, 166753, 209628, 260281, 319600, 388521, 468028, 559153, 662976, 780625, 913276, 1062153, 1228528, 1413721, 1619100, 1846081

Offset: 0

N. J. A. Sloane

The m-th term, for m = A065549(n), is perfect (A000396). - Lekraj Beedassy, Jun 04 2002
Partial sums of A016755. - Lekraj Beedassy, Jan 06 2004
Also, the k-th triangular number, where k = 2n^2 - 1 = A056220(n), i.e., a(n) = A000217(A056220(n)). - Lekraj Beedassy, Jun 11 2004
Also, the j-th hexagonal number, where j = n^2 = A000290(n), i.e., a(n) = A000384(A000290(n)) and a(n) = A056220(n) * A000290(n) or j * k. This sequence is a subsequence of the hexagonal number sequence and retains the aspect intrinsic to the hexagonal number sequence that each number in this sequence can be found by multiplying its triangular number by its hexagonal number. - Bruce J. Nicholson, Aug 22 2017
Odd numbers and their squares both having the form 2x-+1, we may write (2r+1)^3 = (2r+1)*(2s-1), where s = centered squares = (r+1)^2 + r^2. Since 2r+1 = (r+1)^2 - r^2, it follows immediately from summing telescopingly over n-1, the product 2*{(r+1)^4 - r^4} - {(r+1)^2 - r^2}, that Sum_{r=0..n-1} (2r+1)^3 = 2*n^4 - n^2 = n^2*(2n^2 - 1). - Lekraj Beedassy, Jun 16 2004
a(n) is also the starting term in the sum of a number M(n) of consecutive cubed integers equaling a squared integer (A253724) for M(n) equal to twice a squared integer (A001105). Numbers a(n) such that a^3 + (a+1)^3 + ... + (a+M-1)^3 = c^2 has nontrivial solutions over the integers for M equal to twice a squared integer (A001105). If M is twice a squared integer, there always exists at least one nontrivial solution for the sum of M consecutive cubed integers starting from a^3 and equaling a squared integer c^2. For n >= 1, M(n) = 2n^2 (A001105), a(n) = M(M-1)/2 = n^2(2n^2 - 1), and c(n) = sqrt(M/2) (M(M^2-1)/2) = n^3(4n^4 - 1). The trivial solutions with M < 1 and a < 2 are not considered. - Vladimir Pletser, Jan 10 2015
Binomial transform of the sequence with offset 1 is (1, 27, 98, 120, 48, 0, 0, 0, ...). - Gary W. Adamson, Jul 23 2015

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 169, #31.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955. [Annotated scans of just pages 742-743]
Neslihan Kilar, Abdelmejid Bayad, and Yilmaz Simsek, Finite sums involving trigonometric functions and special polynomials: analysis of generating functions and p-adic integrals, Appl. Anal. Disc. Math., hal-04535748, 2024. See p. 22.
Vladimir Pletser, File Triplets (M,a,c) for M=2n^2
Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.
G. Xiao, Sigma Server, Operate on "(2*n-1)^3".
M. J. Zerger, Proof without words: The sum of consecutive odd cubes is a triangular number, Math. Mag., 68 (1995), 371.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A000384, A000447, A000583, A002309, A253724, A253725, A260810.

[n^2*(2*n^2 - 1): n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

A002593:=-z*(z+1)*(z**2+22*z+1)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
a:= n-> n^2*(2*n^2-1): seq(a(n), n=0..50);  # Vladimir Pletser, Jan 10 2015

CoefficientList[Series[(-x^4-23x^3-23x^2-x)/(x-1)^5,{x,0, 80}],x]  (* or *)
Table[ n^2 (2n^2-1),{n,0,80}]  (* Harvey P. Dale, Mar 28 2011 *)
Join[{0},Accumulate[Range[1,91,2]^3]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,28,153,496},40] (* Harvey P. Dale, Mar 22 2017 *)

a(n) = n^2*(2*n^2 - 1) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A000217(A056220(n)). - Lekraj Beedassy, Jun 11 2004
G.f.: (-x^4 - 23*x^3 - 23*x^2 - x)/(x - 1)^5. - Harvey P. Dale, Mar 28 2011
a(n) = n^2*(2n^2 - 1). - Vladimir Pletser, Jan 10 2015
E.g.f.: exp(x)*x*(1 + 13*x + 24*x^2/2! + 12*x^3/3!). - Wolfdieter Lang, Mar 11 2017
a(n) = A000384(A000290(n)) = A056220(n) * A000290(n). - Bruce J. Nicholson, Aug 22 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 - Pi^2/6 - cot(Pi/sqrt(2))*Pi/sqrt(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = cosec(Pi/sqrt(2))*Pi/sqrt(2) - Pi^2/12 - 1. (End)

A011863 Nearest integer to (n/2)^4.

Original entry on oeis.org

0, 0, 1, 5, 16, 39, 81, 150, 256, 410, 625, 915, 1296, 1785, 2401, 3164, 4096, 5220, 6561, 8145, 10000, 12155, 14641, 17490, 20736, 24414, 28561, 33215, 38416, 44205, 50625, 57720, 65536, 74120, 83521, 93789, 104976, 117135, 130321, 144590

Offset: 0

R. K. Guy

First differences are in A019298.
The bisections are A000583 and A219086.
Number of ways to put n-1 copies of 1,2,3 into sets. [Zeilberger?]
s(n) is the number of 4-tuples (w,x,y,z) with all terms in {1,...,n} and |w-x| >= w + |y-z|; see A186707. - Clark Kimberling, May 24 2012

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 2000 terms from Vicenzo Librandi)
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (H_2 for square lattice of Section 6).
Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls?, The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger (2009); Local copy. [PDF file only, no active links]
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A000583, A002817, A019298, A106707, A212714, A219086.

[ (2*n^4-(1-(-1)^n))/32: n in [0..50] ];

Maple
```
seq(round((n/2)^4), n=0..40);
```

Round[(Range[40]/2)^4] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{0,1,5,16,39,81},40] (* Harvey P. Dale, Feb 07 2015 *)

a(n)=round((n/2)^4) \\ Charles R Greathouse IV, Jun 23 2011

G.f.: x^2*(1 + x + x^2)/((1 - x)^5*(1+x)).
a(n) = +4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6). - R. J. Mathar, Dec 07 2010
a(n)+a(n+1) = A002817(n). - R. J. Mathar, Dec 19 2008
a(n) = n^4/16 - 1/32 + (-1)^n/32 - R. J. Mathar, Dec 07 2010, adapted to added a(0) by Hugo Pfoertner, Dec 29 2019
a(n) = (2*A000583(n) + (-1)^n - 1)/32. - Bruno Berselli, Dec 07 2010, adapted to added a(0) by Hugo Pfoertner, Dec 29 2019
n*(n^2+n+2)*a(n+1) = 4*(n^2+2*n+2)*a(n)+(n+2)*(n^2+3*n+4)*a(n-1). Holonomic Ansatz with smallest order of recurrence. - Thotsaporn Thanatipanonda, Dec 12 2010
a(n) = floor(n^4/8)/2. - Gary Detlefs, Feb 19 2011, adapted to added a(0) by Hugo Pfoertner, Dec 29 2019
a(n) = A212714(n)/2, n >= 0. - Wolfdieter Lang, Oct 03 2016, adapted to added a(0) by Hugo Pfoertner, Dec 29 2019
E.g.f.: (1/32)*exp(-x)*(1 + exp(2*x)*(-1 + 2*x + 14*x^2 + 12*x^3 + 2*x^4)). - Stefano Spezia, Dec 29 2019
Sum_{n>=2} 1/a(n) = 6 + Pi^4/90 - 2*Pi*tanh(Pi/2). - Amiram Eldar, Aug 13 2022

Missing a(0) added by N. J. A. Sloane, Dec 29 2019. As a result some of the comments and formulas will need to be adjusted.

cp's OEIS Frontend

A050468 a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.

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A262941 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is an even square or twice a square.

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A270559 Number of ordered ways to write n as x^4 + x^3 + y^2 + z*(z+1)/2, where x, y and z are integers with x nonzero, y nonnegative and z positive.

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A357633 Half-alternating sum of the partition having Heinz number n.

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A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

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A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

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A079901 Triangle of powers, T(n,k) = n^k, 0 <= k <= n, read by rows.

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A002593 a(n) = n^2(2n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.