A244725 -id:A244725 - cp's OEIS frontend

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

A016911 a(n) = (6*n)^3.

Original entry on oeis.org

0, 216, 1728, 5832, 13824, 27000, 46656, 74088, 110592, 157464, 216000, 287496, 373248, 474552, 592704, 729000, 884736, 1061208, 1259712, 1481544, 1728000, 2000376, 2299968, 2628072, 2985984, 3375000, 3796416, 4251528, 4741632, 5268024, 5832000

Offset: 0

N. J. A. Sloane

Volume of a cube with side 6*n. - Wesley Ivan Hurt, Jul 05 2014

			a(1) = (6*1)^3 = 216.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A016923, A016935, A016947, A016959, A016971.

Cf. similar sequences listed in A244725.

[(6*n)^3: n in [0..40]]; // Vincenzo Librandi, May 03 2011

I:=[0,216,1728,5832]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 05 2014

A016911:=n->216*n^3: seq(A016911(n), n=0..40); # Wesley Ivan Hurt, Jul 05 2014

Table[216 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[216 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 05 2014 *)

G.f.: 216*x*(1 + 4*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jul 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Jul 05 2014
a(n) = 216 * A000578(n). - Wesley Ivan Hurt, Jul 05 2014
Sum_{n>=1} 1/a(n) = zeta(3)/216. - Amiram Eldar, Oct 02 2020

A244728 a(n) = 9*n^3.

Original entry on oeis.org

0, 9, 72, 243, 576, 1125, 1944, 3087, 4608, 6561, 9000, 11979, 15552, 19773, 24696, 30375, 36864, 44217, 52488, 61731, 72000, 83349, 95832, 109503, 124416, 140625, 158184, 177147, 197568, 219501, 243000, 268119, 294912, 323433, 353736, 385875, 419904

Offset: 0

Vincenzo Librandi, Jul 05 2014

Volume of a pyramid (square base) with side and height 3*n. - Wesley Ivan Hurt, Aug 25 2014

Volume of the smallest square cuboid containing a ring torus where the tube and hole diameters are both n. - Torlach Rush, Jun 04 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A244725.

Cf. A287335 (see Crossrefs).

List([0..40], n-> 9*n^3); # G. C. Greubel, Jun 30 2019

Magma
```
[9*n^3: n in [0..40]];
```

I:=[0,9,72,243]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

A244728:=n->9*n^3: seq(A244728(n), n=0..40); # Wesley Ivan Hurt, Aug 25 2014

Table[9n^3, {n,0,40}] (* or *) CoefficientList[Series[9*x*(1+4*x+x^2)/(1- x)^4, {x,0,40}], x]

vector(40, n, n--; 9*n^3) \\ G. C. Greubel, Jun 30 2019

[9*n^3 for n in (0..40)] # G. C. Greubel, Jun 30 2019

G.f.: 9*x*(1 + 4*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.
E.g.f.: 9*x*(1 + 3*x + x^2)*exp(x). - G. C. Greubel, Jun 30 2019

A244727 a(n) = 7*n^3.

Original entry on oeis.org

0, 7, 56, 189, 448, 875, 1512, 2401, 3584, 5103, 7000, 9317, 12096, 15379, 19208, 23625, 28672, 34391, 40824, 48013, 56000, 64827, 74536, 85169, 96768, 109375, 123032, 137781, 153664, 170723, 189000, 208537, 229376, 251559, 275128, 300125, 326592, 354571, 384104

Offset: 0

Vincenzo Librandi, Jul 05 2014

After 0, subsequence of A038855.

Volume of a truncated square pyramid with base lengths n and 2n, and height 3n. - Wesley Ivan Hurt, Apr 05 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A038855.

Cf. similar sequences listed in A244725.

Magma
```
[7*n^3: n in [0..40]];
```

I:=[0,7,56,189]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

A244727:=n->7*n^3: seq(A244727(n), n=0..100); # Wesley Ivan Hurt, Apr 05 2016

Table[7 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[7 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]
7 Range[0, 50]^3 (* Wesley Ivan Hurt, Apr 05 2016 *)
LinearRecurrence[{4,-6,4,-1},{0,7,56,189},40] (* Harvey P. Dale, Apr 04 2024 *)

G.f.: 7*x*(1 + 4*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.
E.g.f.: 7*exp(x)*x*(1 + 3*x + x^2). - Stefano Spezia, May 09 2023

A244729 a(n) = 10*n^3.

Original entry on oeis.org

0, 10, 80, 270, 640, 1250, 2160, 3430, 5120, 7290, 10000, 13310, 17280, 21970, 27440, 33750, 40960, 49130, 58320, 68590, 80000, 92610, 106480, 121670, 138240, 156250, 175760, 196830, 219520, 243890, 270000, 297910, 327680, 359370, 393040, 428750, 466560

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A244725.

Magma
```
[10*n^3: n in [0..40]];
```

I:=[0,10,80,270]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Table[10 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[10 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]

G.f.: 10*x*(1 + 4*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.

A016803 a(n) = (4*n)^3.

Original entry on oeis.org

0, 64, 512, 1728, 4096, 8000, 13824, 21952, 32768, 46656, 64000, 85184, 110592, 140608, 175616, 216000, 262144, 314432, 373248, 438976, 512000, 592704, 681472, 778688, 884736, 1000000, 1124864, 1259712, 1404928, 1560896, 1728000, 1906624, 2097152

Offset: 0

N. J. A. Sloane

Volume of a cube with side 4n. - Wesley Ivan Hurt, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578.

Cf. similar sequences listed in A244725.

[64*n^3: n in [0..40]]; // Vincenzo Librandi, Jul 05 2014

A016803:=n->64*n^3: seq(A016803(n), n=0..40); # Wesley Ivan Hurt, Jul 05 2014

Table[64 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[64 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 05 2014 *)

G.f.: 64*x*(1 + 4*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jul 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Jul 05 2014
a(n) = 64 * A000578(n). - Wesley Ivan Hurt, Jul 05 2014

A017523 a(n) = (12*n)^3.

Original entry on oeis.org

0, 1728, 13824, 46656, 110592, 216000, 373248, 592704, 884736, 1259712, 1728000, 2299968, 2985984, 3796416, 4741632, 5832000, 7077888, 8489664, 10077696, 11852352, 13824000, 16003008, 18399744, 21024576, 23887872, 27000000, 30371328, 34012224

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A244725.

[1728*n^3: n in [0..40]]; // Vincenzo Librandi, Jul 05 2014

I:=[0,1728,13824, 46656]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 05 2014

Table[1728 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[1728 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 05 2014 *)

makelist((12*n)^3,n,0,30); /* Martin Ettl, Oct 21 2012 */

G.f.: 1728*x*(1 + 4*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jul 05 2014

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Jul 05 2014

A244726 a(n) = 6*n^3.

Original entry on oeis.org

0, 6, 48, 162, 384, 750, 1296, 2058, 3072, 4374, 6000, 7986, 10368, 13182, 16464, 20250, 24576, 29478, 34992, 41154, 48000, 55566, 63888, 73002, 82944, 93750, 105456, 118098, 131712, 146334, 162000, 178746, 196608, 215622, 235824, 257250, 279936, 303918

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A244725.

Magma
```
[6*n^3: n in [0..40]];
```

I:=[0,6,48,162]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Table[6 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[6 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]

G.f.: 6*x*(1 + 4*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.

A016983 a(n) = (7*n)^3.

Original entry on oeis.org

0, 343, 2744, 9261, 21952, 42875, 74088, 117649, 175616, 250047, 343000, 456533, 592704, 753571, 941192, 1157625, 1404928, 1685159, 2000376, 2352637, 2744000, 3176523, 3652264, 4173281, 4741632, 5359375, 6028568, 6751269, 7529536, 8365427, 9261000, 10218313

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A244725.

[(7*n)^3: n in [0..40]]; // Vincenzo Librandi, May 22 2011

Table[343 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[343 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 05 2014 *)

G.f.: 343*x*(1 + 4*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jul 05 2014

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Vincenzo Librandi, Jul 05 2014

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

cp's OEIS Frontend

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

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A016911 a(n) = (6*n)^3.

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A244728 a(n) = 9*n^3.

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A244727 a(n) = 7*n^3.

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A244729 a(n) = 10*n^3.

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Magma

Magma

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A016803 a(n) = (4*n)^3.

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A017523 a(n) = (12*n)^3.

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