A100009 -id:A100009 - cp's OEIS frontend

A104392 Sums of 2 distinct positive pentatope numbers (A000332).

6, 16, 20, 36, 40, 50, 71, 75, 85, 105, 127, 131, 141, 161, 196, 211, 215, 225, 245, 280, 331, 335, 336, 345, 365, 400, 456, 496, 500, 510, 530, 540, 565, 621, 705, 716, 720, 730, 750, 785, 825, 841, 925, 1002, 1006, 1016, 1036, 1045, 1071, 1127

Offset: 0

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102856.

nn=15; Select[Union[Total/@Subsets[Binomial[Range[4,nn],4],{2}]], #Harvey P. Dale, Mar 13 2011 *)

a(n) = Ptop(i) + Ptop(j) for some positive i=/=j and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104393 Sums of 3 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

21, 41, 51, 55, 76, 86, 90, 106, 110, 120, 132, 142, 146, 162, 166, 176, 197, 201, 211, 216, 226, 230, 231, 246, 250, 260, 281, 285, 295, 315, 336, 337, 341, 346, 350, 351, 366, 370, 371, 380, 401, 405, 406, 415, 435, 457, 461, 471, 491, 501

Offset: 0

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392.

Total/@Subsets[Table[Binomial[n+3,4],{n,10}],{3}]//Sort (* Harvey P. Dale, Feb 14 2018 *)

a(n) = Ptop(i) + Ptop(j) + Ptop(k) for some positive i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104394 Sums of 4 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

56, 91, 111, 121, 125, 147, 167, 177, 181, 202, 212, 216, 231, 232, 236, 246, 251, 261, 265, 286, 296, 300, 316, 320, 330, 342, 351, 352, 356, 371, 372, 376, 381, 385, 386, 406, 407, 411, 416, 420, 421, 436, 440, 441, 450, 462, 472, 476, 492, 496

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393.

a(n) = Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104395 Sums of 5 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

126, 182, 217, 237, 247, 251, 266, 301, 321, 331, 335, 357, 377, 386, 387, 391, 412, 421, 422, 426, 441, 442, 446, 451, 455, 456, 477, 497, 507, 511, 532, 542, 546, 551, 561, 562, 566, 576, 581, 586, 591, 595, 606, 616, 620, 626, 630, 642, 646, 650

Offset: 1

Jonathan Vos Post, Mar 05 2005

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Robert Israel, Table of n, a(n) for n = 1..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394.

N:= 1000: # for terms <= N
ptop:= n -> n*(n+1)*(n+2)*(n+3)/24:
P:= 1:
for i from 1 while ptop(i) < N do
  P:= P * (1 + x*y^ptop(i))
od:
sort(map(degree,convert(convert(series(coeff(P,x,5),y,N+1),polynom),list)));
# Robert Israel, Nov 20 2023

a(n) = Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104396 Sums of 6 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

252, 336, 392, 427, 447, 456, 457, 461, 512, 547, 567, 577, 581, 596, 621, 631, 651, 661, 665, 677, 687, 707, 712, 717, 721, 732, 742, 746, 752, 756, 761, 772, 776, 786, 796, 816, 826, 830, 841, 852, 872, 881, 882, 886, 897, 907, 916, 917, 921, 932

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24.

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394, A104395.

a(n) = Ptop(f) + Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive f=/=g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104397 Sums of 7 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

462, 582, 666, 722, 747, 757, 777, 787, 791, 831, 887, 922, 942, 951, 952, 956, 967, 1007, 1042, 1051, 1062, 1072, 1076, 1091, 1107, 1126, 1142, 1146, 1156, 1160, 1162, 1171, 1172, 1176, 1182, 1202, 1212, 1216, 1227, 1237, 1247, 1251, 1253, 1262, 1267

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24.

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394, A104395, A104396.

a(n) = Ptop(e) + Ptop(f) + Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive e=/=f=/=g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104398 Sums of 8 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

792, 957, 1077, 1161, 1177, 1217, 1252, 1272, 1282, 1286, 1297, 1381, 1437, 1462, 1463, 1472, 1492, 1502, 1506, 1546, 1583, 1602, 1637, 1657, 1666, 1667, 1671, 1722, 1723, 1748, 1757, 1758, 1777, 1778, 1787, 1788, 1791, 1792, 1806, 1827, 1832, 1841

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24.

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394, A104395, A104396, A104397.

Union[Total/@Subsets[Binomial[Range[4,15],4],{8}]] (* Harvey P. Dale, Mar 11 2012 *)

a(n) = Ptop(d) + Ptop(e) + Ptop(f) + Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive d=/=e=/=f=/=g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A100682 Floor of 4th root of pentatope numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31

Offset: 0

Jonathan Vos Post, Dec 06 2004

Conjecture: a(n) = floor((n - 3/2)/24^(1/4)) for n not in {0, 1, 6, 17, 2403, 5318}. - Charles R Greathouse IV, May 01 2012

			a(3) = 1 because floor((3*4*5*6/24)^(1/4)) = floor(15^(1/4)) = floor(1.96798967) = 1.

J. H. Conway and R. K. Guy, The Book of Numbers, pp. 55-57, Copernicus Press, NY, 1996.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number

Cf. A000332, A100009, A007501, A099179.

[Floor(Binomial(n+3, 4)^(1/4)): n in [3..70]]; // Vincenzo Librandi, Dec 14 2015

a:= n-> floor(binomial(n+3, 4)^(1/4)):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 14 2015

a(n)=binomial(n+3,4)^(1/4)\1 \\ Charles R Greathouse IV, May 01 2012

a(n)=sqrtnint(binomial(n+3,4),4) \\ Charles R Greathouse IV, Dec 14 2015

from math import comb
from sympy import integer_nthroot
def A100682(n): return integer_nthroot(comb(n+3,4),4)[0] # Chai Wah Wu, Oct 02 2024

a(n) = floor((A000332(n+3))^(1/4)) = floor(Ptop(n)^(1/4)) = floor(C(n+3, 4)^1/4) = floor((n * (n+1) * (n+2) * (n+3)/4!)^(1/4)).

a(n) = 0.4518... * n + O(1). - Charles R Greathouse IV, Dec 14 2015

A104399 Sums of 9 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

1287, 1507, 1672, 1792, 1793, 1876, 1932, 1958, 1967, 1987, 1997, 2001, 2078, 2157, 2162, 2178, 2218, 2253, 2273, 2283, 2287, 2298, 2322, 2382, 2438, 2442, 2463, 2473, 2493, 2503, 2507, 2526, 2542, 2547, 2582, 2603, 2612, 2617, 2637, 2638, 2647, 2651

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394, A104395, A104396, A104397, A104398.

a(n) = Ptop(c) + Ptop(d) + Ptop(e) + Ptop(f) + Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive c=/=d=/=e=/=f=/=g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104400 Sums of 10 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

2002, 2288, 2508, 2652, 2673, 2793, 2872, 2877, 2933, 2968, 2988, 2998, 3002, 3037, 3107, 3157, 3158, 3241, 3297, 3323, 3327, 3332, 3352, 3362, 3366, 3443, 3492, 3527, 3543, 3583, 3612, 3613, 3618, 3638, 3648, 3652, 3663, 3667, 3696, 3747, 3752, 3778

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24.

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers, but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Robert Israel, Table of n, a(n) for n = 1..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394, A104395, A104396, A104397, A104398, A104399.

N:= 10000: # to get all terms <= N
nmax:= floor(-3/2+1/2*sqrt(5+4*sqrt(1+24*N))):
S:= select(`<=`,{seq(add(s*(s+1)*(s+2)*(s+3)/24,s=c),
     c = combinat:-choose(nmax,10))},N):
sort(convert(S,list)); # Robert Israel, Dec 14 2015

a(n) = Ptop(b) + Ptop(c) + Ptop(d) + Ptop(e) + Ptop(f) + Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive b=/=c=/=d=/=e=/=f=/=g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

cp's OEIS Frontend

A104392 Sums of 2 distinct positive pentatope numbers (A000332).

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A104393 Sums of 3 distinct positive pentatope numbers (A000332).

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A104394 Sums of 4 distinct positive pentatope numbers (A000332).

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A104395 Sums of 5 distinct positive pentatope numbers (A000332).

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A104396 Sums of 6 distinct positive pentatope numbers (A000332).

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A104397 Sums of 7 distinct positive pentatope numbers (A000332).

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A104398 Sums of 8 distinct positive pentatope numbers (A000332).

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A100682 Floor of 4th root of pentatope numbers.