A005904 -id:A005904 - cp's OEIS frontend

A104012 Indices of centered dodecahedral numbers (A005904) which are semiprimes (A001358).

1, 2, 3, 5, 6, 11, 14, 15, 21, 26, 30, 35, 36, 44, 54, 63, 69, 74, 81, 114, 128, 131, 135, 138, 153, 165, 168, 191, 194, 209, 216, 224, 228, 231, 239, 261, 270, 303, 315, 321, 323, 326, 330, 336, 345, 363, 380, 384, 398, 404, 410, 411, 414, 429, 440, 443, 455, 468, 470

Offset: 1

Jonathan Vos Post, Feb 24 2005

Because the cubic polynomial for centered dodecahedral numbers factors into n time an irreducible quadratic, the dodecahedral numbers can never be prime, but can be semiprime iff (2*n+1) is prime and (5*n^2+5*n+1) is prime. Centered dodecahedral numbers (A005904) are not to be confused with dodecahedral numbers (A006566) = n(3n-1)(3n-2)/2, nor with rhombic dodecahedral numbers (A005917).

Intersection of A005097 and A090563. - Michel Marcus, Apr 30 2016

			a(1) = 1 because A005904(1) = 33 = 3 * 11, which is semiprime.
a(2) = 2 because A005904(2) = 155 = 5 * 31, which is semiprime.
a(3) = 3 because A005904(3) = 427 = 7 * 61, which is semiprime.
a(4) = 5 because A005904(5) = 1661 = 11 * 151.
194 is in this sequence because A005904(194) = 73579739 = 389 * 189151, which is semiprime.

Zak Seidov, Table of n, a(n) for n = 1..1000
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.

Cf. A001222, A001358, A005904, A090563, A104011.

isok(n) = isprime(2*n+1) && isprime(5*n^2+5*n+1); \\ Michel Marcus, Apr 30 2016

n such that A001222(A005904(n)) = 2. n such that Bigomega((2*n+1)*(5*n^2 + 5*n + 1)) is 2. n such that A104011(n) = 2.

A104011 Number of prime factors (with multiplicity) of centered dodecahedral numbers (A005904).

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 2, 3, 3, 3, 4, 2, 4, 4, 2, 2, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 2, 4, 4, 3, 2, 6, 3, 3, 4, 2, 2, 5, 3, 3, 6, 3, 4, 3, 2, 4, 4, 4, 3, 4, 3, 3, 4, 3, 2, 3, 3, 4, 5, 4, 3, 3, 4, 2, 5, 3, 3, 7, 3, 2, 3, 3, 4, 4, 2, 3, 5, 4, 3, 3, 3, 2, 4, 3, 4, 4, 4, 4, 3, 4, 3, 4, 4, 3, 5, 3, 3, 6, 3, 3

Offset: 0

Jonathan Vos Post, Feb 24 2005

When a(n) = 2, n is a term of A104012: indices of centered dodecahedral numbers (A005904) which are semiprimes.

			a(9) = 3 because A005904(9) = 8569 = 11 * 19 * 41, which has 3 prime factors (which happen to have the same number of digits).
a(18) = 3 because A005904(18) = 63307 = 29 * 37 * 59.
a(96) = 3 because A005904(96) = 8986273 = 101 * 193 * 461.
a(126) = 5 because A005904(126) = 20242783 = 11 * 23 * 29 * 31 * 89, which has 5 prime factors (which happen to have the same number of digits).

Amiram Eldar, Table of n, a(n) for n = 0..10000
Boon K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558; author's copy.

Cf. A001222, A005904, A104012.

PrimeOmega[(2*n+1)*(5*n^2+5*n+1)] /. n -> Range[0, 99] (* Giovanni Resta, Jun 17 2016 *)

a(n) = A001222(A005904(n)).

a(n) = Bigomega((2*n+1)*(5*n^2 + 5*n + 1)).

A missing term inserted by Giovanni Resta, Jun 17 2016

A329658 Numbers that are sums of consecutive centered dodecahedral numbers (A005904).

Original entry on oeis.org

1, 33, 34, 155, 188, 189, 427, 582, 615, 616, 909, 1336, 1491, 1524, 1525, 1661, 2570, 2743, 2997, 3152, 3185, 3186, 4215, 4404, 5313, 5740, 5895, 5928, 5929, 6137, 6958, 8569, 8619, 9528, 9955, 10110, 10143, 10144, 10352, 11571, 13095, 14706, 14756, 15203, 15665, 16092

Offset: 1

Ilya Gutkovskiy, Nov 18 2019

nonn

Cf. A005902, A269237, A329599, A329634, A329635, A329636, A329650.

A362863 Centered hecatonicosachoral numbers.

Original entry on oeis.org

1, 1441, 11521, 44641, 122401, 273601, 534241, 947521, 1563841, 2440801, 3643201, 5243041, 7319521, 9959041, 13255201, 17308801, 22227841, 28127521, 35130241, 43365601, 52970401, 64088641, 76871521, 91477441, 108072001, 126828001, 147925441, 171551521, 197900641

Offset: 1

Léo Cymrot Cymbalista, May 06 2023

A hecatonicosachoral number is a centered figurate number that represents a hecatonicosachoron, which is a four-dimensional regular polytope composed of 120 cells.

One of the 6 centered regular polichoral (centered pentachoral, centered hexadecachoral, centered octachoral, centered icositetrachoral, centered hexacosichoral and centered hecatonicosachoral) numbers.

Eric Weisstein's World of Mathematics, Figurate Number.
Wikipedia, 120-cell.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005891 (2D), A005904 (3D), A006322, A151989.

Table[300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1, {n, 1, 100}]

a(n) = 300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1.
a(n) = 1440*A006322(n-1) + 1 for n > 1.
a(n) = 288*(A151989(n-1)-1)/25 + 1.
G.f.: x*(1 + 1436*x + 4326*x^2 + 1436*x^3 + x^4)/(1 - x)^5. - Stefano Spezia, May 12 2023

A269237 a(n) = (n + 1)^2(5n^2 + 10*n + 2)/2.

Original entry on oeis.org

1, 34, 189, 616, 1525, 3186, 5929, 10144, 16281, 24850, 36421, 51624, 71149, 95746, 126225, 163456, 208369, 261954, 325261, 399400, 485541, 584914, 698809, 828576, 975625, 1141426, 1327509, 1535464, 1766941, 2023650, 2307361, 2619904, 2963169, 3339106, 3749725, 4197096

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

Partial sums of centered dodecahedral numbers (A005904).

OEIS Wiki, Centered Platonic numbers
Eric Weisstein's World of Mathematics, Platonic Solid
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A005904, A006566, A008354, A014820, A037270, A132366.

[(n + 1)^2*(5*n^2 + 10*n + 2)/2 : n in [0..50]]; // Wesley Ivan Hurt, Oct 15 2017

A269237:=n->(n + 1)^2*(5*n^2 + 10*n + 2)/2: seq(A269237(n), n=0..50); # Wesley Ivan Hurt, Oct 15 2017

Table[(n + 1)^2 ((5 n^2 + 10 n + 2)/2), {n, 0, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 34, 189, 616, 1525}, 36]

x='x+O('x^99); Vec((1+29*x+29*x^2+x^3)/(1-x)^5) \\ Altug Alkan, Apr 10 2016

G.f.: (1 + 29*x + 29*x^2 + x^3)/(1 - x)^5.
E.g.f.: exp(x)*(2 + 66*x + 122*x^2 + 50*x^3 + 5*x^4)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
Sum_{n>=0} 1/a(n) = (5 - Pi^2 - sqrt(15)*Pi*cot(sqrt(3/5)*Pi))/9 = 1.0377796966... . - Vaclav Kotesovec, Apr 10 2016

A271532 a(n) = (-1)^n(n + 1)(5n^2 + 10n + 1).

Original entry on oeis.org

1, -32, 123, -304, 605, -1056, 1687, -2528, 3609, -4960, 6611, -8592, 10933, -13664, 16815, -20416, 24497, -29088, 34219, -39920, 46221, -53152, 60743, -69024, 78025, -87776, 98307, -109648, 121829, -134880, 148831, -163712, 179553, -196384, 214235, -233136, 253117, -274208, 296439

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

Alternating sum of centered dodecahedral numbers (A005904).

Without signs and up to offset, this is row 5 of the array A284873. - Andrey Zabolotskiy, Oct 10 2017

OEIS Wiki, Centered Platonic numbers
Eric Weisstein's World of Mathematics, Platonic Solid
Index entries for linear recurrences with constant coefficients, signature (-4,-6,-4,-1)

Cf. A000578, A004466, A005904, A006527, A005900, A006566.

Table[(-1)^n (n + 1) (5 n^2 + 10 n + 1), {n, 0, 38}]
LinearRecurrence[{-4, -6, -4, -1}, {1, -32, 123, -304}, 39]

a(n)=(-1)^n*(n+1)*(5*n^2+10*n+1) \\ Charles R Greathouse IV, Jul 26 2016

for n in range(0,10**3):print((-1)**n*(n+1)*(5*n**2+10*n+1)) # Soumil Mandal, Apr 10 2016

G.f.: (1 - 28*x + x^2)/(1 + x)^4.
E.g.f.: exp(-x)*(1 - 31*x + 30*x^2 - 5*x^3).
a(n) = -4*a(n-1) - 6*a(n-2) - 4*a(n-3) - a(n-4).

A284742 Centered Platonic numbers.

Original entry on oeis.org

1, 5, 7, 9, 13, 15, 25, 33, 35, 55, 63, 69, 91, 121, 129, 147, 155, 189, 195, 231, 295, 309, 341, 377, 425, 427, 559, 561, 575, 589, 791, 833, 855, 909, 923, 1035, 1159, 1241, 1325, 1415, 1561, 1661, 1665, 1729, 2047, 2057, 2059, 2331, 2511, 2625, 2743, 2869, 3025, 3059, 3303, 3605, 3871, 3925, 4089, 4215, 4255

Offset: 1

Ilya Gutkovskiy, Apr 01 2017

Union of centered tetrahedral numbers (A005894), centered octahedral numbers (A001845), centered cube numbers (A005898), centered icosahedral numbers (A005902) and centered dodecahedral numbers (A005904).

OEIS Wiki, Centered Platonic numbers [Unapproved page]

Cf. A001845, A005894, A005898, A005902, A005904, A053012.

nn = 18; t1 = Table[(2 n + 1) (n^2 + n + 3)/3, {n, 0, nn}]; t2 = Table[(2 n + 1) (2 n^2 + 2 n + 3)/3, {n, 0, nn}]; t3 = Table[n^3 + (n + 1)^3, {n, 0, nn}]; t4 = Table[(2 n + 1) (5 n^2 + 5 n + 3)/3, {n, 0, nn}]; t5 = Table[(2 n + 1) (5 n^2 + 5 n + 1), {n, 0,  nn}]; Select[Union[t1, t2, t3, t4, t5], # <= t1[[-1]] &]

cp's OEIS Frontend

A104012 Indices of centered dodecahedral numbers (A005904) which are semiprimes (A001358).

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A104011 Number of prime factors (with multiplicity) of centered dodecahedral numbers (A005904).

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A329658 Numbers that are sums of consecutive centered dodecahedral numbers (A005904).

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A362863 Centered hecatonicosachoral numbers.

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A269237 a(n) = (n + 1)^2*(5*n^2 + 10*n + 2)/2.

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A271532 a(n) = (-1)^n*(n + 1)*(5*n^2 + 10*n + 1).

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A284742 Centered Platonic numbers.

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Mathematica

A269237 a(n) = (n + 1)^2(5n^2 + 10*n + 2)/2.

A271532 a(n) = (-1)^n(n + 1)(5n^2 + 10n + 1).