A005904 -id:A005904

A104012

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

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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.

FORMULA

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.

EXAMPLE

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.

PROG

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Feb 24 2005

STATUS

approved

A104011

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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.

FORMULA

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

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

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

Cf. A001222, A005904, A104012.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Feb 24 2005

EXTENSIONS

A missing term inserted by Giovanni Resta, Jun 17 2016

STATUS

approved

A329658

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..46.

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Nov 18 2019

STATUS

approved

A362863

Centered hecatonicosachoral numbers.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..29.

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).

FORMULA

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Léo Cymrot Cymbalista, May 06 2023

STATUS

approved

A269237

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Partial sums of centered dodecahedral numbers (A005904).

LINKS

Table of n, a(n) for n=0..35.

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)

FORMULA

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

MAPLE

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

MATHEMATICA

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]

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,easy,changed

AUTHOR

Ilya Gutkovskiy, Apr 09 2016

STATUS

approved

A271532

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..38.

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)

FORMULA

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).

MATHEMATICA

Table[(-1)^n (n + 1) (5 n^2 + 10 n + 1), {n, 0, 38}]

LinearRecurrence[{-4, -6, -4, -1}, {1, -32, 123, -304}, 39]

PROG

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

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

CROSSREFS

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

KEYWORD

sign,easy,changed

AUTHOR

Ilya Gutkovskiy, Apr 09 2016

STATUS

approved

A284742

Centered Platonic numbers.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..61.

OEIS Wiki, Centered Platonic numbers [Unapproved page]

MATHEMATICA

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]] &]

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Apr 01 2017

STATUS

approved