A000567 -id:A000567 - cp's OEIS frontend

A259167 Positive octagonal numbers (A000567) that are squares (A000290) divided by 2.

8, 78408, 752875208, 7229107670408, 69413891098384008, 666512175097575576008, 6399849835873029582446408, 61451357457540654953074835208, 590055927907455532986394985222408, 5665716958316030570194709695030728008, 54402213643694597627554069505290065112008

Offset: 1

Colin Barker, Jun 19 2015

Intersection of A000567 and A001105. - Michel Marcus, Jun 20 2015

			8 is in the sequence because 8 is the 2nd octagonal number, and 2*8 is the 4th square.

Colin Barker, Table of n, a(n) for n = 1..251
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Cf. A000290, A000567, A001105, A259156-A259166.

I:=[8, 78408, 752875208]; [n le 3 select I[n] else 9603*Self(n-1)-9603*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 20 2015

LinearRecurrence[{9603, -9603, 1}, {8, 78408, 752875208}, 20] (* Vincenzo Librandi, Jun 20 2015 *)

Vec(-8*x*(x^2+198*x+1)/((x-1)*(x^2-9602*x+1)) + O(x^20))

G.f.: -8*x*(x^2+198*x+1) / ((x-1)*(x^2-9602*x+1)).

A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

Original entry on oeis.org

1045, 2465, 2821, 15841, 20501, 34133, 51221, 68101, 89441, 116033, 118405, 162401, 170885, 216545, 300833, 364705, 439301, 472033, 530881, 642181, 687365, 746005, 970145, 976981, 997633, 1104133, 1148245, 1193221, 1231361, 1239061, 1398101, 1654661, 1971541

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

2465 is also a Carmichael number (A002997).
2821 is also a primary Carmichael number (A324316).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019

			A324315(4) = 1045 = 5 * 11 * 19 = 19 * (3 * 19 - 2) = A000567(19), so 1045 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A000567, A002997, A324315, A324316, A324317, A324318, A324319, A324369, A324370, A324371, A324404, A324405, A324973.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
ON[n_] := n(3n - 2);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[ON@ Prime[Range[100]], TestS[#] &]

More terms from Amiram Eldar, Dec 05 2020

A244645 Decimal expansion of the sum of the reciprocals of the octagonal numbers (A000567).

Original entry on oeis.org

1, 2, 7, 7, 4, 0, 9, 0, 5, 7, 5, 5, 9, 6, 3, 6, 7, 3, 1, 1, 9, 4, 9, 5, 3, 4, 9, 2, 1, 0, 2, 4, 3, 3, 2, 1, 1, 5, 5, 6, 6, 3, 4, 4, 8, 0, 3, 9, 0, 2, 4, 7, 2, 3, 2, 6, 9, 3, 4, 9, 1, 9, 8, 4, 0, 7, 5, 1, 5, 1, 5, 1, 5, 1, 9, 5, 5, 4, 5, 1, 9, 6, 0, 7, 6, 2, 4, 3, 0, 6, 3, 1, 6, 3, 3, 1, 4, 1, 0, 8, 8, 0, 5, 0, 3

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.2774090575596367311949534921024332115566344803902472326934919840751515151955452...

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Wikipedia, Polygonal number

Cf. A000038, A013661, A244639, A244644, A244646, A244647, A244648, A244649, A000567.

RealDigits[ Sum[1/(3n^2 - 2n), {n, 1 , Infinity}], 10, 111][[1]]

sumpos(n=1, 1/(3*n^2 - 2*n)) \\ Michel Marcus, Sep 12 2016

sumnumrat(1/(3*n-2)/n,1) \\ Charles R Greathouse IV, Feb 08 2023

Equals Sum_{n>=1} 1/(3*n^2 - 2*n).

Equals Pi/(4*sqrt(3)) + 3*log(3)/4. - Vaclav Kotesovec, Jul 05 2014

A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

Original entry on oeis.org

0, 1, 8, 9, 21, 29, 30, 40, 61, 65, 69, 70, 96, 105, 126, 133, 134, 135, 161, 176, 201, 222, 225, 229, 230, 231, 280, 294, 309, 334, 341, 355, 363, 364, 401, 405, 408, 470, 481, 505, 510, 531, 534, 539, 540, 560, 621, 630, 645, 681, 695, 735, 736, 749, 756, 764, 765, 814, 833, 846

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A002414, A034705, A034706, A319184, A319185, A322636.

N:= 1000: # for terms up to N
Octa:= [seq(n*(3*n-2),n=0..floor((1+sqrt(1+3*N))/3))]:
PS:= ListTools:-PartialSums(Octa):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025

terms = 60;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(3n-2), {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A344376 Numbers that are both octagonal numbers (A000567) and octagonal pyramidal numbers (A002414).

Original entry on oeis.org

0, 1, 1045, 5985, 123395663059845, 774611255177760

Offset: 1

Seiichi Manyama, May 16 2021

Terms correspond to integral points on an elliptic curve, which allows all of them to be found efficiently. - Max Alekseyev, Feb 19 2024

Intersection of A000567 and A002414.
Cf. A027568, A334012.
Cf. A378361 (octagonal indices), A378918 (octagonal pyramidal indices).

for(k=0, 1e5, if(ispolygonal(m=k*(k+1)*(2*k-1)/2, 8), print1(m", ")))

Keywords 'fini' and 'full' added by Max Alekseyev, Feb 19 2024

A258128 Octagonal numbers (A000567) that are the sum of two consecutive octagonal numbers.

Original entry on oeis.org

5461, 813281, 7272157205, 1083057360705, 9684433559760981, 1442322650052752161, 12896895753596262561301, 1920761265591267733640321, 17174976631595008767000306005, 2557904668044167195987033355105, 22872156829955018609383449248248341

Offset: 1

Colin Barker, May 21 2015

			5461 is in the sequence because Oct(43) = 5461 = 2640 + 2821 = Oct(30) + Oct(31).

Colin Barker, Table of n, a(n) for n = 1..326
Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).

Cf. A000567, A258129, A258130, A258131, A258132.

CoefficientList[Series[-x*(x^4 +20*x^3 -1146230*x^2 +807820*x +5461)/((x-1)*(x^2 -1154*x +1)*(x^2 +1154*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1,1331714,-1331714,-1,1},{5461,813281,7272157205,1083057360705,9684433559760981},20] (* Harvey P. Dale, Feb 19 2018 *)

Vec(-x*(x^4 +20*x^3 -1146230*x^2 +807820*x +5461)/((x-1)*(x^2 -1154*x +1)*(x^2 +1154*x +1)) + O('x^20))

G.f.: -x*(x^4+20*x^3-1146230*x^2+807820*x+5461) / ((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)).

A258129 Octagonal numbers (A000567) that are the sum of three consecutive octagonal numbers.

Original entry on oeis.org

698901, 5102520783381, 37252493940331837461, 271973082264557457061125141, 1985621622943208359132836202790421, 14496630316026749501691464257547633057301, 105837027604506739193825102426073141683789429781, 772695182809023513889440668692977953487035688873891861

Offset: 1

Colin Barker, May 21 2015

			698901 is in the sequence because Oct(483) = 698901 = 231296 + 232965 + 234640 = Oct(278) + Oct(279) + Oct(280).

Colin Barker, Table of n, a(n) for n = 1..145
Index entries for linear recurrences with constant coefficients, signature (7300803,-7300803,1).

Cf. A000567, A258128, A258130, A258131, A258132.

CoefficientList[Series[-21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{7300803,-7300803,1},{698901,5102520783381,37252493940331837461},20] (* Harvey P. Dale, Sep 16 2018 *)

Vec(-21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)) + O('x^20))

G.f.: -21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)).

A258130 Octagonal numbers (A000567) that are the sum of ten consecutive octagonal numbers.

Original entry on oeis.org

1045, 1325345, 1910970885, 2755618515265, 3973599987865685, 5729928426883626945, 8262552817966202013445, 11914595433578836419585185, 17180838352667864150839647765, 24774756989951626526674352316385, 35725182398671892783600265200403845

Offset: 1

Colin Barker, May 21 2015

			1045 is in the sequence because Oct(19) = 1045 = 1+8+21+40+65+96+133+176+225+280 = Oct(1) + ... + Oct(10).

Colin Barker, Table of n, a(n) for n = 1..316
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Cf. A000567, A258128, A258129, A258131, A258132.

CoefficientList[Series[-95 x (63 x^2 - 1922 x + 11)/((x - 1) (x^2 - 1442 x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1443,-1443,1},{1045,1325345,1910970885},20] (* Harvey P. Dale, Jul 25 2019 *)

Vec(-95*x*(63*x^2-1922*x+11)/((x-1)*(x^2-1442*x+1)) + O('x^20))

G.f.: -95*x*(63*x^2 - 1922*x + 11)/((x - 1)*(x^2 - 1442*x + 1)).

A258131 Octagonal numbers (A000567) that are the sum of eleven consecutive octagonal numbers.

Original entry on oeis.org

49665, 348161, 19701781, 138502485, 7841194625, 55123576321, 3120775694421, 21939044808725, 1242060885120385, 8731684710231681, 494337111502154261, 3475188575627335765, 196744928316972210945, 1383116321414969338241, 78303987133043437737301

Offset: 1

Colin Barker, May 21 2015

			49665 is in the sequence because Oct(129) = 49665 = 3400 + 3605 + 3816 + 4033 + 4256 + 4485 + 4720 + 4961 + 5208 + 5461 + 5720 = Oct(34) + ... + Oct(44).

Colin Barker, Table of n, a(n) for n = 1..766
Index entries for linear recurrences with constant coefficients, signature (1,398,-398,-1,1).

Cf. A000567, A258128, A258129, A258130, A258132.

CoefficientList[Series[-11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
Select[Total/@Partition[Table[n(3n-2),{n,5*10^6}],11,1],IntegerQ[(Sqrt[1+3#]+1)/3]&] (* Harvey P. Dale, Aug 31 2018 *)

Vec(-11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)) + O('x^20))

G.f.: -11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)).

A258132 Octagonal numbers (A000567) that are the sum of fifteen consecutive octagonal numbers.

Original entry on oeis.org

4715040, 8463840, 1122749669280, 2015496399840, 267373851637578960, 479974343542849680, 63672943775553479639280, 114302050117965712710960, 15163202176330482896520455040, 27220118818712616412771202880, 3610995292612020914167620625112640

Offset: 1

Colin Barker, May 21 2015

			4715040 is in the sequence because Oct(1254) = 4715040 = 300833 + 302736 + 304645 + 306560 + 308481 + 310408 + 312341 + 314280 + 316225 + 318176 + 320133 + 322096 + 324065 + 326040 + 328021 = Oct(317) + ... + Oct(331).

Colin Barker, Table of n, a(n) for n = 1..370
Index entries for linear recurrences with constant coefficients, signature (1,238142,-238142,-1,1).

Cf. A000567, A258128, A258129, A258130, A258131.

CoefficientList[Series[-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)

Vec(-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)) + O('x^20))

G.f.: -240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)).

cp's OEIS Frontend

A259167 Positive octagonal numbers (A000567) that are squares (A000290) divided by 2.

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A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

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A244645 Decimal expansion of the sum of the reciprocals of the octagonal numbers (A000567).

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A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

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A344376 Numbers that are both octagonal numbers (A000567) and octagonal pyramidal numbers (A002414).

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A258128 Octagonal numbers (A000567) that are the sum of two consecutive octagonal numbers.

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A258129 Octagonal numbers (A000567) that are the sum of three consecutive octagonal numbers.

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A258130 Octagonal numbers (A000567) that are the sum of ten consecutive octagonal numbers.

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