A100752 -id:A100752 - cp's OEIS frontend

A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n(7n+5)/2.

0, 6, 19, 39, 66, 100, 141, 189, 244, 306, 375, 451, 534, 624, 721, 825, 936, 1054, 1179, 1311, 1450, 1596, 1749, 1909, 2076, 2250, 2431, 2619, 2814, 3016, 3225, 3441, 3664, 3894, 4131, 4375, 4626, 4884, 5149, 5421, 5700, 5986, 6279, 6579, 6886

Offset: 0

Bruno Berselli, Jan 13 2011

This sequence is a bisection of A118277 (even part).
Sequence found by reading the line from 0, in the direction 0, 19... and the line from 6, in the direction 6, 39,..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - Omar E. Pol, Jul 24 2012
The early part of this sequence is a strikingly close approximation to the early part of A100752. - Peter Munn, Nov 14 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. second k-gonal numbers: A005449 (k=5), A014105 (k=6), A147875 (k=7), A045944 (k=8), this sequence (k=9), A033954 (k=10), A062728 (k=11), A135705 (k=12).

Cf. A001106, A022264, A022265, A024966, A055998, A100752, A174738, A186029, A218471.

[n*(7*n+5)/2: n in [0..50]]; // Bruno Berselli, Sep 23 2016

I:=[0, 6, 19]; [n le 3 select I[n] else 3*Self(n-1) -3*Self(n-2) +Self(n-3): n in [1..60]]; // Vincenzo Librandi, Oct 15 2012

f[n_] := n (7 n + 5)/2; f[Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3, -3, 1}, {0, 6, 19}, 60] (* or *) Array[(#(7# + 5))/2&, 60, 0] (* Harvey P. Dale, Aug 19 2011 *)
CoefficientList[Series[x (6 + x)/(1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 15 2012 *)

a(n)=n*(7*n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(6 + x)/(1 - x)^3.
a(n) = Sum_{i=0..(n-1)} A017053(i) for n>0.
a(-n) = A001106(n).
Sum_{i=0..n} (a(n)+i)^2 = ( Sum_{i=(n+1)..2*n} (a(n)+i)^2 ) + 21*A000217(n)^2 for n>0.
a(n) = a(n-1)+7*n-1 for n>0, with a(0)=0. - Vincenzo Librandi, Feb 05 2011
a(0)=0, a(1)=6, a(2)=19; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Aug 19 2011
a(n) = A174738(7n+5). - Philippe Deléham, Mar 26 2013
a(n) = A001477(n) + 2*A000290(n) + 3*A000217(n). - J. M. Bergot, Apr 25 2014
a(n) = A055998(4*n) - A055998(3*n). - Bruno Berselli, Sep 23 2016
E.g.f.: (x/2)*(12 + 7*x)*exp(x). - G. C. Greubel, Aug 19 2017

A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 4, 1, 7, 7, 1, 9, 20, 10, 1, 10, 34, 40, 14, 1, 10, 46, 86, 67, 17, 1, 10, 55, 141, 175, 101, 20, 1, 10, 62, 192, 338, 313, 142, 24, 1, 10, 67, 242, 522, 694, 507, 190, 27, 1, 10, 72, 287, 733, 1197, 1273, 768, 244, 30

Offset: 1

L. Edson Jeffery, Jan 07 2015

Or a(n,k) = the number of positive integers less than or equal to 10^k that are divisible by no prime exceeding prime(n).

			Array begins:
{1,  4,  7,  10,   14,   17,    20,    24,    27,     30, ...}
{1,  7, 20,  40,   67,  101,   142,   190,   244,    306, ...}
{1,  9, 34,  86,  175,  313,   507,   768,  1105,   1530, ...}
{1, 10, 46, 141,  338,  694,  1273,  2155,  3427,   5194, ...}
{1, 10, 55, 192,  522, 1197,  2432,  4520,  7838,  12867, ...}
{1, 10, 62, 242,  733, 1848,  4106,  8289, 15519,  27365, ...}
{1, 10, 67, 287,  945, 2579,  6179, 13389, 26809,  50351, ...}
{1, 10, 72, 331, 1169, 3419,  8751, 20198, 42950,  85411, ...}
{1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, ...}
{1, 10, 79, 402, 1581, 5158, 14697, 37627, 88415, 193571, ...}

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A108277 (rows 1-9).

Cf. A011557, A066343, A253572, A253573.

r = 10; y[1] = t = Table[2^j, {j, 0, 39}]; max = 10^13; len = 10^10; prev = 0; For[n = 2, n <= r, n++, next = 0; For[k = 1, k <= 43, k++, If[Prime[n]^k < max, t = Union[t, Prime[n]*t]; s = FirstPosition[t, v_ /; v > len, 0]; t = Take[t, s[[1]] - 1]; If[t[[-1]] > len, t = Delete[t, -1]]; next = Length[t]; If[next == prev, Break, prev = next], Break]]; y[n] = t]; b[i_, j_] := FirstPosition[y[i], v_ /; v > 10^j][[1]]; a253635[n_, j_] := If[IntegerQ[b[n, j]], b[n, j] - 1, 0]; Flatten[Table[a253635[n - j, j], {n, r}, {j, 0, n - 1}]] (* array antidiagonals flattened *)

A108276 Number of positive integers <= 10^n that are divisible by no prime exceeding 19.

Original entry on oeis.org

1, 10, 72, 331, 1169, 3419, 8751, 20198, 42950, 85411, 160626, 288126, 496303, 825326, 1330766, 2088013, 3197529, 4791093, 7039193, 10159603, 14427309, 20186026, 27861175, 37974797, 51162295, 68191379, 89983125, 117635672

Offset: 0

Robert G. Wilson v, May 31 2005

nonn

Row 8 of A253635.

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108277, A011557, A080682.

n = 9; t = Select[ Flatten[ Table[19^h*Select[ Flatten[ Table[17^g*Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &], {h, 0, n*Log[19, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

a(10)-a(18) from Donovan Johnson, Sep 16 2009

a(19)-a(27) from Max Alekseyev, Apr 28 2010

A108277 Number of positive integers <= 10^n that are divisible by no prime exceeding 23.

Original entry on oeis.org

1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, 263529, 495412, 892644, 1550012, 2605342, 4254753, 6771752, 10531080, 16038303, 23965659, 35195450, 50872227, 72464493, 101837746, 141340075, 193902062, 263152095, 353549942

Offset: 0

Robert G. Wilson v, May 31 2005

nonn

Row 9 of A253635.

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A011557, A080683.

n = 6; t = Select[ Flatten[ Table[23^i*Select[ Flatten[ Table[19^h*Select[ Flatten[ Table[17^g*Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &], {h, 0, n*Log[19, 10]}]], # <= 10^n &], {i, 0, n*Log[23, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

a(7)-a(18) from Donovan Johnson, Sep 16 2009

a(19)-a(27) from Max Alekseyev, Apr 28 2010

cp's OEIS Frontend

A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n(7n+5)/2.

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A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

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A108276 Number of positive integers <= 10^n that are divisible by no prime exceeding 19.

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A108277 Number of positive integers <= 10^n that are divisible by no prime exceeding 23.

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cp's OEIS Frontend

A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n*(7*n+5)/2.

Views

Author

Keywords

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Programs

Magma

Magma

Mathematica

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Formula

A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A108276 Number of positive integers <= 10^n that are divisible by no prime exceeding 19.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A108277 Number of positive integers <= 10^n that are divisible by no prime exceeding 23.

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Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n(7n+5)/2.