William A. Tedeschi - cp's OEIS frontend

A180446 Number of non-pentagonal numbers <= n.

0, 0, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 65, 66

Offset: 0

William A. Tedeschi, Sep 07 2010

			a(5) = 5 - floor((sqrt(24*5+1)+1)/6) = 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

f[n_] := n - Floor[(Sqrt[24 n + 1] + 1)/6]; Array[f, 74, 0] (* Robert G. Wilson v, Sep 10 2010 *)
Accumulate[Table[If[IntegerQ[(1+Sqrt[1+24n])/6],0,1],{n,0,80}]]-1 (* Harvey P. Dale, May 22 2023 *)

l = [n-floor((sqrt(24*n+1)+1)/6) for n in range(0,101)]

from math import isqrt
def A180446(n): return n-(m:=isqrt((k:=n<<1)//3))-(k>m*(3*m+5)) # Chai Wah Wu, Nov 04 2024

a(n) = n - floor((sqrt(24n+1)+1)/6) = n - A180447(n).

More terms from Robert G. Wilson v, Sep 10 2010

A180447 n appears 3n+1 times.

Original entry on oeis.org

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

Offset: 0

William A. Tedeschi, Sep 07 2010

			a(5) = floor((sqrt(24*5+1)+1)/6) = 2.

Kevin Ryde, Table of n, a(n) for n = 0..10000

Cf. A000326 (indices of run starts), A016655, A180446.

f[n_] := Floor[(Sqrt[24 n + 1] + 1)/6]; Array[f, 105, 0] (* Robert G. Wilson v, Sep 10 2010 *)

a(n) = (sqrtint(24*n+1)+1)\6; \\ Kevin Ryde, Apr 21 2021

l = [floor((sqrt(24*n+1)+1)/6) for n in range(0,101)]

from math import isqrt
def A180447(n): return (m:=isqrt((k:=n<<1)//3))+(k>m*(3*m+5)) # Chai Wah Wu, Nov 04 2024

a(n) = floor((sqrt(24n+1)+1)/6).
a(n) = m+1 if 2n>m*(3m+5) and a(n) = m otherwise where m = floor(sqrt(2n/3)). For n>0, a(n) = k+1 if 2n>=(k+1)(3k+2) and a(n) = k otherwise where k = floor(sqrt(2(n-1)/3)). - Chai Wah Wu, Nov 04 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 (= A016655 / 10). - Amiram Eldar, Jun 30 2025

More terms from Robert G. Wilson v, Sep 10 2010

A180448 Primes of the form floor(k^sqrt(2)).

Original entry on oeis.org

2, 7, 29, 37, 41, 59, 79, 89, 197, 281, 311, 431, 491, 571, 607, 617, 673, 683, 751, 997, 1019, 1051, 1117, 1217, 1229, 1297, 1321, 1367, 1571, 1583, 1607, 1657, 1669, 1871, 1949, 2309, 2447, 2503, 2531, 2687, 2903, 3413, 3613, 3739, 3881, 3929, 4057, 4073, 4219

Offset: 1

William A. Tedeschi, Sep 07 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Select[Floor[Range[0,500]^Sqrt[2]],PrimeQ] (* Harvey P. Dale, May 20 2011 *)

for(n=1, 154392, if(ispseudoprime(t=floor(n^sqrt(2))), print1(t", "))); v \\ Charles R Greathouse IV, Feb 18 2011

A134886 INTERSECT A000040.

Formula rewritten by R. J. Mathar, Sep 09 2010

A180449 Primes of the form floor( (k(sqrt(3)k-1))/sqrt(2) ).

Original entry on oeis.org

3, 167, 197, 577, 631, 809, 1009, 1231, 1741, 1931, 2029, 2339, 3533, 4079, 7207, 10301, 11933, 14741, 17551, 18743, 24943, 26003, 32027, 37813, 42239, 45013, 49831, 51827, 54377, 61843, 76369, 81973, 122849, 128339, 130729, 145531, 154097, 171047, 172883

Offset: 1

William A. Tedeschi, Sep 07 2010, typo in definition corrected Sep 09 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Select[With[{c3=Sqrt[3],c2=Sqrt[2]},Table[Floor[n (c3 n-1)/c2], {n,500}]],PrimeQ] (* Harvey P. Dale, May 05 2011 *)

for(n=1, 1e4, if(ispseudoprime(t=floor((n*(sqrt(3)*n-1))/sqrt(2))), print1(t", "))); v \\ Charles R Greathouse IV, Feb 18 2011

A180450 Primes of the form floor( (k^sqrt(2) + k)/sqrt(2) ).

Original entry on oeis.org

3, 5, 7, 13, 19, 43, 67, 71, 89, 103, 107, 127, 137, 163, 191, 311, 317, 337, 383, 397, 431, 547, 569, 577, 599, 607, 653, 661, 677, 701, 709, 733, 757, 823, 857, 977, 1021, 1039, 1129, 1193, 1249, 1277, 1381, 1459, 1699, 1709, 1823, 1949, 2099, 2131, 2153, 2521, 2647

Offset: 1

William A. Tedeschi, Sep 07 2010

Intersection of A000040 with the sequence 1, 3, 5, 7, 10, 13, 16, 19, 22, 25, 28, 32, 35, ... defined by the floor function.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

select(isprime,[seq(floor((n^sqrt(2)+n)/sqrt(2)),n=1..350)]); # Muniru A Asiru, Sep 29 2018

Select[With[{b = Sqrt[2]}, Table[Floor[(n^b + n)/b], {n, 500}]], PrimeQ] (* G. C. Greubel, Sep 29 2018 *)

for(n=1, 148438, if(ispseudoprime(t=floor((n^sqrt(2)+n)/sqrt(2))), print1(t", "))); v \\ Charles R Greathouse IV, Feb 18 2011

Formula replaced by a comment - R. J. Mathar, Sep 09 2010

A180451 Primes of the form floor(2*k^sqrt(2) - k).

Original entry on oeis.org

3, 19, 29, 41, 101, 109, 127, 281, 293, 353, 509, 523, 743, 821, 853, 919, 1019, 1193, 1229, 1283, 1301, 2503, 2683, 2797, 2843, 3221, 3637, 3863, 4093, 4327, 4889, 4943, 5897, 6451, 6481, 6659, 6689, 6719, 6779, 6869, 6899, 7541, 7603, 7759, 7853, 8423

Offset: 1

William A. Tedeschi, Sep 07 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Select[Table[Floor[2n^Sqrt[2] -n],{n,1000}],PrimeQ] (* Harvey P. Dale, Jun 23 2013 *)

for(n=1, 1e4, if(ispseudoprime(t=floor(2*n^sqrt(2)-n)), print1(t", "))); v \\ Charles R Greathouse IV, Feb 18 2011

Sequence renamed by R. J. Mathar, Sep 09 2010

A180452 Primes of the form floor(k^sqrt(Pi)).

Original entry on oeis.org

3, 7, 11, 17, 23, 31, 59, 107, 151, 167, 239, 367, 439, 491, 601, 631, 691, 919, 1063, 2309, 2909, 3083, 3823, 4019, 4219, 4423, 5431, 5507, 5659, 5813, 6047, 7451, 9551, 10037, 10837, 10939, 14071, 16673, 17291, 19073, 19993, 21067, 21613, 25163

Offset: 1

William A. Tedeschi, Sep 07 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A134707 INTERSECT A000040.

Select[Floor[Range[350]^Sqrt[Pi]],PrimeQ] (* Harvey P. Dale, Apr 23 2022 *)

for(n=1, 1e4, if(ispseudoprime(t=floor(n^sqrt(Pi))), print1(t", "))); v \\ Charles R Greathouse IV, Feb 18 2011

Formula rewritten by R. J. Mathar, Sep 09 2010

A137936 a(n) = 5*mod(n,5) + floor(n/5).

Original entry on oeis.org

0, 5, 10, 15, 20, 1, 6, 11, 16, 21, 2, 7, 12, 17, 22, 3, 8, 13, 18, 23, 4, 9, 14, 19, 24, 5, 10, 15, 20, 25, 6, 11, 16, 21, 26, 7, 12, 17, 22, 27, 8, 13, 18, 23, 28, 9, 14, 19, 24, 29, 10, 15, 20, 25, 30, 11, 16, 21, 26, 31, 12, 17, 22, 27, 32, 13, 18, 23, 28, 33, 14, 19, 24, 29, 34

Offset: 0

William A. Tedeschi, Mar 06 2008

			a(0) = 5*mod(0,5) + floor(0/5) = 0
a(3) = 5*mod(3,5) + floor(3/5) = 15

Cf. A010874, A002266.

Python
```
a = lambda n: 5*(n%5) + floor(n/5)
```

a(n) = 5*mod(n,5) + floor(n/5) = 5*A010874(n) + A002266(n)

O.g.f.: -x(-5x^3+19x^4-5x^2-5x-5)/[(-1+x)^2*(x^3+x^4+x^2+x+1)] . - R. J. Mathar, Mar 07 2008

A137797 a(n) = 2( (n+1) mod 5 ) - 2( (n+1) mod 2 ).

Original entry on oeis.org

0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6, 0, 0, 4, 4, 8, -2, 2, 2, 6, 6

Offset: 0

William A. Tedeschi, Feb 10 2008

The sequence is periodic with period 10. - Colin Barker, Dec 16 2014

			a(2) = 2*((2+1) mod 5) - 2*((2+1) mod 2) = 2*(3 mod 5) - 2*(3 mod 2) = 4.

Index entries for linear recurrences with constant coefficients, signature (-1,0,0,0,1,1).

Suggested by A010700. Used in A137798.

LinearRecurrence[{-1,0,0,0,1,1},{0,4,4,8,-2,2},100] (* Harvey P. Dale, Jun 08 2015 *)

concat(0, Vec(-2*x*(3*x^3+6*x^2+4*x+2)/((x-1)*(x+1)*(x^4+x^3+x^2+x+1)) + O(x^100))) \\ Colin Barker, Dec 16 2014

a(n) = -a(n-1)+a(n-5)+a(n-6) for n>5. - Colin Barker, Dec 16 2014

G.f.: -2*x*(3*x^3+6*x^2+4*x+2) / ((x-1)*(x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Dec 16 2014

A137935 a(n) = 5n + 26*floor(n/5).

Original entry on oeis.org

0, 5, 10, 15, 20, 51, 56, 61, 66, 71, 102, 107, 112, 117, 122, 153, 158, 163, 168, 173, 204, 209, 214, 219, 224, 255, 260, 265, 270, 275, 306, 311, 316, 321, 326, 357, 362, 367, 372, 377, 408, 413, 418, 423, 428, 459, 464, 469, 474, 479, 510, 515, 520, 525, 530, 561, 566

Offset: 0

William A. Tedeschi, Mar 06 2008

			a(0) = 5(0) + 26*floor(0/5) = 0
a(3) = 5(3) + 26*floor(3/5) = 15

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A002266.

seq(5*n + 26*floor(n/5), n=0..200); # Robert Israel, Apr 02 2017

Python
```
a = lambda n: 5*n + 26*floor(n/5)
```

a(n) = 5n + 26*floor(n/5) = 5n + 26*A002266(n)

G.f.: (5*x+5*x^2+5*x^3+5*x^4+31*x^5)/(1-x-x^5+x^6). - Robert Israel, Apr 02 2017

cp's OEIS Frontend

User: William A. Tedeschi

A180446 Number of non-pentagonal numbers <= n.

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A180447 n appears 3n+1 times.

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A180448 Primes of the form floor(k^sqrt(2)).

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A180449 Primes of the form floor( (k*(sqrt(3)*k-1))/sqrt(2) ).

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A180450 Primes of the form floor( (k^sqrt(2) + k)/sqrt(2) ).

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A180451 Primes of the form floor(2*k^sqrt(2) - k).

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A180452 Primes of the form floor(k^sqrt(Pi)).

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A137936 a(n) = 5*mod(n,5) + floor(n/5).

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A180449 Primes of the form floor( (k(sqrt(3)k-1))/sqrt(2) ).

A137797 a(n) = 2( (n+1) mod 5 ) - 2( (n+1) mod 2 ).