A016945 -id:A016945 - cp's OEIS frontend

A016950 a(n) = (6*n + 3)^6.

729, 531441, 11390625, 85766121, 387420489, 1291467969, 3518743761, 8303765625, 17596287801, 34296447249, 62523502209, 107918163081, 177978515625, 282429536481, 433626201009, 646990183449, 941480149401, 1340095640625, 1870414552161, 2565164201769, 3462825991689

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A016945, A016946, A016947, A016948, A016949.

Subsequence of A001014.

[(6*n+3)^6: n in [0..40]]; // Vincenzo Librandi, May 05 2011

(3 + 6Range[0, 20])^6 (* Harvey P. Dale, Jan 31 2011 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^6 = A016946(n)^3 = A016947(n)^2.
a(n) = 3^6*A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/699840. (End)

A030140 The nonsquares squared.

Original entry on oeis.org

4, 9, 25, 36, 49, 64, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

N. J. A. Sloane

The complement of the fourth powers A000583 within the squares A000290. - Peter Munn, Aug 20 2019

			a(1)=2^2, a(2)=3^2, a(3)=5^2, a(4)=6^2, a(5)=7^2, ..., a(n)=(integer which is not a perfect square)^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000037, A000290, A000583, A013661, A013662, A062503.
Positions of 2's in A352080.
Related to A016945 via A225546.

[(n + Floor(1/2 + Sqrt(n)))^2: n in [1..60]]; // Vincenzo Librandi, Apr 06 2020

a:=proc(n) if type(sqrt(n),integer)=false then n^2 else fi end: seq(a(n),n=1..70); # Emeric Deutsch, Apr 11 2007

a[n_] := (n + Floor[1/2 + Sqrt[n]])^2;
Array[a, 50] (* Jean-François Alcover, Apr 05 2020 *)

from math import isqrt
def A030140(n): return (n+(k:=isqrt(n))+int(n>=k*(k+1)+1))**2 # Chai Wah Wu, Jun 17 2024

a(n) = A000037(n)^2.
Sum_{n>=1} 1/a(n) = zeta(2) - zeta(4) = A013661 - A013662 = 0.5626108331... - Amiram Eldar, Nov 14 2020
{a(n) : n >= 1} = {A225546(6m+3) : m >= 0}. - Peter Munn, Nov 17 2022

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A109162 a(1) = 1; for n > 1, a(n) = A019565(a(n-1)).

Original entry on oeis.org

1, 2, 3, 6, 15, 210, 10659, 54230826, 249853434654335387610276087

Offset: 1

Leroy Quet, Aug 18 2005

nonn

After the initial 1, even-indexed terms are of the form 4k+2 (members of A016825) and odd-indexed terms are of the form 6k+3 (members of A016945). However, not all even terms after 2 are multiples of three, because not all odd-indexed terms are of the form 4k+3. For example, because a(11) is of the form 4k+1, a(12) cannot be a multiple of three. - Antti Karttunen, Jun 18 2017

			a(4) = 6, which is 110 in binary. So a(5) is the product of the primes corresponding to the 1's of 110, 3*5 = 15.

Frank Adams-Watters, Table of n, a(n) for n = 1..11

Cf. A019565, A285320 (a left inverse).
The left edge of A285332 and A285333.
Cf. A153013, A328316 for similar iteration sequences, and also A376406, A376407, A376408.

NestList[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, 1, 11] (* Michael De Vlieger, Aug 20 2017 *)

More terms from Franklin T. Adams-Watters, Aug 29 2006

A165351 Numerator of 3*n/2.

Original entry on oeis.org

0, 3, 3, 9, 6, 15, 9, 21, 12, 27, 15, 33, 18, 39, 21, 45, 24, 51, 27, 57, 30, 63, 33, 69, 36, 75, 39, 81, 42, 87, 45, 93, 48, 99, 51, 105, 54, 111, 57, 117, 60, 123, 63, 129, 66, 135, 69, 141, 72, 147, 75, 153, 78, 159, 81, 165, 84, 171, 87, 177, 90, 183, 93, 189, 96, 195

Offset: 0

Paul Curtz, Sep 16 2009

First trisection of A026741. The other trisections are A165355 and A165367.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000034 (denominator).

Cf. A008585, A016945, A026741, A098832, A165355, A165367.

[Numerator(3*n/2): n in [0..100]]; // Vincenzo Librandi, Mar 03 2014

A165351:=n->numer(3*n/2); seq(A165351(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013

Table[Numerator[3n/2], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 11 2013 *)
CoefficientList[Series[3*x*(1+x+x^2)/(1-x^2)^2, {x, 0, 70}], x] (* Vincenzo Librandi, Mar 03 2014 *)
LinearRecurrence[{0,2,0,-1},{0,3,3,9},70] (* Harvey P. Dale, Jun 20 2021 *)

[3*n*(3-(-1)^n)/4 for n in (0..100)] # G. C. Greubel, Jul 31 2022

a(n) = A026741(3*n) = 3*A026741(n).
a(2n) = A008585(n).
a(2n+1) = A016945(n).
G.f.: 3*x*(1+x+x^2)/((1-x)^2 * (1+x)^2).
a(n) = numerator(3n/2). - Wesley Ivan Hurt, Oct 11 2013
a(n) = 3*n / (1 + ((n+1) mod 2)). - Wesley Ivan Hurt, Feb 25 2014
From G. C. Greubel, Jul 31 2022: (Start)
a(n) = 3*n*(3 - (-1)^n)/4.
E.g.f.: (3*x/2)*( 2*cosh(x) + sinh(x) ). (End)

Edited and extended by R. J. Mathar, Sep 26 2009

New name from Wesley Ivan Hurt, Oct 13 2013

A249734 Even bisection of A003961: Replace in 2n each prime factor p(k) with prime p(k+1).

Original entry on oeis.org

3, 9, 15, 27, 21, 45, 33, 81, 75, 63, 39, 135, 51, 99, 105, 243, 57, 225, 69, 189, 165, 117, 87, 405, 147, 153, 375, 297, 93, 315, 111, 729, 195, 171, 231, 675, 123, 207, 255, 567, 129, 495, 141, 351, 525, 261, 159, 1215, 363, 441, 285, 459, 177, 1125, 273, 891, 345, 279, 183, 945, 201, 333, 825, 2187, 357, 585, 213, 513, 435, 693, 219, 2025, 237, 369

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Row 2 of A246278.
Cf. A249735 (the other bisection of A003961).
Cf. A016945, A048673, A064216, A064989, A249827, A243501, A243502.
Cf. also A000079, A000244.

a(n) = A003961(2*n).
a(n) = 3 * A003961(n).
a(n) = A064989(A249827(n)).
a(n) = A003961(A243501(A064216(n))).
a(n) = A003961(A243502(A048673(n))).
a(n) = A016945(A048673(n)-1). [Permutation of A016945, 6n+3.]
Other identities. For all n >= 1:
a(A000079(n-1)) = A000244(n). [Maps each 2^n to 3^(n+1).]

A016951 a(n) = (6*n + 3)^7.

Original entry on oeis.org

2187, 4782969, 170859375, 1801088541, 10460353203, 42618442977, 137231006679, 373669453125, 897410677851, 1954897493193, 3938980639167, 7446353252589, 13348388671875, 22876792454961, 37725479487783, 60170087060757, 93206534790699, 140710042265625, 207616015289871

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A016759, A016945, A016946, A016947, A016948, A016949, A016950.

Subsequence of A001015.

[(6*n+3)^7: n in [0..40]]; // Vincenzo Librandi, May 05 2011

a[n_] := (6*n + 3)^7; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^7.
a(n) = 3^7*A016759(n).
Sum_{n>=0} 1/a(n) = 127*zeta(7)/279936.
Sum_{n>=0} (-1)^n/a(n) = 61*Pi^7/403107840. (End)

A047263 Numbers that are congruent to {0, 1, 2, 4, 5} mod 6.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane, Dec 11 1999

Complement of A016945. - R. J. Mathar, Feb 25 2008
Nonnegative integers m such that floor(2*m^2/12) = 2*floor(m^2/12). See the Crossrefs field of A265187 for similar sequences. - Bruno Berselli, Dec 08 2015
Also, numbers k such that Fibonacci(k) mod 4 = 0, 1 or 3. - Bruno Berselli, Oct 17 2017

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

Cf. A016945, A265187.

[n : n in [0..100] | n mod 6 in [0, 1, 2, 4, 5]]; // Wesley Ivan Hurt, Aug 16 2016

for n from 0 to 200 do if n mod 6 <> 3 then printf(`%d,`,n) fi: od:
A047263:=n->6*floor(n/5)+[0, 1, 2, 4, 5][(n mod 5)+1]: seq(A047263(n), n=0..100); # Wesley Ivan Hurt, Aug 16 2016

Select[Range[0,100], Mod[#,6]!=3&] (* Harvey P. Dale, May 17 2011 *)
LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,4,5,6},90] (* Harvey P. Dale, Oct 05 2014 *)

first(n) = {select(x->(x%6!=3), vector(6*n\5, i, i-1))} \\ David A. Corneth, Oct 17 2017

O.g.f.: x*(x^2+1)*(x^2+x+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). - R. J. Mathar, Feb 25 2008
a(n) = a(n-5) + 6 for n > 5. - R. J. Mathar, Feb 25 2008
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6. - Harvey P. Dale, Oct 05 2014
From Wesley Ivan Hurt, Aug 16 2016: (Start)
a(n) = n + floor((n-4)/5).
a(n) = (6*n - 4 - ((n+1) mod 5))/5.
a(5k) = 6k-1, a(5k-1) = 6k-2, a(5k-2) = 6k-4, a(5k-3) = 6k-5, a(5k-4) = 6k-6. (End)
Sum_{n>=2} (-1)^n/a(n) = log(2+sqrt(3))/sqrt(3) - log(2)/6. - Amiram Eldar, Dec 17 2021

More terms from James Sellers, Feb 19 2001

A064539 Numbers n such that 2^n + n^2 is prime.

Original entry on oeis.org

1, 3, 9, 15, 21, 33, 2007, 2127, 3759, 29355, 34653, 57285, 99069, 1933695

Offset: 1

Jason Earls, Oct 16 2001

Values 2^2007+2007^2, 2^2127+2127^2, 2^3759+3759^2 were proved prime with Primo.
n is always an odd multiple of 3 (except for the first term), i.e., a(n) is a subsequence of A016945. - Lekraj Beedassy, Jan 01 2007
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 165 pp. 30, 160, Ellipses Paris 2004.

Henri Lifchitz, Renaud Lifchitz, PRP Top Records. 2^n+n^2.

Cf. A001580, A016945, A061119, A072180, A094133.

for(n=1,5000, if(isprime(2^n+n^2),print(n)))

a(10)-a(13) from Hugo Pfoertner, Jun 26 2004

a(14) from Ryan Propper, May 11 2023. n=1933695 corresponds to a probable prime with 582101 digits, and was PRP tested with PFGW.

A106833 3n and 2n, alternating.

Original entry on oeis.org

3, 4, 9, 8, 15, 12, 21, 16, 27, 20, 33, 24, 39, 28, 45, 32, 51, 36, 57, 40, 63, 44, 69, 48, 75, 52, 81, 56, 87, 60, 93, 64, 99, 68, 105, 72, 111, 76, 117, 80, 123, 84, 129, 88, 135, 92, 141, 96, 147, 100, 153, 104, 159, 108, 165, 112, 171, 116, 177, 120, 183

Offset: 1

Zak Seidov, May 19 2005

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A118402 (first differences).

Mathematica
```
Table[n(2 + Mod[n, 2]), {n, 50}]
```

a(n)=sumdiv(n,d,moebius(d)*sigma(2*n/d)) \\ Benoit Cloitre, Oct 18 2009

a(n) = n*(2 + (n mod 2)).
a(2*n) = 6*n + 3 = A016945(n). - Paul Curtz, Nov 23 2008
a(2*n+1) = A008586(n+1).
From R. J. Mathar, Apr 08 2009: (Start)
G.f.: x*(3+4*x+3*x^2)/((x-1)^2*(1+x)^2).
a(n) = 2*a(n-2) - a(n-4). (End)
a(n) = Sum_{d|n} mu(d)*sigma(2*n/d). - Benoit Cloitre, Oct 18 2009
a(n) = n*(5-(-1)^n)/2. - Wesley Ivan Hurt, May 14 2014

More terms from Michel Marcus, May 17 2014

A177713 Sums of two or more positive consecutive odd numbers.

Original entry on oeis.org

4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 39, 40, 44, 45, 48, 49, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 111, 112, 115, 116, 117, 119, 120, 121, 123, 124, 125, 128

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 11 2010

Sums of two or more positive consecutive odd numbers are Sum_{k=0..m} (2*j+i+2*k) = (m+1)*(m+2*j+i) with m >= 1 and 2*j+i >=1. Testing a number n against being a member can be done by scanning all divisors d, building m=d-1, if this is >= 1 building n/d-m, and testing this for being an odd number >= 1. - R. J. Mathar, Jan 25 2011

The sums of two positive consecutive odd numbers are A008586 (without 0), the sums of three positive consecutive odd numbers are A016945 (without 3), etc.

			1+3=4, 3+5=8, 1+3+5=9, 5+7=12, 3+5+7=15, 7+9=16, ...

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 3G, p. 179.

Erzsébet Orosz, On odd-summing numbers, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 125-129.

Cf. A008586, A016945.

isA177713 := proc(n) local d,l; for d in numtheory[divisors](n) do l := d-1 ; if l >=1 then l := n/d -l; if type(l,'odd') and l>=1 then return true; end if; end if; end do: return false; end proc:
for n from 2 to 130 do if isA177713(n) then printf("%d,",n) ; end if; end do; # R. J. Mathar, Jan 25 2011

z=200; lst={}; Do[c=a; Do[c+=b; If[c<=2*z, AppendTo[lst,c]], {b,a-2,1,-2}],{a,1,z,2}]; Union@lst

cp's OEIS Frontend

A016950 a(n) = (6*n + 3)^6.

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A030140 The nonsquares squared.

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A109162 a(1) = 1; for n > 1, a(n) = A019565(a(n-1)).

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A165351 Numerator of 3*n/2.

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A249734 Even bisection of A003961: Replace in 2n each prime factor p(k) with prime p(k+1).

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A016951 a(n) = (6*n + 3)^7.

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A047263 Numbers that are congruent to {0, 1, 2, 4, 5} mod 6.

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