A059826 -id:A059826 - cp's OEIS frontend

A131473 a(n) = n^6 - n.

0, 0, 62, 726, 4092, 15620, 46650, 117642, 262136, 531432, 999990, 1771550, 2985972, 4826796, 7529522, 11390610, 16777200, 24137552, 34012206, 47045862, 63999980, 85766100, 113379882, 148035866, 191102952, 244140600, 308915750

Offset: 0

Mohammad K. Azarian, Jul 27 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6-n: n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A131473:=n->n^6-n; seq(A131473(n), n=0..30); # Wesley Ivan Hurt, Feb 25 2014

f[n_]:=n^6-n;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
Array[#^6-#&,60,0] (* Harvey P. Dale, Aug 10 2011 *)

[lucas_number1(3,n^3,n) for n in range(0, 27)] # Zerinvary Lajos, May 16 2009

A219619 a(n) = n! * (n^4 + n^2 + 1).

Original entry on oeis.org

1, 3, 42, 546, 6552, 78120, 959760, 12353040, 167771520, 2410611840, 36654508800, 589291718400, 10002032409600, 178908534604800, 3366215358105600, 66496549287168000, 1376573115101184000, 29810519036153856000, 674176353586864128000, 15896946656727392256000

Offset: 0

Franz Vrabec, Nov 24 2012

Sum_{n>=0} 1/a(n) = e/2.

			a(3) = 3!*(3^4 + 3^2 + 1) = 6*91 = 546.

Iain Fox, Table of n, a(n) for n = 0..445
G. I. Senum, A Series for e, Problem E3352, Amer. Math. Monthly 98 (1991), No. 4, pp. 369-370.

Cf. A000142 (n!), A059826 (n^4 + n^2 + 1).

Array[#!*(#^4 + #^2 + 1) &, 20, 0] (* Michael De Vlieger, Jan 29 2021 *)
nmax = 20; CoefficientList[Series[(1 - 2*x + 16*x^2 + 6*x^3 + 3*x^4) / (1 - x)^5, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 29 2021 *)

a(n) = n! * (n^4 + n^2 + 1); \\ Michel Marcus, Nov 19 2017

first(n) = { my(x='x+O('x^n)); Vec(serlaplace((-3*x^4-6*x^3-16*x^2+2*x-1)/(x-1)^5)); } \\ Iain Fox, Nov 19 2017

a(n) = A000142(n)*A059826(n). - Michel Marcus, Nov 19 2017

E.g.f.: (-3*x^4 - 6*x^3 - 16*x^2 + 2*x - 1)/(x - 1)^5. - Iain Fox, Nov 19 2017

A257874 Numbers n such that the largest prime divisor of n^4 + n^2 + 1 is equal to the largest prime divisor of (n+1)^4 + (n+1)^2 + 1.

Original entry on oeis.org

3, 6, 8, 10, 12, 15, 17, 21, 24, 27, 31, 33, 38, 41, 43, 48, 50, 52, 54, 57, 59, 62, 66, 69, 71, 73, 75, 78, 80, 82, 85, 90, 93, 95, 97, 99, 101, 103, 105, 111, 115, 117, 119, 124, 127, 131, 133, 136, 138, 141, 143, 145, 147, 150, 153, 155, 157, 162, 164, 168

Offset: 1

Michel Lagneau, May 11 2015

nonn

The sequence is infinite. Proof:
Let p(n) denote the largest prime divisor of n^4 + n^2 + 1 and let q(n) denote the largest prime divisor of n^2 + n + 1. Then p(n) = q(n^2), and from
n^4 + n^2 + 1 = (n^2+1)^2 - n^2 = (n^2-n+1)(n^2+n+1)= ((n-1)^2 + (n-1)+1)(n^2+n+1) it follows that p(n) = max{q(n),q(n-1)} for n>=2.
Keeping in mind that n^2-n+1 is odd, we have
gcd(n^2+n+1,n^2-n+1) = gcd(2n,n^2-n+1)= gcd(n,n^2-n+1)= 1.
Therefore q(n) is different from q(n-1).
To prove the result, it suffices to show that the set
S = {n in Z | n>=2 and q(n) > q(n-1) and q(n) > q(n+1)}
is infinite, since for each n in S one has
p(n) = max{q(n),q(n-1)} = q(n) = max{q(n),q(n+1)} = p(n+1).
Suppose on the contrary that S is finite. Since q(2) = 7 < 13 = q(3) and q(3) = 13 > 7 = q(4), the set S is nonempty. Since it is finite, we can consider its largest element, say m.
Note that it is impossible that q(m)>q(m+1)>q(m+2)>... because all these numbers are positive integers, so there exists a number k>=m such that q(k)= k+1 such that q(l)>q(l+1). By the minimality of l, we have q(l-1)= k + 1 > k >= m, this contradicts the maximality of m, and hence S is indeed infinite.

			3 is in the sequence because 3^4+3^2+1 = 91 = 13*7 and 4^4+4^2+1 = 273 = 13*7*3, so 13 is the greatest prime factor of both expressions.

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

Cf. A002061, A059826.

with(numtheory):for n from 1 to 400 do:x1:=n^4 + n^2 + 1:x2:=(n+1)^4 + (n+1)^2+1:y1:=factorset(x1):n1:=nops(y1):y2:=factorset(x2):n2:=nops(y2):if y1[n1]=y2[n2] then printf(`%d, `,n):else fi:od:

fQ[n_]:=Last[FactorInteger[n^4+n^2+1]][[1]]==Last[FactorInteger[(n+1)^4+(n+1)^2+1]][[1]];Select[Range[168],fQ[#]&] (* Ivan N. Ianakiev, Jun 11 2015 *)
SequencePosition[Table[FactorInteger[n^4+n^2+1][[-1,1]],{n,200}],{x_,x_}][[;;,1]] (* Harvey P. Dale, Nov 06 2024 *)

gpf(n)=if(n<4, return(n)); my(f=factor(n)[,1]); f[#f]
is(n)=my(a=n^4+n^2+1, b=(n+1)^4 +(n+1)^2+1, g=gcd(a,b), p=gpf(g)); g>1 && p>=gpf(a/g) && p>=gpf(b/g) \\ Charles R Greathouse IV, May 11 2015

A260534 Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 7, 2, 1, 1, 1, 6, 1, 13, 5, 3, 1, 1, 1, 7, 1, 21, 10, 11, 1, 1, 1, 1, 8, 1, 31, 17, 31, 1, 4, 1, 1, 1, 9, 1, 43, 26, 69, 1, 23, 3, 1, 1, 1, 10, 1, 57, 37, 131, 1, 94, 21, 5, 1, 1, 1, 11

Offset: 0

Peter Luschny, Sep 20 2015

A parametrization of Stern's diatomic series (which is here T(1,k)). (For other generalizations of Dijkstra's fusc function see the Luschny link.)

			Array starts:
n\k[0, 1,  2, 3,  4,  5,   6, 7,    8,    9,    10]
[0] 1, 1,  1, 1,  1,  1,   1, 1,    1,    1,     1, ...
[1] 1, 1,  2, 1,  3,  2,   3, 1,    4,    3,     5, ... [A002487]
[2] 1, 1,  3, 1,  7,  5,  11, 1,   23,   21,    59, ... [A101624]
[3] 1, 1,  4, 1, 13, 10,  31, 1,   94,   91,   355, ...
[4] 1, 1,  5, 1, 21, 17,  69, 1,  277,  273,  1349, ... [A101625]
[5] 1, 1,  6, 1, 31, 26, 131, 1,  656,  651,  3881, ...
[6] 1, 1,  7, 1, 43, 37, 223, 1, 1339, 1333,  9295, ...
[7] 1, 1,  8, 1, 57, 50, 351, 1, 2458, 2451, 19559, ...
[8] 1, 1,  9, 1, 73, 65, 521, 1, 4169, 4161, 37385, ...
-,-,-,-,A002061,A002522,A071568,-,-,A059826,-,A002523,

Chai Wah Wu, Table of n, a(n) for n = 0..10010
Peter Luschny, Rational Trees and Binary Partitions.

Cf. A002061, A002487, A002522, A002523, A059826, A071568, A101624, A101625.

T := (n,k) -> add(modp(binomial(k-j,j),2)*n^j, j=0..k):
seq(lprint(seq(T(n,k),k=0..10)),n=0..5);

Table[If[(n - k) == 0, 1, Sum[(n - k)^j Mod[Binomial[k - j, j], 2], {j, 0, k}]], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, Sep 21 2015 *)

def A260534_T(n,k):
    return sum(0 if ~(k-j) & j else n**j for j in range(k+1)) # Chai Wah Wu, Feb 08 2016

A203173 Central polygonal numbers that are nontrivially the product of two central polygonal numbers.

Original entry on oeis.org

21, 91, 273, 651, 931, 1333, 2451, 3783, 4161, 4557, 6643, 10101, 14763, 20881, 22351, 28731, 31863, 38613, 50851, 52671, 65793, 83811, 99541, 105301, 130683, 139503, 160401, 194923, 221371, 234741, 235711, 280371, 316407, 332353, 391251, 427063, 457653, 532171, 615441

Offset: 1

Franklin T. Adams-Watters, Dec 30 2011

nonn

Central polygonal numbers are those of the form n^2-n+1, or equivalently n^2+n+1. We exclude factorizations where one of the factors is 1.

			21 = 4^2+4+1 = 7*3 = (2^2+2+1)*(1^2+1+1), so 21 is in the sequence.

Cf. A002061 (central polygonal numbers), A059826 (a subsequence except for first two terms).

iscpn(n)=local(r=sqrtint(n-1));n==r^2+r+1
iscpnprod(n)=local(x,y);for(i=1,n,x=i^2+i+1;y=n\x;if(x>y,return(0));if(n==x*y&&iscpn(y),return(1)));0
ap(n)=for(k=1,n,if(iscpnprod(k^2+k+1),print1(k^2+k+1", ")))

A241855 Array t(n,k) of sum of successive even powers of primes, where t(n,k) = sum_(j=0..k-1) prime(n)^(2j), with n>=1 and k>=0, read by ascending antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 5, 0, 1, 10, 21, 0, 1, 26, 91, 85, 0, 1, 50, 651, 820, 341, 0, 1, 122, 2451, 16276, 7381, 1365, 0, 1, 170, 14763, 120100, 406901, 66430, 5461, 0, 1, 290, 28731, 1786324, 5884901, 10172526, 597871, 21845, 0, 1, 362, 83811, 4855540, 216145205, 288360150, 254313151, 5380840, 87381

Offset: 1

Jean-François Alcover, Apr 30 2014

Conjecture: any term, except 0 and 1, is never a square.
Row n=1 is A002450,
row n=2 is A002452,
row n=3 is A218728,
row n=4 is A218753,
rows n>=5 are not in the OEIS,
column k=2 is A066872,
columns k>=3 are not in the OEIS.

			Array begins:
0,  1,   5,    21,      85,       341,        1365, ...
0,  1,  10,    91,     820,      7381,       66430, ...
0,  1,  26,   651,   16276,    406901,    10172526, ...
0,  1,  50,  2451,  120100,   5884901,   288360150, ...
0,  1, 122, 14763, 1786324, 216145205, 26153569806, ...
etc.

Cf. A002450, A002452, A066872, A059826, A059830, A059839, A218728, A218753.

t[n_, k_] := ((Prime[n]^2)^k-1)/(Prime[n]^2-1); Table[t[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten

t(n,k) = ((prime(n)^2)^k-1)/(prime(n)^2-1).

cp's OEIS Frontend

A131473 a(n) = n^6 - n.

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A219619 a(n) = n! * (n^4 + n^2 + 1).

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A257874 Numbers n such that the largest prime divisor of n^4 + n^2 + 1 is equal to the largest prime divisor of (n+1)^4 + (n+1)^2 + 1.

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A260534 Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

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A203173 Central polygonal numbers that are nontrivially the product of two central polygonal numbers.

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PARI

A241855 Array t(n,k) of sum of successive even powers of primes, where t(n,k) = sum_(j=0..k-1) prime(n)^(2j), with n>=1 and k>=0, read by ascending antidiagonals.

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