A010804 -id:A010804 - cp's OEIS frontend

A022532 Nexus numbers (n+1)^16-n^16.

1, 65535, 42981185, 4251920575, 148292923329, 2668522016831, 30411820662145, 248242046141055, 1571545212141185, 8146979811148159, 35949729863572161, 138934529031464255, 480532350288143425, 1512536728626191295, 4390455017903519489, 11878335717996660991

Offset: 0

N. J. A. Sloane

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Column k=15 of A047969.

Cf. A010804 (n^16).

[(n+1)^16-n^16: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

b:=16: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Table[(n+1)^16-n^16,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

for(n=0,20, print1((n+1)^16 - n^16, ", ")) \\ G. C. Greubel, Feb 27 2018

a(n) = A010804(n+1) - A010804(n). - Michel Marcus, Feb 28 2018

More terms added by G. C. Greubel, Feb 27 2018

A170780 a(n) = n^8*(n^8 + 1)/2.

Original entry on oeis.org

0, 1, 32896, 21526641, 2147516416, 76294140625, 1410555793536, 16616468167201, 140737496743936, 926510115949281, 5000000050000000, 22974865038965521, 92442129662509056, 332708304999455281, 1088976669642580096

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 16 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=32896, there are 2^16=65536 oriented arrangements of two colors. Of these, 2^8=256 are achiral. That leaves (65536-256)/2=32640 chiral pairs. Adding achiral and chiral, we get 32896. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Row 16 of A277504.

Cf. A010804 (oriented), A001016 (achiral).

List([0..30], n -> n^8*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

[n^8*(n^8+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 26 2011

Table[n^8*(n^8+1)/2, {n, 0, 30}] (* G. C. Greubel, Dec 05 2017 *)

for(n=0, 30, print1(n^8*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 05 2017

for n in range(0,20): print(int(n**8*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^8*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 32879*x^2 + 20967545*x^3 + 1786036695*x^4 + 42691617829* x^5 + 391057805899*x^6 + 1603741496717*x^7 + 3191399514435*x^8 + 3191399514435*x^9 + 1603741496717*x^10 + 391057805899*x^11 + 42691617829*x^12 + 1786036695*x^13 + 20967545*x^14 + 32879*x^15 + x^16) /(1-x)^17. - G. C. Greubel, Dec 05 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010804(n) + A001016(n)) / 2 = (n^16 + n^8) / 2.
G.f.: (Sum_{j=1..16} S2(16,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..15} A145882(16,k) * x^k / (1-x)^17.
E.g.f.: (Sum_{k=1..16} S2(16,k)*x^k + Sum_{k=1..8} S2(8,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>16, a(n) = Sum_{j=1..17} -binomial(j-18,j) * a(n-j). (End)

A309012 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that (i^16 mod prime(n)) > (j^16 mod prime(n)).

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 0, 16, 21, 43, 30, 62, 77, 99, 129, 146, 203, 187, 228, 245, 252, 345, 372, 382, 402, 558, 570, 631, 663, 756, 901, 1114, 961, 1325, 1398, 1253, 1571, 1470, 1601, 1795, 2024, 1988, 2349, 2014, 2184, 2200, 2728, 3054, 3084, 3718, 3386, 3224, 3018, 3861, 3866, 4258, 4361, 4418, 5110, 4724

Offset: 1

Zhi-Wei Sun, Jul 06 2019

nonn

Conjecture : Let p be an odd prime, and let N be the number of ordered pairs (i,j) with 0 < i < j < p/2 and (i^16 mod p) > (j^16 mod p). When p == 1 (mod 16), we have 2 | N. Also, N == |{0

			a(5) = 3 with prime(5) = 11, and (2^16 mod 11) = 9 greater than (3^16 mod 11) = 3, (4^16 mod 11) = 4 and (5^16 mod 11) = 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Quadratic residues and related permutations and identities, arXiv:1809.07766 [math.NT], 2018-2019.

Cf. A000040, A010804, A319311, A319882, A319894, A319903, A320044.

r[p_]:=r[p]=Sum[Boole[PowerMod[j,16,p]>PowerMod[k,16,p]],{k,2,p/2},{j,1,k-1}];
Print[Table[r[Prime[n]],{n,1,60}]]

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A022532 Nexus numbers (n+1)^16-n^16.

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A170780 a(n) = n^8*(n^8 + 1)/2.

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A309012 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that (i^16 mod prime(n)) > (j^16 mod prime(n)).

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