A056221 -id:A056221 - cp's OEIS frontend

A056222 Image of partition numbers (A000041) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

-1, 1, -1, 4, -6, 16, -17, 34, -24, 84, -98, 273, -194, 449, -209, 1089, -726, 2695, -1295, 5049, -990, 10044, -3125, 23335, -2936, 40516, 3052, 82874, 3553, 171086, 25168, 307395, 104259, 577831, 206819, 1149058, 488114, 2030380, 1156155, 3805389, 2233077, 7109113

Offset: 0

N. J. A. Sloane, Aug 06 2000

sign

Stefano Spezia, Table of n, a(n) for n = 0..10000

Cf. A000041, A056221.

a[n_]:=PartitionsP[n+1]^2-PartitionsP[n]PartitionsP[n+2]; Array[a,40,0] (* Stefano Spezia, Jul 15 2024 *)

a(40)-a(41) from Stefano Spezia, Jul 15 2024

A056141 a(n) = primefloor(n)primeceiling(n) - previousprime(n)nextprime(n).

Original entry on oeis.org

-1, 0, 4, 0, -6, 0, 0, 0, 30, 0, -18, 0, 0, 0, 42, 0, -30, 0, 0, 0, -22, 0, 0, 0, 0, 0, 128, 0, -112, 0, 0, 0, 0, 0, 98, 0, 0, 0, 90, 0, -78, 0, 0, 0, -70, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 248, 0, -232, 0, 0, 0, 0, 0, 158, 0, 0, 0, 150, 0, -280, 0, 0, 0, 0, 0, 182

Offset: 3

Henry Bottomley, Jun 15 2000

			a(3)=3*3-2*5=-1, a(4)=3*5-3*5=0

Cf. A056139, A056140.

Cf. A056221 (nonzero terms).

a(n) = if (isprime(n), n^2 - precprime(n-1)*nextprime(n+1), 0); \\ Michel Marcus, Mar 22 2020

a(n) = A007917(n)*A007918(n) - A007917(n-1)*A007918(n+1).
a(n) = A030664(n) - A013638(n).
a(n) = A056140(n) - A056139(n).
a(n) = A056140(n) if n is prime, a(n)=0 otherwise.

More terms from Michel Marcus, Mar 22 2020

A324795 a(n) = 2p(n)p(n+2) - p(n+1)^2 where p(k) = k-th prime.

Original entry on oeis.org

11, 17, 61, 61, 205, 205, 421, 573, 585, 1185, 1173, 1501, 2005, 2349, 2737, 2985, 4185, 4173, 4741, 5889, 5877, 7173, 8181, 8569, 9781, 11005, 11005, 12301, 14917, 13477, 17637, 17649, 21505, 19777, 23985, 24577, 25869, 28509, 29857, 30585, 35617

Offset: 1

N. J. A. Sloane, Sep 10 2019

nonn

Theorem: a(n) > 0. Proof: Use p(n+1) <= 2 p(n)^2 for n > 4. (See Sándor et al.) QED

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VII, p. 247, section VII.18.b.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A056221 (if leading coefficient 2 is changed to 1), A327447 or A309487 (if 2 is changed to 4).

With[{p = Prime[Range[50]]}, 2 * p[[1;;-3]] * p[[3;;-1]] - p[[2;;-2]]^2] (* Amiram Eldar, Apr 25 2024 *)

A342567 a(n) = (prime(n)^2 - prime(n-1)*prime(n+1))/2, n >= 3.

Original entry on oeis.org

2, -3, 15, -9, 21, -15, -11, 64, -56, 49, 45, -39, -35, 18, 124, -116, 79, 75, -140, 91, -71, -65, 210, 105, -99, 111, -105, -537, 663, -119, 280, -546, 606, -296, 18, 175, -155, 18, 364, -714, 774, -189, 201, -983, 72, 916, 231, -225, -221, 484, -954, 532, 18, 18

Offset: 3

Hugo Pfoertner, Jun 20 2021

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A056221.

a[n_]:=(Prime[n]^2 - Prime[n-1]*Prime[n+1])/2; Array[a,54,3] (* Stefano Spezia, Jul 15 2024 *)

forprime(p=5,265,my(pp=precprime(p-1),pn=nextprime(p+1));print1((p^2-pp*pn)/2,", "))

from primesieve.numpy import n_primes
primesarray = numpy.array(n_primes(10005,1))
for i in range (2, 10003):
     print(((primesarray[i]**2)-(primesarray[i-1]*primesarray[i+1]))//2)
     # Karl-Heinz Hofmann, Jun 20 2021

a(n) = A056221(n-1)/2 for n >= 3.

A350200 Array read by antidiagonals: T(n,k) is the determinant of the Hankel matrix of the 2*n-1 consecutive primes starting at the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 5, -4, -2, 1, 7, 6, 12, 0, 1, 11, -30, -72, 144, 288, 1, 13, 18, 72, 0, 576, -1728, 1, 17, -42, -72, 288, 1152, -7104, -26240, 1, 19, 30, -96, 144, -1248, -11712, 45248, 222272, 1, 23, 22, -188, 488, -112, -11360, 21184, 450432, 1636864

Offset: 0

Pontus von Brömssen, Dec 19 2021

			Array begins:
  n\k|      1      2       3       4       5       6       7       8
  ---+--------------------------------------------------------------
   0 |      1      1       1       1       1       1       1       1
   1 |      2      3       5       7      11      13      17      19
   2 |      1     -4       6     -30      18     -42      30      22
   3 |     -2     12     -72      72     -72     -96    -188    -480
   4 |      0    144       0     288     144     488    1800    2280
   5 |    288    576    1152   -1248    -112    4432   -1552   15952
   6 |  -1728  -7104  -11712  -11360  -10816   29952  -89152  -57088
   7 | -26240  45248   21184 -103168  -43264 -605440 -379264  271552
   8 | 222272 450432 1068800 2022912 3927552 5399552 6315904 6861312
T(3,2) = 12, the determinant of the Hankel matrix
  [3  5  7]
  [5  7 11]
  [7 11 13].

Wikipedia, Hankel matrix

Cf. A350201.

Cf. A000012 (row n = 0), A000040 (row n = 1), A056221 (row n = 2 with opposite sign), A024356 (column k = 1), A071543 (column k = 2).

from sympy import Matrix,prime,nextprime
def A350200(n,k):
    p = [prime(k)] if n > 0 else []
    for i in range(2*n-2): p.append(nextprime(p[-1]))
    return Matrix(n,n,lambda i,j:p[i+j]).det()

Offset corrected by Pontus von Brömssen, Aug 25 2022

A092386 Let p(n) = n-th prime; a(n) = largest prime factor of (p(n+1)^2 - p(n)*p(n+2)).

Original entry on oeis.org

1, 2, 3, 5, 3, 7, 5, 11, 2, 7, 7, 5, 13, 7, 3, 31, 29, 79, 5, 7, 13, 71, 13, 7, 7, 11, 37, 7, 179, 17, 17, 7, 13, 101, 37, 3, 7, 31, 3, 13, 17, 43, 7, 67, 983, 3, 229, 11, 5, 17, 11, 53, 19, 3, 3, 17, 67, 17, 19, 17, 43, 521, 7, 103, 173, 683, 23, 233, 23, 31, 67, 23, 3, 23, 53, 73, 5

Offset: 0

Frank Schwellinger (nummer_eins(AT)web.de), Mar 20 2004

Ivan Neretin, Table of n, a(n) for n = 0..9999

Cf. A006530, A056221.

Table[ FactorInteger[Prime[n + 1]^2 - Prime[n]Prime[n + 2]][[ -1, 1]], {n, 80}] (* Robert G. Wilson v, Mar 24 2004 *)

a(n) = A006530(A056221(n)).

More terms from Robert G. Wilson v, Mar 24 2004

A106671 a(n) = ( prime(n + 2) * prime(n) - prime(n + 1)^2 ) modulo 3.

Original entry on oeis.org

1, 2, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1

Offset: 1

Roger L. Bagula, May 13 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A010872, A056221.

a = Table[Mod[Prime[n + 2]*Prime[n] - Prime[n + 1]^2, 3], {n, 1, 200}]
Mod[#[[3]]#[[1]]-#[[2]]^2,3]&/@Partition[Prime[Range[110]],3,1] (* Harvey P. Dale, Oct 03 2015 *)

a(n) = (prime(n + 2)*prime(n) - prime(n + 1)^2) % 3; \\ Michel Marcus, Apr 21 2017

a(n) = A056221(n) mod 3 = A010872(A056221(n)). - Michel Marcus, Apr 21 2017

A327434 Prime(n)^2 - prime(n-1)*prime(n+1) for n in A046868.

Original entry on oeis.org

4, 30, 42, 128, 98, 90, 36, 248, 158, 150, 182, 420, 210, 222, 1326, 560, 1212, 36, 350, 36, 728, 1548, 402, 144, 1832, 462, 968, 1064, 36, 36, 1088, 578, 570, 3126, 630, 2732, 2796, 782, 36, 782, 1620, 3372, 3468, 902, 1860, 930, 2012, 1980, 2028, 5234, 6600

Offset: 1

Chai Wah Wu, Sep 10 2019

nonn

Positive terms in A056221.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

CF. A046868, A056221.

A139312 Characteristic function of the good primes (version 1).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0

Offset: 2

Roger L. Bagula, Jun 07 2008

a(n)=1 if prime(n)^2 - prime(n-1)*prime(n+1) >=0, else a(n)=0.

Cf. A056221, A046869, A068828.

f[n_] := If[ Prime[n]^2 - Prime[n - 1]*Prime[n + 1] > 0, 1, 0]; Array[f, 105, 2] (*alternative formula: derived*) Solve[x^2 - (x - a)*(x + b) == 0, x]; a = -Prime[n - 1] + Prime[n]; b = -Prime[n] + Prime[n + 1]; f[n_] = If[-Prime[-1 + n] + 2 Prime[n] - Prime[1 + n] == 0, 0, a*b/(b - a)]; Table[ If[ f[n] > 0, 0, 1], {n, 2, 106}]
If[#[[2]]^2-(#[[1]]#[[3]])>=0,1,0]&/@Partition[Prime[Range[110]],3,1] (* Harvey P. Dale, Jan 25 2015 *)

a(n)=my(p=prime(n));p^2>=precprime(p-1)*nextprime(p+1) \\ Charles R Greathouse IV, Jun 24 2011

a(n) = 1 if A056221(n-1)<=0, else a(n)=0.

All entries corrected. - R. J. Mathar, Charles R Greathouse IV Robert G. Wilson v, Jun 16 2011

A381374 Little Hankel transform of A317614: a(n) = A317614(n+1)^2 - A317614(n)*A317614(n+2).

Original entry on oeis.org

1, 1, 97, 49, 769, 289, 2977, 961, 8161, 2401, 18241, 5041, 35617, 9409, 63169, 16129, 104257, 25921, 162721, 39601, 242881, 58081, 349537, 82369, 487969, 113569, 663937, 152881, 883681, 201601, 1153921, 261121, 1481857, 332929, 1875169, 418609, 2342017, 519841, 2891041

Offset: 1

Stefano Spezia, Feb 21 2025

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A317614.

Cf. A056221, A056222, A239607, A374668.

LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{1,97,49,769,289,2977,961,8161,2401,18241},38]

a(n) = (10 + 6*(-1)^n + 4*n*(n + 2)*(3*(n + 1)^2 + (-1)^n*(2*n^2 + 4*n + 5)))/16.
a(n) = 5*a(n-2) - 10*a(n-4) + 10*a(n-6) - 5*a(n-8) + a(n-10) for n > 10.
G.f.: (1 + x + 92*x^2 + 44*x^3 + 294*x^4 + 54*x^5 + 92*x^6 - 4*x^7 + x^8 + x^9)/(1 - x^2)^5.
E.g.f.: ((4 + 3*x + 123*x^2 + 10*x^3 + 5*x^4)*cosh(x) + (1 + 69*x + 21*x^2 + 50*x^3 + x^4)*sinh(x))/4.
a(2*n) = A239607(n).

cp's OEIS Frontend

A056222 Image of partition numbers (A000041) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

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A056141 a(n) = primefloor(n)*primeceiling(n) - previousprime(n)*nextprime(n).

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A324795 a(n) = 2*p(n)*p(n+2) - p(n+1)^2 where p(k) = k-th prime.

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A342567 a(n) = (prime(n)^2 - prime(n-1)*prime(n+1))/2, n >= 3.

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A350200 Array read by antidiagonals: T(n,k) is the determinant of the Hankel matrix of the 2*n-1 consecutive primes starting at the k-th prime, n >= 0, k >= 1.

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A092386 Let p(n) = n-th prime; a(n) = largest prime factor of (p(n+1)^2 - p(n)*p(n+2)).

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Mathematica

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A106671 a(n) = ( prime(n + 2) * prime(n) - prime(n + 1)^2 ) modulo 3.

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Programs

Mathematica

PARI

Formula

A327434 Prime(n)^2 - prime(n-1)*prime(n+1) for n in A046868.

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A139312 Characteristic function of the good primes (version 1).

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A056141 a(n) = primefloor(n)primeceiling(n) - previousprime(n)nextprime(n).

A324795 a(n) = 2p(n)p(n+2) - p(n+1)^2 where p(k) = k-th prime.