A074170 -id:A074170 - cp's OEIS frontend

A074171 a(1) = 1. For n >= 2, a(n) is either a(n-1)+n or a(n-1)-n; we use the minus sign only if a(n-1) is prime. E.g., since a(2)=3 is prime, a(3)=a(2)-3=0.

1, 3, 0, 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, 99, 114, 130, 147, 165, 184, 204, 225, 247, 270, 294, 319, 345, 372, 400, 429, 459, 490, 522, 555, 589, 624, 660, 697, 735, 774, 814, 855, 897, 940, 984, 1029, 1075, 1122, 1170, 1219, 1269, 1320, 1372, 1425, 1479

Offset: 1

Amarnath Murthy, Aug 30 2002

In spite of the definition, this is simply 1, 3, then numbers of the form n*(n+7)/2 (A055999). In other words, a(n) = (n-3)*(n+4)/2 for n >= 3. The proof is by induction: For n > 3, a(n-1) = (n-4)*(n+3)/2 is composite, so a(n) = a(n-1) + n = (n-3)*(n+4)/2. - Dean Hickerson, T. D. Noe, Paul C. Leopardi, Labos Elemer and others, Oct 04 2004

If a 2-set Y and a 3-set Z, having one element in common, are subsets of an n-set X then a(n) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007

			a(1) = 1
a(2) = a(1) + 2 = 3, which is prime, so
a(3) = a(2) - 3 = 0, which is not prime, so
a(4) = a(3) + 4 = 4, which is not prime, etc.

Milan Janjic, Two Enumerative Functions.

Cf. A074170, A055999.

{ta={1, 3}, tb={{0}}};Do[s=Last[ta]; If[PrimeQ[s], ta=Append[ta, s-n]]; If[ !PrimeQ[s], ta=Append[ta, s+n]]; Print[{a=Last[ta], b=(n-3)*(n+4)/2, a-b}]; tb=Append[tb, a-b], {n, 3, 100000}]; {ta, {tb, Union[tb]}} (* Labos Elemer, Oct 07 2004 *)

a(1) = 1, a(2) = 3; a(n+1) = a(n)+n if a(n) is not a prime; a(n+1) = a(n)-n if a(n) is prime.

More terms from Jason Earls, Sep 01 2002

More terms from Labos Elemer, Oct 07 2004

A212427 a(n) = 17*n + A000217(n-1).

Original entry on oeis.org

0, 17, 35, 54, 74, 95, 117, 140, 164, 189, 215, 242, 270, 299, 329, 360, 392, 425, 459, 494, 530, 567, 605, 644, 684, 725, 767, 810, 854, 899, 945, 992, 1040, 1089, 1139, 1190, 1242, 1295, 1349, 1404, 1460, 1517, 1575, 1634, 1694, 1755, 1817, 1880, 1944, 2009

Offset: 0

Jesse Han, May 16 2012

Generalization: T(n,i) = A000217(i-1+n) - A000217(i-1) = i*n + A000217(n-1); in this case is i=17. See also the comment in A212428.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000096, A001008, A002805, A055998-A056000, A056115, A056119, A056121, A056126, A051942, A101859, A132754-A132758, A212428.

For n > 22, T(n,17) matches A074170 but with opposite sign.

[n*(n+33)/2: n in [0..49]]; // Bruno Berselli, Jun 22 2012

Table[-17 (17 - 1)/2 + (17 + n) (16 + n)/2, {n, 0, 100}]

a(n)=n*(n+33)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (16+n)*(17+n)/2 - 16*17/2 = 17*n + (n-1)*n/2 = n*(n+33)/2.
G.f.: x*(17-16*x)/(1-x)^3. - Bruno Berselli, Jun 22 2012
a(n) = 17*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
From Amiram Eldar, Jan 11 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*A001008(33)/(33*A002805(33)) = 53676090078349/216605329340400.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/33 - 14606816124167/340379803249200. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: exp(x)*x*(34 + x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A076543 a(n) = Sum_{k=1..n} k*sqf(k) where sqf(k)=1 if k is squarefree and sqf(k)=-1 otherwise.

Original entry on oeis.org

1, 3, 6, 2, 7, 13, 20, 12, 3, 13, 24, 12, 25, 39, 54, 38, 55, 37, 56, 36, 57, 79, 102, 78, 53, 79, 52, 24, 53, 83, 114, 82, 115, 149, 184, 148, 185, 223, 262, 222, 263, 305, 348, 304, 259, 305, 352, 304, 255, 205, 256, 204, 257, 203, 258, 202, 259, 317, 376, 316, 377

Offset: 1

Zak Seidov, Oct 19 2002

Surprisingly, first 12 terms are also in A074170.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A074170.

Accumulate[Table[If[SquareFreeQ[n],n,-n],{n,70}]] (* Harvey P. Dale, Mar 18 2015 *)

a(n) = sum(k = 1, n, if (issquarefree(k), k, -k)); \\ Michel Marcus, Oct 02 2013

cp's OEIS Frontend

A074171 a(1) = 1. For n >= 2, a(n) is either a(n-1)+n or a(n-1)-n; we use the minus sign only if a(n-1) is prime. E.g., since a(2)=3 is prime, a(3)=a(2)-3=0.

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A212427 a(n) = 17*n + A000217(n-1).

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A076543 a(n) = Sum_{k=1..n} k*sqf(k) where sqf(k)=1 if k is squarefree and sqf(k)=-1 otherwise.

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