A258087 -id:A258087 - cp's OEIS frontend

A272058 Start with all terms set to 0. Then add n to the next n+3 terms for n=0,1,2,... .

0, 0, 1, 3, 6, 10, 14, 20, 25, 33, 39, 49, 56, 68, 76, 90, 99, 115, 125, 143, 154, 174, 186, 208, 221, 245, 259, 285, 300, 328, 344, 374, 391, 423, 441, 475, 494, 530, 550, 588, 609, 649, 671, 713, 736, 780, 804, 850, 875, 923, 949, 999, 1026, 1078, 1106

Offset: 0

Wesley Ivan Hurt, Apr 19 2016

			n  | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10, ...
__________________________________________
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
+       0, 0, 0
+          1, 1, 1, 1,
+             2, 2, 2, 2, 2
+                3, 3, 3, 3, 3, 3
+                   4, 4, 4, 4, 4, 4, 4
+                      5, 5, 5, 5, 5, 5, 5, 5
+                         6, 6, 6, 6, 6, 6, 6, 6, 6
+                            7, 7, 7, 7, 7, 7, 7, 7, 7, 7
+                               8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
+                                  9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
+                                     ...
__________________________________________
a(n)|0, 0, 1, 3, 6,10,14,20,25,33,39, ...

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

Cf. A095794, A258087.

[0,0] cat [(6*n^2+6*n-23+(7-2*n)*(-1)^n)/16 : n in [2..100]];

A272058:=n->(6*n^2+6*n-23+(7-2*n)*(-1)^n)/16: 0,0,seq(A272058(n),n=2..100);

CoefficientList[Series[x^2*(1 + 2 x + x^2 - x^4)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1, 0, 0}, {0, 0, 1, 3, 6, 10, 14}, 60]

G.f.: x^2*(1 + 2*x + x^2 - x^4)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 6.
a(n) = (6*n^2 + 6*n - 23 + (7 - 2*n)*(-1)^n)/16 for n > 1.
a(n) = floor((n+3)/4) * floor((3*n-4)/2) + (floor((n-1)/2) mod 2) * floor((3*n-3)/4) for n > 1.
For n > 1, a(2n) = A095794(n). - Jon E. Schoenfield, Feb 19 2022

A288656 a(n) = Sum_{k=1..n} Sum_{i=floor((k-1)/2)..k-1} i * c(i), where c is the prime characteristic (A010051).

Original entry on oeis.org

0, 0, 0, 2, 7, 12, 22, 30, 45, 57, 69, 81, 104, 122, 153, 184, 215, 239, 280, 321, 381, 441, 501, 561, 644, 716, 788, 860, 932, 991, 1079, 1167, 1286, 1405, 1524, 1643, 1762, 1864, 2003, 2142, 2281, 2401, 2562, 2723, 2927, 3131, 3335, 3539, 3790, 4018, 4246, 4474, 4702, 4930, 5211, 5492, 5773, 6054, 6335, 6616, 6956, 7267, 7639

Offset: 0

Wesley Ivan Hurt, Jun 12 2017

Sum of all the primes appearing in columns less than or equal to n from the example in A258087.

Cf. A010051, A258087.

Partial Sums of A288726.

with(numtheory): A288656:=n->add(add(i*(pi(i)-pi(i-1)), i=floor((k-1)/2)..k-1), k=1..n): seq(A288656(n), n=0..100);

Table[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, Floor[(k - 1)/2], k - 1}], {k, n}], {n, 0, 62}] (* Michael De Vlieger, Jun 15 2017 *)

A288726 a(n) = Sum_{i=floor((n-1)/2)..n-1} i * c(i), where c is the prime characteristic (A010051).

Original entry on oeis.org

0, 0, 0, 2, 5, 5, 10, 8, 15, 12, 12, 12, 23, 18, 31, 31, 31, 24, 41, 41, 60, 60, 60, 60, 83, 72, 72, 72, 72, 59, 88, 88, 119, 119, 119, 119, 119, 102, 139, 139, 139, 120, 161, 161, 204, 204, 204, 204, 251, 228, 228, 228, 228, 228, 281, 281, 281, 281, 281, 281, 340, 311, 372, 372, 372, 341, 341, 341, 408

Offset: 0

Wesley Ivan Hurt, Jun 14 2017

Sum of the primes in the n-th column of the example in A258087.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A010051, A258087, A288656.

with(numtheory): A288726:=n->add(i*(pi(i)-pi(i-1)), i=floor((n-1)/2)..n-1): seq(A288726(n), n=0..100);
# Alternative:
M:= 100: # to get a(0) to a(2*M+1)
A:= Array(0..2*M+1):
A[3]:= 2:
for k from 2 to M do
   if isprime(2*k-1) then A[2*k]:= A[2*k-1]+2*k-1 else A[2*k]:=A[2*k-1] fi;
   if isprime(k-1) then A[2*k+1]:= A[2*k]-(k-1) else A[2*k+1]:= A[2*k] fi;
od:
convert(A,list); # Robert Israel, Jun 16 2017

Table[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, Floor[(n - 1)/2], n - 1}], {n, 0, 68}] (* Michael De Vlieger, Jun 14 2017 *)

a(n) = sum(i=floor((n-1)/2), n-1, i*isprime(i)) \\ Felix Fröhlich, Jun 16 2017

From Robert Israel, Jun 16 2017: (Start)
For k >= 2, a(2*k+1) - a(2*k) = 1-k if k-1 is prime, otherwise 0.
a(2*k+2) - a(2*k+1) = 2*k+1 if 2*k+1 is prime, otherwise 0. (End)
a(n) = A288656(n) - A288656(n-1), n>=1. - Wesley Ivan Hurt, Dec 26 2023

cp's OEIS Frontend

A272058 Start with all terms set to 0. Then add n to the next n+3 terms for n=0,1,2,... .

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A288656 a(n) = Sum_{k=1..n} Sum_{i=floor((k-1)/2)..k-1} i * c(i), where c is the prime characteristic (A010051).

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A288726 a(n) = Sum_{i=floor((n-1)/2)..n-1} i * c(i), where c is the prime characteristic (A010051).

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