

A294095


Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part squarefree and smaller part prime.


2



0, 0, 0, 0, 1, 0, 3, 6, 8, 4, 1, 10, 16, 8, 16, 28, 27, 24, 24, 30, 24, 36, 19, 64, 43, 52, 18, 64, 38, 28, 48, 70, 86, 72, 65, 122, 102, 88, 82, 152, 121, 136, 98, 182, 141, 164, 71, 188, 164, 160, 83, 136, 141, 192, 101, 176, 118, 236, 114, 304, 150, 240
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OFFSET

1,7


COMMENTS

It appears that the four bands visible in the scatterplot correspond, from bottom to top, to n == 1 or 5 (mod 6), n == 3 (mod 6), n == 2 or 4 (mod 6), and n == 0 (mod 6) respectively.  Robert Israel, Mar 13 2018
Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*xx^2 at prime values of x such that nx is squarefree for x in 0 < x <= floor(n/2). For example, d/dx n*xx^2 = n2x. So for a(12), the prime values of x which make 12x squarefree are x=2,5 and so a(12) = 122*2 + 122*5 = 8 + 2 = 10.  Wesley Ivan Hurt, Mar 24 2018


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions


FORMULA

a(n) = Sum_{i=1..floor(n/2)} (n  2i) * A010051(i) * A008966(n  i).


EXAMPLE

For n=13, the partitions of n of the form (larger squarefree + smaller prime) are 11 + 2 and 10 + 3. So a(13) = (11  2) + (10  3) = 16.  Michael B. Porter, Mar 14 2018


MAPLE

N:= 1000: # to get a(1)..a(N)
P:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
nP:= nops(P):
f:= proc(n) local j, t;
t:= 0:
for j from 1 to nP while 2*P[j]<n do
if numtheory:issqrfree(nP[j]) then t:= t+n2*P[j] fi
od;
t
end proc:
map(f, [$1..N]); # Robert Israel, Mar 13 2018


MATHEMATICA

Table[Sum[(n  2 i) (PrimePi[i]  PrimePi[i  1]) MoebiusMu[n  i]^2, {i, Floor[n/2]}], {n, 80}]
Table[Total[Abs[Flatten[Differences/@Select[IntegerPartitions[n, {2}], SquareFreeQ[ #[[1]]]&&PrimeQ[#[[2]]]&]]]], {n, 70}] (* Harvey P. Dale, Aug 02 2021 *)


PROG

(PARI) a(n) = sum(i=1, n\2, (n  2*i)*isprime(i)*issquarefree(n  i)); \\ Michel Marcus, Mar 24 2018


CROSSREFS

Cf. A010051, A008966, A294096.
Sequence in context: A010621 A258232 A296568 * A306633 A096416 A232717
Adjacent sequences: A294092 A294093 A294094 * A294096 A294097 A294098


KEYWORD

easy,nonn,look


AUTHOR

Wesley Ivan Hurt, Oct 22 2017


STATUS

approved



