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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.
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%I #31 Aug 02 2021 11:06:56

%S 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,

%T 64,38,28,48,70,86,72,65,122,102,88,82,152,121,136,98,182,141,164,71,

%U 188,164,160,83,136,141,192,101,176,118,236,114,304,150,240

%N 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.

%C It appears that the four bands visible in the scatter-plot 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

%C Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*x-x^2 at prime values of x such that n-x is squarefree for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(12), the prime values of x which make 12-x squarefree are x=2,5 and so a(12) = 12-2*2 + 12-2*5 = 8 + 2 = 10. - _Wesley Ivan Hurt_, Mar 24 2018

%H Robert Israel, <a href="/A294095/b294095.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e 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

%p N:= 1000: # to get a(1)..a(N)

%p P:= select(isprime, [2,seq(i,i=3..N/2,2)]):

%p nP:= nops(P):

%p f:= proc(n) local j,t;

%p t:= 0:

%p for j from 1 to nP while 2*P[j]<n do

%p if numtheory:-issqrfree(n-P[j]) then t:= t+n-2*P[j] fi

%p od;

%p t

%p end proc:

%p map(f, [$1..N]); # _Robert Israel_, Mar 13 2018

%t Table[Sum[(n - 2 i) (PrimePi[i] - PrimePi[i - 1]) MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 80}]

%t Table[Total[Abs[Flatten[Differences/@Select[IntegerPartitions[n,{2}],SquareFreeQ[ #[[1]]]&&PrimeQ[#[[2]]]&]]]],{n,70}] (* _Harvey P. Dale_, Aug 02 2021 *)

%o (PARI) a(n) = sum(i=1, n\2, (n - 2*i)*isprime(i)*issquarefree(n - i)); \\ _Michel Marcus_, Mar 24 2018

%Y Cf. A010051, A008966, A294096.

%K easy,nonn,look

%O 1,7

%A _Wesley Ivan Hurt_, Oct 22 2017