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A294095
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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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2
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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
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1,7
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COMMENTS
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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
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
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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(n-P[j]) then t:= t+n-2*P[j] fi
od;
t
end proc:
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = sum(i=1, n\2, (n - 2*i)*isprime(i)*issquarefree(n - i)); \\ Michel Marcus, Mar 24 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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