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A063960
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Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.
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2
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0, 0, 0, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 17, 17, 17, 17, 17, 17, 17, 17, 28, 28, 28, 28, 41, 41, 41, 41, 41, 41, 41, 41, 58, 58, 58, 58, 77, 77, 77, 77, 77, 77, 77, 77, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 129, 129, 129, 129, 160, 160, 160, 160
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OFFSET
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1,4
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COMMENTS
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Sum of the prime numbers among the smallest parts of the partitions of n into two parts. For example, a(8)=5; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). The prime numbers among the smallest parts are 2 and 3, so 2 + 3 = 5. - Wesley Ivan Hurt, Nov 01 2017
Number of distinct rectangles with integer length and prime width such that L + W = n, W <= L. For a(14)=17; the rectangles are 2 X 12, 3 X 11, 5 X 9, and 7 X 7. The sum of the lengths are then 2+3+5+7 = 17. - Wesley Ivan Hurt, Nov 08 2017
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LINKS
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FORMULA
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EXAMPLE
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20! = (2^18)*(3^8)*(5^4)*(7^2)*11*13*17*19, the non-unitary prime divisors are {2, 3, 5, 7}, so a(20) = 2 + 3 + 5 + 7 = 17.
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MAPLE
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seq(add(j, j=select(isprime, [$1..iquo(n, 2)])), n=1..65); # Peter Luschny, Nov 28 2022
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MATHEMATICA
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Join[{0, 0, 0}, Table[Total[Transpose[Select[FactorInteger[n!], Last[#]>1&]][[1]]], {n, 4, 70}]] (* Harvey P. Dale, Jun 19 2013 *)
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PROG
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(PARI) { for (n=1, 1000, f=factor(n!)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063960.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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