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A161670 Sum of largest prime factor of composite(k) for k from smallest prime factor of composite(n) to largest prime factor of composite(n). 0

%I #7 Sep 08 2022 08:45:45

%S 3,5,3,2,13,5,23,10,3,5,13,20,38,5,5,56,2,23,13,3,35,80,15,5,92,53,13,

%T 23,38,10,129,5,7,13,77,56,5,30,23,89,187,13,215,20,3,48,38,80,126,23,

%U 5,263,10,92,22,56,13,2,329,23,72,365,184,38,13,40,129,212,398,84,5,23,35

%N Sum of largest prime factor of composite(k) for k from smallest prime factor of composite(n) to largest prime factor of composite(n).

%C "composite(n)" stands for "n-th composite number", so composite(1) to composite(8) are 4, 6, 8, 9, 10, 12, 14, 15.

%e composite(1) = 4; (smallest prime factor of 4) = (largest prime factor of 4) = 2. composite(2) = 6, (largest prime factor of 6) = 3. Hence a(1) = 3.

%e composite(5) = 10; (smallest prime factor of 10) = 2, (largest prime factor of 10) = 5. composite(2) to composite(5) are 6, 8, 9, 10, largest prime factors are 3, 2, 3, 5. Hence a(5) = 3+2+3+5 = 13.

%e composite(7) = 14; (smallest prime factor of 14) = 2, (largest prime factor of 14) = 7. composite(2) to composite(7) are 6, 8, 9, 10, 12, 14, largest prime factors are 3, 2, 3, 5, 3, 7. Hence a(5) = 3+2+3+5+3+7 = 23.

%o (Magma) Composites:=[ j: j in [4..100] | not IsPrime(j) ];

%o [ &+[ E[ #E] where E is PrimeDivisors(Composites[k]): k in [D[1]..D[ #D]] where D is PrimeDivisors(Composites[n]) ]: n in [1..73] ]; // _Klaus Brockhaus_, Jun 25 2009

%Y Cf. A002808 (composite numbers), A111426 (difference between largest and smallest prime factor of composite(n)).

%K nonn

%O 1,1

%A _Juri-Stepan Gerasimov_, Jun 16 2009, Jun 18 2009

%E Edited, corrected (a(39)=33 replaced by 23, a(40)=84 replaced by 89) and extended by _Klaus Brockhaus_, Jun 25 2009

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