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%I #27 May 30 2023 07:45:38
%S 1,2,0,0,4,6,9,10,15,14,21,0,8,22,33,26,12,0,65,34,51,18,57,0,20,46,
%T 69,27,115,0,161,30,16,62,93,0,155,0,217,45,111,42,185,82,24,50,129,0,
%U 44,94,141,63,235,0,329,75,52,0,265,70,36,66,177,122,183,0,305,0,40,134
%N Anti-derivative of n: or the first occurrence of n in A003415, or zero if impossible.
%C With Goldbach's conjecture, any even integer n = 2k > 2 can be written as sum of two primes, n = p + q, and therefore admits N = pq as (not necessarily smallest) anti-derivative, so a(2k) > 0, and a(2k) <= pq <= k^2. [Remark inspired by L. Polidori.] - _M. F. Hasler_, Apr 09 2015
%C a(n) <= n^2/4 for n > 1. This is because if A003415(x) = n > 1, x = a*b for some a,b > 1, and then n = A003415(x) = a*A003415(b) + A003415(a)*b >= a + x/a >= 2*sqrt(x), i.e. x <= (n/2)^2. - _Robert Israel_, May 29 2023
%H T. D. Noe, <a href="/A098699/b098699.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) = n for { 4, 27, 3125, 823543, ... } = { p^p; p prime } = A051674.
%p ader:= proc(n) local t;
%p n * add(t[2]/t[1], t = ifactors(n)[2])
%p end proc:
%p N:= 100: # for a(0) .. a(N)
%p V:= Array(0..N): count:= 0:
%p for x from 1 to N^2/4 while count < 100 do
%p v:= ader(x);
%p if v > 0 and v <= 100 and V[v] = 0 then
%p count:= count+1; V[v]:= x;
%p fi;
%p od:
%p convert(V,list); # _Robert Israel_, May 29 2023
%t a[1] = 0; a[n_] := Block[{f = Transpose[ FactorInteger[ n]]}, If[ PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; b = Table[0, {70}]; b[[1]] = 1; Do[c = a[n]; If[c < 70 && b[[c + 1]] == 0, b[[c + 1]] = n], {n, 10^3}]; b
%o (PARI) A098699(n)=for(k=1,(n\2)^2+2,A003415(k)==n&&return(k)) \\ _M. F. Hasler_, Apr 09 2015
%o (Python)
%o from sympy import factorint
%o def A098699(n):
%o if n < 2:
%o return n+1
%o for m in range(1,(n**2>>2)+1):
%o if sum((m*e//p for p,e in factorint(m).items())) == n:
%o return m
%o return 0 # _Chai Wah Wu_, Sep 12 2022
%Y Cf. A003415, A051674, zeros in A098700.
%K nonn
%O 0,2
%A _Robert G. Wilson v_, Sep 21 2004
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