A098699 Anti-derivative of n: or the first occurrence of n in A003415, or zero if impossible.

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, 69, 27, 115, 0, 161, 30, 16, 62, 93, 0, 155, 0, 217, 45, 111, 42, 185, 82, 24, 50, 129, 0, 44, 94, 141, 63, 235, 0, 329, 75, 52, 0, 265, 70, 36, 66, 177, 122, 183, 0, 305, 0, 40, 134

Offset: 0

Author

Robert G. Wilson v, Sep 21 2004

Keywords

• nonn

Comments

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

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A003415, A051674, zeros in A098700.

Programs

Maple

ader:= proc(n) local t;
  n * add(t[2]/t[1], t = ifactors(n)[2])
end proc:
N:= 100: # for a(0) .. a(N)
V:= Array(0..N): count:= 0:
for x from 1 to N^2/4 while count < 100 do
  v:= ader(x);
  if v > 0 and v <= 100 and V[v] = 0 then
    count:= count+1; V[v]:= x;
  fi;
od:
convert(V,list); # Robert Israel, May 29 2023

Mathematica

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

PARI

A098699(n)=for(k=1,(n\2)^2+2,A003415(k)==n&&return(k)) \\ M. F. Hasler, Apr 09 2015

Python

from sympy import factorint
def A098699(n):
    if n < 2:
        return n+1
    for m in range(1,(n**2>>2)+1):
        if sum((m*e//p for p,e in factorint(m).items())) == n:
            return m
    return 0 # Chai Wah Wu, Sep 12 2022

Formula

a(n) = n for { 4, 27, 3125, 823543, ... } = { p^p; p prime } = A051674.