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

0,2

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

Formula

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

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

Prog

(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

Crossrefs

Cf. A003415, A051674, zeros in A098700.

Sequence in context: A333706 A056676 A352450 * A021837 A236934 A155719

Adjacent sequences: A098696 A098697 A098698 * A098700 A098701 A098702

Keyword

nonn

Author

Robert G. Wilson v, Sep 21 2004

Status

approved