A273286 Positive integers n such that n=p+q for some primes p,q with pi(p)*pi(q) = sigma(n).

92, 130, 132, 136, 154, 270, 286, 384, 398, 456, 546, 608, 630, 636, 702, 934, 944, 2730, 4394, 4470, 4556, 5544, 12084, 14320, 17572, 22632, 27808, 27930, 31150, 31284, 32534, 36346, 41004, 41544, 42274, 56916, 58552, 61680, 66654, 74826, 86200

Offset: 1

Author

Giuseppe Coppoletta, Jun 20 2016

Keywords

• nonn

Comments

Equivalently, integers n such that sigma(n) = i*j for some i,j with prime(i)+prime(j) = n. Each term is necessarily even, otherwise if n is odd n=2+q < sigma(2+q) = pi(2)*pi(q) = pi(q) < q which is absurd. Also p and q cannot be equal, otherwise sigma(2*p) = 3*(p+1) = pi(p)^2 with no solution.

Conjecture: the sequence is infinite and each term has only one decomposition into a sum of suitable primes p,q.

Using Rosser's theorem we can show that the primes p,q >= 19 and each of them can only occur for a finite number of terms n. - Robert Israel, Jun 30 2016

Examples

			92 = 19 + 73 with pi(19) * pi(73) = 8 * 21 = 168 = sigma(92).
		

Links

• Eric Weisstein's World of Mathematics, Rosser's Theorem

Crossrefs

Cf. A000203, A000720, A272860, A272862.

Programs

Maple

N:= 10^6: # to use primes up to N
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
filter:= proc(n) local s,i,j;
  s:= numtheory:-sigma(n);
  for i in select(`>=`,numtheory:-divisors(s), ceil(sqrt(s))) minus {s} do
     if i > nops(Primes) then return FAIL
     elif Primes[i] + Primes[s/i] = n then return true fi
  od:
  false
end proc:
A:= NULL:
for n from 2 by 2 do
  v:= filter(n);
  if v = FAIL then break
  elif v then A:= A, n
  fi
od:
A; # Robert Israel, Jun 30 2016

Mathematica

Select[Range[10^3], Function[n, Length@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], And[Times @@ Boole@ PrimeQ@ {First@ #, Last@ #} == 1, DivisorSigma[1, First@ # + Last@ #] == PrimePi[First@ #] PrimePi[Last@ #]] &] > 0]] (* Michael De Vlieger, Jun 30 2016 *)

PARI

is(n) = if(n%2==1, return(0), my(x=n-1, y=1); while(x > y, if(ispseudoprime(x) && ispseudoprime(y) && sigma(x+y)==primepi(x)*primepi(y), return(1)); x--; y++); return(0)) \\ Felix Fröhlich, Jun 28 2016

PARI

is(n) = my( d=divisors(sigma(n))); for(i=1,ceil(#d/2), if(prime(d[i]) + prime(d[#d + 1-i]) == n, return(1))); return(0) \\ David A. Corneth, Jun 30 2016

Sage

def sol(n): return [j for j in divisors(sigma(n)) if j^2<= sigma(n) and is_prime(n-nth_prime(j)) and j * prime_pi(n-nth_prime(j))==sigma(n)]
v=[n for n in range(2,100000,2) if sol(n)]
print('list_n =',v)
w=[sigma(n) for n in v]; print('list_sigma(n) =',w)
list_pi(p)=flatten([sol(n) for n in range(2,100000,2) if sol(n)])
print('list_pi(p) =',list_pi(p))
list_pi(q)=[w[n]/list_pi[n] for n in range(len(v))]
print('list_pi(q) =',list_pi(q))

Formula

Integers n such that sigma(n) = pi(q) * pi(n-q) for some prime q.