A272860 -id:A272860 - cp's OEIS frontend

A272861 Sum of two integers when equal to the product of their prime-counting functions.

12, 16, 18, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 116, 120, 280, 310, 325, 330, 942, 948, 954, 960, 966, 972, 984, 990, 996, 2968, 3003, 8224, 8232, 8240, 8248, 8280, 8288, 8304, 8312, 8360, 8408, 23499, 23508, 23589

Offset: 1

Giuseppe Coppoletta, Jun 18 2016

nonn

The sums are listed in increasing order. The only term with equal addends is a(2)= 16 = 8 + 8 = pi(8)^2, Indeed j=8 is the only solution to pi(j)^2 = 2*j, which is easily seen using pi(j) > j/log(j) for j>16.

			32 is a term because 32 = 10 + 22 = 4 * 8 = pi(10) * pi(22).

Giovanni Resta, Table of n, a(n) for n = 1..320 (terms < 4*10^9)
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A272860, A273286, A000720.

nn = 10^3; Select[Range@ nn, Function[k, IntegerQ@ SelectFirst[Range@ nn, k == PrimePi[#] PrimePi[k - #] &]]] (* Version 10, or *)
Select[Range[10^3], Function[n, Length@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], n == PrimePi[First@ #] PrimePi[Last@ #] &] > 0]] (* Michael De Vlieger, Jun 30 2016 *)

def g(n): return [k for k in (3..n/2) if n==prime_pi(k)*prime_pi(n-k)]
[n for n in range(2,1000) if len(g(n))>0]

Positive integers n such that n = pi(j) * pi(n-j) for some j.

a(50)-a(53) from Giovanni Resta, Jun 29 2016

A272862 Positive integers j such that prime(i) + prime(j) = i*j for some i <= j.

Original entry on oeis.org

4, 6, 8, 24, 29, 30, 164, 165, 166, 1051, 2624, 2638, 2650, 2670, 2674, 2676, 40027, 40028, 40112, 251701, 251703, 251706, 251751, 637144, 637202, 637216, 637220, 1617162, 1617165, 4124694, 10553383, 10553408, 10553464, 10553533, 10553535, 10553839, 69709686

Offset: 1

Giuseppe Coppoletta, Jul 25 2016

nonn

Also pi(q) for primes q verifying p+q = pi(p)*pi(q) for some prime p <= q.

The list of products i*j gives A272860. See also comments there.

			8 is a term as prime(3) + prime(8) = 3*8.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..43

Cf. A272860, A272861, A000720, A000040.

Select[Range[3000], Function[j, Total@ Boole@ Map[Prime@ # + Prime@ j == # j &, Range@ j] > 0]] (* Michael De Vlieger, Jul 28 2016 *)

is(n) = for(i=1, n, if(prime(i)+prime(n)==i*n, return(1))); return(0) \\ Felix Fröhlich, Jul 27 2016

is(n,p=prime(n))=my(i); forprime(q=2,p, if(i++*n==p+q, return(1))); 0
v=List(); n=0; forprime(p=2,1e6, if(is(n++,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jul 28 2016

def sol(n):
    if n<5: a=n
    else: a=exp(n+1)/(n+1)
    b=(n-1)/n^2*exp(n^2/(n-1.1))
    return [j for j in range(a,b) if is_prime(n*j-nth_prime(n)) and prime_pi(n*j-nth_prime(n))==j]
flatten([sol(i) for i in (1..15) if len(sol(i))>0]) #

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

Original entry on oeis.org

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

Giuseppe Coppoletta, Jun 20 2016

nonn

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

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

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

Cf. A000203, A000720, A272860, A272862.

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

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 *)

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

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

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))

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

cp's OEIS Frontend

A272861 Sum of two integers when equal to the product of their prime-counting functions.

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A272862 Positive integers j such that prime(i) + prime(j) = i*j for some i <= j.

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

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