A111181 -id:A111181 - cp's OEIS frontend

A065046 Numbers k such that prime(k) - pi(k) is a prime.

1, 2, 3, 4, 7, 9, 14, 15, 16, 19, 22, 30, 38, 44, 55, 64, 79, 81, 90, 91, 93, 94, 96, 101, 113, 115, 117, 118, 121, 122, 123, 124, 125, 133, 142, 148, 163, 173, 176, 178, 182, 183, 185, 189, 190, 199, 205, 206, 208, 223, 226, 232, 239, 240, 251, 253, 255, 263

Offset: 1

Robert G. Wilson v, Nov 05 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A065042, A111181.

Select[Range[300], PrimeQ[Prime[ # ] - PrimePi[ # ]] & ]

isok(m) = isprime(prime(m) - primepi(m)); \\ Harry J. Smith, Oct 04 2009

A111183 a(n) = prime(x) - pi(x) where x is the least x such that (prime(x+1) - pi(x+1)) - (prime(x) - pi(x)) = n.

Original entry on oeis.org

2, 3, 5, 15, 47, 19, 339, 80, 168, 128, 185, 196, 103, 275, 1771, 1871, 1028, 498, 3004, 851, 3641, 1087, 11845, 1613, 5402, 2404, 3182, 2889, 5225, 4190, 5461, 10585, 16958, 1280, 22444, 9357, 56241, 30129, 24857, 19006, 34461, 15852, 224417, 15401

Offset: 1

Cino Hilliard, Oct 22 2005

nonn

Conjecture: a(n) exists for every n.

Robert Israel, Table of n, a(n) for n = 1..262

Cf. A000040, A000720, A111181.

N:= 100: # for a(1) .. a(N)
p:= 2: m:= 0: b:= 2:V:= Vector(N): count:= 0:
for x from 2 while count < N do
  p:= nextprime(p);
  if isprime(x) then m:= m+1 fi;
  bp:= b; b:= p-m;
  v:= b-bp;
  if v >= 1 and v <= N and V[v] = 0 then V[v]:= bp; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Mar 16 2025

a(n) = { for(x=1, oo, my(y=prime(x)-primepi(x), z=prime(x+1)-primepi(x+1)); if(z-y == n,return(y)) ); }

A064777 Numbers k such that prime(k) - pi(k) is divisible by k.

Original entry on oeis.org

1, 2, 3, 18, 42, 95, 524, 273585, 1735537, 4406057, 4406063, 4406188, 4406196, 4406341, 4406539, 4406541, 28703894, 73694240, 73694281, 73694287, 73694360, 73694363, 73694410, 3287860772, 3287860773, 3287860880, 3287860889, 3287860895, 3287860897

Offset: 1

Labos Elemer and Robert G. Wilson v, Oct 18 2001

nonn

			k = 18 is a term: prime(18) = 61, pi(18) = 7, and (61-7)/18 = 54/18 = 3.

Cf. A000040, A000720, A065139, A111181.

Do[ If[ IntegerQ[ (Prime[n] - PrimePi[n])/n ], Print[n]], {n, 1, 6*10^7} ]
Select[Range[5000000],Divisible[Prime[#]-PrimePi[#],#]&] (* Harvey P. Dale, Aug 12 2013 *)

isok(k) = ((prime(k) - primepi(k)) % k) == 0; \\ Michel Marcus, Jun 15 2021

a(18)-a(23) from Donovan Johnson, Apr 23 2010

a(24)-a(29) from Donovan Johnson, Oct 17 2012

cp's OEIS Frontend

A065046 Numbers k such that prime(k) - pi(k) is a prime.

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A111183 a(n) = prime(x) - pi(x) where x is the least x such that (prime(x+1) - pi(x+1)) - (prime(x) - pi(x)) = n.

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A064777 Numbers k such that prime(k) - pi(k) is divisible by k.

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