A085342 -id:A085342 - cp's OEIS frontend

A085341 Number of primes between sigma(n) and n inclusive.

0, 1, 0, 2, 0, 2, 0, 2, 2, 3, 0, 4, 0, 3, 3, 5, 0, 5, 0, 5, 3, 3, 0, 8, 2, 4, 3, 7, 0, 10, 0, 7, 4, 5, 4, 13, 0, 5, 4, 12, 0, 11, 0, 9, 7, 6, 0, 15, 1, 9, 5, 10, 0, 14, 4, 14, 6, 8, 0, 22, 0, 6, 9, 13, 5, 16, 0, 11, 5, 15, 0, 24, 0, 9, 9, 13, 3, 18, 0, 20, 8

Offset: 1

Labos Elemer, Jul 10 2003

nonn

			n = 12: sigma(n) = 28, pi(28) - pi(12) = 9 - 5 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000720, A000203, A070803, A070804, A085342-A085347.

Table[PrimePi[DivisorSigma[1,n]]-PrimePi[n],{n,90}] (* Harvey P. Dale, Aug 18 2015 *)

a(n) = primepi(sigma(n)) - primepi(n); \\ Michel Marcus, Dec 15 2013

a(n) = pi(sigma(n)) - pi(n) = A000720(A000203(n)) - A000720(n).

A074398 Number of primes between n and phi(n), with neither n nor phi(n) included in the count in case they are primes.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 3, 0, 3, 2, 2, 0, 4, 0, 4, 3, 4, 0, 5, 1, 4, 2, 4, 0, 6, 0, 5, 3, 5, 2, 6, 0, 5, 3, 6, 0, 8, 0, 6, 5, 6, 0, 9, 2, 7, 4, 6, 0, 9, 4, 7, 5, 7, 0, 11, 0, 8, 7, 7, 3, 10, 0, 8, 5, 10, 0, 11, 0, 10, 9, 10, 4, 12, 0, 11, 6, 10, 0, 14, 5, 10, 7, 11, 0, 15, 4, 10, 7, 10, 4, 13, 0

Offset: 1

Joseph L. Pe, Sep 24 2002

			phi(6) = 2 and there are 2 primes between 2 and 6 (endpoints are excluded), namely 3, 5. Hence a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A000720, A010051, A085342.

(*gives number of primes < n*) f[n_] := Module[{r, i}, r = 0; i = 1; While[Prime[i] < n, i++ ]; i - 1]; (*gives number of primes between m and n with endpoints excluded*) g[m_, n_] := Module[{r}, r = f[m] - f[n]; If[PrimeQ[n], r = r - 1]; r]; Table[g[n, EulerPhi[n]], {n, 1, 100}]
(* Second program: *)
Array[PrimePi@ # - PrimePi@ EulerPhi@ # - Boole@ PrimeQ@ # &, 96] (* or *) Array[Count[Range[EulerPhi@ # + 1, # - 1], ?PrimeQ] &, 96] (* _Michael De Vlieger, Dec 16 2017 *)

A074398(n) = (primepi(n) - primepi(eulerphi(n)) - isprime(n)); \\ Antti Karttunen, Dec 16 2017

a(n) = A085342(n) - A010051(n) = A000720(n) - A000720(A000010(n)) - A010051(n). - Antti Karttunen, Dec 16 2017

Name clarified by Antti Karttunen, Dec 16 2017

A085345 Least number x such that number of primes between phi(x) and x equals n.

Original entry on oeis.org

2, 6, 12, 18, 24, 30, 50, 42, 48, 66, 60, 78, 96, 84, 90, 130, 108, 176, 114, 132, 156, 182, 150, 168, 186, 180, 216, 198, 228, 429, 210, 258, 308, 240, 276, 282, 270, 306, 294, 300, 354, 366, 336, 330, 384, 378, 396, 360, 432, 438, 622, 444, 390, 490, 474, 498

Offset: 1

Labos Elemer, Jul 10 2003

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A000010, A000203, A070803, A070804, A085341-A085347.

N:= 200: # for a(1)..a(N)
V:= Vector(N):
count:= 0:
for x from 1 while count < N do
  v:= numtheory:-pi(x) - numtheory:-pi(numtheory:-phi(x));
  if v >= 1 and v <= N and V[v] = 0 then
    V[v]:= x; count:= count+1;
  fi
od:
convert(V,list); # Robert Israel, Aug 23 2018

a(n)=Min{x; A085342(x)=n}

A085344 Least number x such that number of primes between sigma(x) and x equals n.

Original entry on oeis.org

2, 4, 10, 12, 16, 46, 28, 24, 44, 30, 42, 40, 36, 54, 48, 66, 178, 78, 104, 80, 102, 60, 128, 72, 84, 152, 90, 138, 255, 96, 108, 174, 140, 126, 132, 266, 160, 150, 248, 222, 156, 120, 246, 200, 144, 198, 634, 224, 220, 204, 370, 260, 168, 376, 555, 430, 354, 308

Offset: 1

Labos Elemer, Jul 10 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A000010, A000203, A070803, A070804.

Cf. A085341, A085342, A085343, A085345, A085346, A085347.

m = 100; seq = Table[0, {m}]; c = 0; n = 0; While[c < m, n++; i = PrimePi[ DivisorSigma[1, n]] - PrimePi[n]; If[i <= m && seq[[i]] == 0, c++; seq[[i]] = n]]; seq (* Amiram Eldar, Mar 01 2020 *)

a(n) = {my(x=1); while (primepi(sigma(x)) - primepi(x) != n, x++); x;} \\ Michel Marcus, Mar 01 2020

a(n) = Min{x; A085341(x)=n}.

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A085341 Number of primes between sigma(n) and n inclusive.

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A074398 Number of primes between n and phi(n), with neither n nor phi(n) included in the count in case they are primes.

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A085345 Least number x such that number of primes between phi(x) and x equals n.

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A085344 Least number x such that number of primes between sigma(x) and x equals n.

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