A070804 -id:A070804 - cp's OEIS frontend

A070803 Number of primes not exceeding sum of divisors of n.

0, 2, 2, 4, 3, 5, 4, 6, 6, 7, 5, 9, 6, 9, 9, 11, 7, 12, 8, 13, 11, 11, 9, 17, 11, 13, 12, 16, 10, 20, 11, 18, 15, 16, 15, 24, 12, 17, 16, 24, 13, 24, 14, 23, 21, 20, 15, 30, 16, 24, 20, 25, 16, 30, 20, 30, 22, 24, 17, 39, 18, 24, 27, 31, 23, 34, 19, 30, 24, 34, 20, 44, 21, 30, 30

Offset: 1

Labos Elemer, May 08 2002

			n=50: sigma(50) = 93, pi(93) = 24 = a(50).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A007818, A000203, A000720, A070800-A070804.

[#PrimesUpTo(SumOfDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Feb 06 2017

Table[PrimePi[DivisorSigma[1, n]], {n, 1, 256}]

A070803(n) = primepi(sigma(n)) \\ Michael B. Porter, Jan 28 2010

[prime_pi(sigma(n,1)) for n in range(1, 76)] # - Zerinvary Lajos, Jun 06 2009

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

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

Original entry on oeis.org

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).

A085347 Least number x so that number of primes not exceeding sigma(x) equals n.

Original entry on oeis.org

0, 2, 5, 4, 6, 8, 10, 19, 12, 29, 16, 18, 20, 43, 33, 28, 24, 32, 67, 30, 45, 57, 44, 36, 52, 101, 63, 85, 91, 48, 64, 86, 137, 66, 149, 99, 106, 163, 60, 129, 88, 80, 105, 72, 197, 104, 100, 84, 147, 229, 90, 114, 112, 96, 171, 148, 136, 201, 108, 281, 138, 152, 172

Offset: 1

Labos Elemer, Jul 10 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..500

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

{0}~Join~Table[x = 1; While[PrimePi@ DivisorSigma[1, x] != n, x++]; x, {n, 2, 63}] (* Michael De Vlieger, Feb 05 2017 *)

a(n) = if (n==1, 0, x = 1; while(primepi(sigma(x))!= n,  x++); x); \\ Michel Marcus, Dec 11 2013

a(n) = Min{x; A070803(x)=n};

A085342 Number of primes between phi(n) and n, where n is included in the count if it is a prime, while phi(n) is never included in the count even if it is a prime.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 10 2003

Number of primes in (phi(n), n]. - Charles R Greathouse IV, Dec 26 2013

			n=12: phi(n)=4, pi(12)-pi(4)=5-2=3.

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

Cf. A000720, A000010, A010051, A070803, A070804, A074398, A085341-A085347.

Array[PrimePi[#] - PrimePi@ EulerPhi@ # &, 97] (* Michael De Vlieger, Dec 16 2017 *)

a(n) = primepi(n) - primepi(eulerphi(n)); \\ Michel Marcus, Dec 26 2013

a(n) = pi(n) - pi(phi(n)) = A000720(n) - A000720(A000010(n)).

a(n) = A074398(n) + A010051(n). - Antti Karttunen, Dec 16 2017

Name clarified by Antti Karttunen, Dec 16 2017

A085343 Number of primes between sigma(n) and phi(n) inclusive.

Original entry on oeis.org

0, 2, 1, 3, 1, 4, 1, 4, 3, 5, 1, 7, 1, 6, 5, 7, 1, 9, 1, 9, 6, 7, 1, 13, 3, 8, 5, 11, 1, 16, 1, 12, 7, 10, 6, 19, 1, 10, 7, 18, 1, 19, 1, 15, 12, 12, 1, 24, 3, 16, 9, 16, 1, 23, 8, 21, 11, 15, 1, 33, 1, 14, 16, 20, 8, 26, 1, 19, 10, 25, 1, 35, 1, 19, 18, 23, 7, 30, 1, 31, 14, 18, 1, 39, 10, 19

Offset: 1

Labos Elemer, Jul 10 2003

nonn

a(p) = 1 for prime p > 2. Since phi(p) = p - 1 and sigma(p) = p + 1, the largest prime q < p - 1 must be the prime previous to p, while p itself is the largest prime less than p + 1 for p > 2. - Michael De Vlieger, Jan 22 2020

			n=12: sigma(12)=28, phi(n)=4, Pi(28)-Pi(4)=9-2=7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000720, A000010, A000203, A070803, A070804, A085122, A085341-A085347.

Array[Subtract @@ PrimePi@{DivisorSigma[1, #], EulerPhi@ #} &, 86] (* Michael De Vlieger, Jan 22 2020 *)

a(n) = primepi(sigma(n)) - primepi(eulerphi(n)); \\ Michel Marcus, Aug 29 2019

a(n) = pi(sigma(n)) - pi(phi(n)) = A000720(A000203(n)) - A000720(A000010(n)).

a(n) = A070803(n) - A070804(n). - Antti Karttunen, Jan 22 2020

A085122 a(n) = PrimePi(sigma(n)-phi(n)) - (PrimePi(sigma(n)) - PrimePi(phi(n))), where PrimePi = A000720, sigma = A000203 and phi = A000010.

Original entry on oeis.org

0, -1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 3, 3, 0, 2, 0, 3, 2, 2, 3, 3, 0, 3, 4, 3, 0, 4, 0, 3, 4, 3, 0, 4, 3, 5, 3, 5, 0, 3, 3, 3, 3, 3, 0, 3, 0, 4, 3, 4, 3, 4, 0, 5, 5, 5, 0, 4, 0, 2, 5, 4, 4, 4, 0, 5, 5, 5, 0, 7, 4, 5, 4, 5, 0, 4, 3, 5, 5, 5, 5, 4, 0, 5, 5, 5, 0, 6, 0, 6, 6, 7, 0, 5, 0, 5, 6, 8, 0, 5, 5, 6, 7, 5, 5, 5, 6, 5, 6, 7, 5, 5, 0, 7

Offset: 1

Labos Elemer, Jul 11 2003

Scatterplot of this sequence shows interesting strata. - Antti Karttunen, Jan 22 2020

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

Cf. A000010, A000203, A000720, A051612, A070803, A070804, A085343.

t=Table[PrimePi[DivisorSigma[1, w]-EulerPhi[w]]- (PrimePi[DivisorSigma[1, w]]-PrimePi[EulerPhi[w]]), {w, 1, 10000}]

A085122(n) = (primepi(sigma(n)-eulerphi(n)) - (primepi(sigma(n))-primepi(eulerphi(n)))); \\ Antti Karttunen, Jan 22 2020

a(n) = A000720(A051612(n)) - (A070803(n) - A070804(n)) = A000720(A051612(n)) - A085343(n). - Antti Karttunen, Jan 22 2020

Name edited by Antti Karttunen, Jan 22 2020

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}

A085346 Least number x so that number of primes between sigma(x) and phi(x) equals n.

Original entry on oeis.org

3, 2, 4, 6, 10, 14, 12, 26, 18, 34, 28, 32, 24, 62, 44, 30, 123, 40, 36, 64, 56, 106, 54, 48, 70, 66, 146, 105, 88, 78, 80, 135, 60, 178, 72, 102, 202, 112, 84, 164, 114, 90, 96, 154, 695, 231, 138, 108, 184, 1141, 176, 140, 126, 244, 132, 160, 326, 232, 186, 208, 120

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, A085341-A085347.

s(n) = my(f = factor(n)); primepi(sigma(f)) - primepi(eulerphi(f));
list(len) = {my(v = vector(len), k = 1, c = 0, i); while(c < len, i = s(k); if(i > 0 && i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Dec 20 2024

a(n) = Min{x; A085343(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}.

A247087 a(n) = pi(phi(p(P(n)))) = A000720(A000010(A000041(A000040(n)))).

Original entry on oeis.org

0, 1, 3, 4, 9, 25, 41, 39, 168, 462, 442, 1939, 2571, 3998, 5123, 17040, 24853, 38887, 195022, 183430, 404386, 381060, 1162366, 2105509, 1799881, 5966593, 5380661, 14184985, 10473967, 22631261, 135452589, 109540327, 244730051, 487610708, 604467085, 671043205, 3350187738

Offset: 1

Alois P. Heinz, Mar 14 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..90 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000010, A000040, A000041, A000720, A058697, A058698, A070804.

with(numtheory): with(combinat): p:=numbpart: P:=ithprime:
a:= n-> pi(phi(p(P(n)))):
seq(a(n), n=1..20);

a[n_] := PrimePi @ EulerPhi @ PartitionsP @ Prime @ n;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 25 2017 *)

a(n) = A070804(A058698(n)) = A000720(A000010(A000041(A000040(n)))).

a(31)-a(37) from Amiram Eldar, Sep 03 2024

cp's OEIS Frontend

A070803 Number of primes not exceeding sum of divisors of n.

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

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A085347 Least number x so that number of primes not exceeding sigma(x) equals n.

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A085342 Number of primes between phi(n) and n, where n is included in the count if it is a prime, while phi(n) is never included in the count even if it is a prime.

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A085343 Number of primes between sigma(n) and phi(n) inclusive.

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A085122 a(n) = PrimePi(sigma(n)-phi(n)) - (PrimePi(sigma(n)) - PrimePi(phi(n))), where PrimePi = A000720, sigma = A000203 and phi = A000010.

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

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