A000849 -id:A000849 - cp's OEIS frontend

A285905 a(n) = A275768(A002110(n)).

0, 0, 5, 26, 124, 852, 7550, 86125, 1250924, 23748764

Offset: 1

Michael De Vlieger and Jamie Morken, May 03 2017

The number of ways to express primorial p_n# = A002110(n) as (prime(i) + prime(j))/2 when (prime(i) - prime(j))/2 also is prime.
Let p_n < q <= prime(pi(p_n#)), with pi(p_n#) = A000849(n). All such primes q are coprime to primorial p_n# since they are larger than the greatest prime factor of p_n#. One of the two primes counted by a(n) must be a prime q, the other a prime r = (2p_n# - q). Further, (r - q) must be prime to be counted by a(n). Therefore an efficient method of computing a(n) begins with generating the range of prime totatives prime(n + 1) <= q <= prime(pi(p_n#)) of primorial p_n#, the number of which is given by A048862(n).
a(n) < A048862(n) < A000849(n) for n > 2.

			a(3) = 5 since there are 5 ways to express A002110(3) = 30 as (prime(i) + prime(j))/2 with (prime(i) - prime(j))/2 also prime:
  (53 + 7)/2 = 30, (53 - 7)/2 = 46/2 = 23
  (47 + 13)/2 = 30, (47 - 13)/2 = 34/2 = 17
  (43 + 17)/2 = 30, (43 - 17)/2 = 26/2 = 13
  (41 + 19)/2 = 30, (41 - 19)/2 = 22/2 = 11
  (37 + 23)/2 = 30, (37 - 23)/2 = 14/2 = 7.

Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Totative

Cf. A000849, A002110, A048862, A275768.

With[{j = 10^3}, Do[Module[{P = Times @@ Prime@ Range@ n, m}, m = PrimePi@ P; Print@ Total@ Reap[Do[Sow@ Count[Map[{2 P - #, #} &, Prime@ Range[Max[n, k], Min[k + j - 1, m]]], w_ /; And[PrimeQ@ First@ w, PrimeQ[(Subtract @@ w)/2]]], {k, 1, m, j}]][[-1, 1]]], {n, 9}]] (* or *)
Table[Function[P, Count[Map[{2 P - #, #} &, #], w_ /; And[PrimeQ@ First@ w, PrimeQ[(Subtract @@ w)/2]]] &@ Flatten@ Select[Prime@ Range[n + 1, PrimePi[P]], Times @@ Boole@ Map[PrimeQ, {#, P - #}] == 1 &]]@ Product[Prime@ i, {i, n}], {n, 9}] (* Michael De Vlieger, May 03 2017 *)
countOfPrimes = 0
countOfPrimes2 = 0
countOfPrimes3 = 0
Pn10 = 2*3*5*7*11*13*17*19*23*29
PnToUse = Pn10
distanceToCheck = PnToUse
For[i=0,iJamie Morken, May 05 2017 *)

A343119 Number of compositions (ordered partitions) of the n-th primorial into distinct parts.

Original entry on oeis.org

1, 1, 11, 41867, 517934206090276988507, 42635439758725572299058305546953458030363703549127905691758491973278624456679699932948789006991639715987

Offset: 0

Alois P. Heinz, Apr 09 2021

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..6

Cf. A002110, A032020, A000849, A005867, A054640, A058254, A062447, A342996, A343147.

b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*ithprime(n)) end:
g:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
     `if`(k=0, `if`(n=0, 1, 0), g(n-k, k)+k*g(n-k, k-1)))
    end:
a:= n-> add(g(b(n), k), k=0..floor((sqrt(8*b(n)+1)-1)/2)):
seq(a(n), n=0..5);

$RecursionLimit = 5000;
b[n_] := If[n == 0, 1, b[n - 1]*Prime[n]];
g[n_, k_] := g[n, k] = If[k < 0 || n < 0, 0,
     If[k == 0, If[n == 0, 1, 0], g[n - k, k] + k*g[n - k, k - 1]]];
a[n_] := Sum[g[b[n], k], {k, 0, Floor[(Sqrt[8*b[n] + 1] - 1)/2]}];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

a(n) = A032020(A002110(n)).

A372730 Number of primes <= A005867(n).

Original entry on oeis.org

0, 0, 1, 4, 15, 92, 757, 8899, 125261, 2232782, 51902553, 1327191561, 41351244491, 1452937916515, 54332144724834, 2246960940148460, 105818707666943651, 5595105626396158784, 308241771351984486729, 18772520681296116861073

Offset: 0

Alexandre Herrera, May 11 2024

			a(3) = 4 because there are 4 primes less than A005867(3) = 8: 2, 3, 5 and 7.

Cf. A000720, A005867, A000849.

A372730(n) = primepi(prod(k=1, n, prime(k)-1)); \\ Antti Karttunen, May 22 2024

from sympy import prime,primepi
p = 1
l = [0]
for i in range(1,12):
    p *= (prime(i) - 1)
    l.append(primepi(p))
print(l)

a(n) = A000720(A005867(n)).

a(9)-a(11) from Antti Karttunen, May 22 2024
a(12)-a(16) from Amiram Eldar, May 22 2024
a(17)-a(18) from Chai Wah Wu, Jun 04 2024
a(19) from Chai Wah Wu, Jun 05 2024

A048980 Difference between number of nonprimes and primes in reduced residue system of primorial numbers.

Original entry on oeis.org

1, 1, 0, -6, -36, -196, -724, 7512, 366838, 11928316, 421130508, 14598816402, 584642184936, 25314953837836, 1128885572358548, 54492272309366314, 2950485568862138250, 213151926413154110951

Offset: 0

Labos Elemer

sign

			n=4, Q(4)=2*3*5*7=210, reduced residue system includes 48 terms:42 primes and 6 composites and 1: a(4)=6-42=-36.

Cf. A048868, A048867, A048597, A002110, A048862, A048863, A048866.

Table[Function[P, EulerPhi@ P - 2 # &[PrimePi@ P - n]]@ Product[Prime@ i, {i, n}], {n, 0, 12}] (* Michael De Vlieger, May 08 2017 *)

a(n) = A048863(n) - A048862(n) = A048866(A002110(n)).

a(n) = A005867(n) - 2*A000849(n) + 2*n.

Corrected and extended by Max Alekseyev, Feb 22 2016

A065898 Which composite number is the product of first n primes (the n-th primorial number)?: a(n) = k such that A002808(k) = A002110(n), or 0 if A002110(n) is not composite.

Original entry on oeis.org

0, 2, 19, 163, 1966, 26781, 468178, 9053660, 210809338, 6169323433, 192531847119, 7161249384065, 294835346718114, 12720163581273289, 599492054060678551, 31846920298131321838, 1882691381652701947175, 115037118886354670022443, 7718754971134321663159676

Offset: 1

Labos Elemer, Nov 28 2001

nonn

			a(3) = 19 because 2*3*5 = 30 = A002808(19) is the 19th composite number.

Cf. A000720, A002808, A002110, A000849.

Table[q = Product[ Prime[i], {i, 1, n}]; q - PrimePi[q] - 1, {n, 1, 12}]

a(n) = my(m = vecprod(primes(n))); m - primepi(m) - 1; \\ Amiram Eldar, Aug 09 2024

a(n) = A002110(n) - A000849(n) - 1. - Amiram Eldar, Aug 09 2024

More terms from Robert G. Wilson v, Nov 29 2001

Name clarified and a(13)-a(19) calculated from the data at A000849 and added by Amiram Eldar, Aug 09 2024

A066264 Number of composites < primorial(p) with all prime factors > p.

Original entry on oeis.org

0, 0, 0, 5, 141, 2517, 49835, 1012858, 24211837, 721500293, 22627459400, 844130935667, 34729870646917, 1491483322755273, 69890000837179156

Offset: 1

Patrick De Geest, Dec 10 2001

nonn

There is a simple relationship between this sequence and the number of primes < primorial(p), as given by A000849 and sequence A005867 which gives the number of composites in primorial(p+1) having (p+1) as their lowest prime factor: a(n) = n + A005867(n) - A000849(n) - 1. - Dennis Martin (dennis.martin(AT)dptechnology.com), Apr 15 2007

			There are 5 composites < primorial(7) or 210 and whose prime factors are all larger than 7: 121 (11*11), 143 (11*13), 169 (13*13), 187 (11*17) and 209 (11*19).

Eric Weisstein's World of Mathematics, Primorial

Cf. A000849, A002110, A005867.

Array[#1 + EulerPhi@ #2 - PrimePi@ #2 - 1 & @@ {#, Product[Prime@ i, {i, #}]} &, 12] (* Michael De Vlieger, Apr 03 2019 *)

a(n) = n + A005867(n) - A000849(n) - 1. - Michael De Vlieger, Apr 03 2019, citing Dennis Martin's comment above.

More terms from Dennis Martin (dennis.martin(AT)dptechnology.com), Apr 15 2007
Offset corrected by Charles J. Daniels (chajadan(AT)gmail.com), Dec 06 2009
a(14)-a(15) from Donovan Johnson, May 03 2010

A106558 a(n)=1+floor(sqrt(Pi/2)*p(n)#).

Original entry on oeis.org

3, 8, 38, 264, 2896, 37638, 639830, 12156759, 279605448, 8108557990, 251365297667, 9300516013674

Offset: 1

Pierre CAMI, May 09 2005

nonn

a(n) is an approximation to the number of primes less than p(n+1)#

Cf. A000849.

A166678 a(n) = pi((sqrt(P(n))+1)^2) - pi(P(n)), where pi(n) = number of primes <= n and P(n) = n-th primorial.

Original entry on oeis.org

2, 2, 3, 6, 14, 34, 110, 384, 1540, 7019, 34501, 183439, 1045196, 6164423, 38285946

Offset: 1

Daniel Tisdale, Oct 18 2009, Oct 23 2009

Conjecture: pi((sqrt(P(n))+1)^2) - pi(P(n)) >= n.

Cf. A000720 (pi), A002110 (primorials), A000849 (pi(primorials)).

a[n_] := Product[Prime[k], {k, 1, n}]; Table[PrimePi[(Sqrt[a[n]] + 1)^2] - PrimePi[a[n]], {n, 1, 12}] (* G. C. Greubel, May 22 2016 *)

a(n) = my(P=vecprod(primes(n))); primepi((sqrt(P)+1)^2) - primepi(P); \\ Michel Marcus, Aug 15 2022

a(13)-a(15) from Ray Chandler, May 10 2010

Name edited by Michel Marcus, Aug 15 2022

A276497 Number of noncomposites in the reduced residue system of n-th primorial number, A002110(n).

Original entry on oeis.org

1, 2, 8, 43, 339, 3243, 42325, 646022, 12283523, 300369787, 8028643000, 25948875073, 9414916809083, 362597750396727, 15397728527812844, 742238179058722876, 40068968501510691878

Offset: 1

Andres Cicuttin, Sep 29 2016

Cf. A000849, A002110, A048862.

Primorial[n_] := Product[Prime[j], {j, 1, n}];
Table[PrimePi[Primorial[n]] - n + 1, {n, 1, 12}]

a(n) = primepi(prod(k=1,n,prime(k))) - n + 1; \\ Michel Marcus, Oct 03 2016

a(n) = pi(primorial(n)) - n + 1, n > 0.
a(n) = A000849(n) - n + 1, n > 0.
a(n) = A048862(n) + 1. - Michel Marcus, Oct 03 2016

Definition aligned with formulas and data by Peter Munn, Sep 06 2023

A286424 Number of partitions of p_n# into parts (q, k) both coprime to p_n#, with q prime and k nonprime, where p_n# = A002110(n).

Original entry on oeis.org

0, 0, 1, 1, 4, 110, 1432, 23338, 397661, 8193828, 212858328, 5941706227

Offset: 0

Michael De Vlieger, May 08 2017

Number of totative pairs (q, k) such that prime q + k nonprime = p_n# and both gcd(q, p_n#) = 1 and gcd(k, p_n#) = 1, with p_n < q <= pi(p_n#), where pi(p_n#) = A000849(n) - n = A048862(n).
Primes p_n < q <= pi(p_n#) are greater than the greatest prime factor of p_n# = p_n, and are thus coprime to p_n#. By the definition of primorial, we need not consider p >= p_n, as these p are divisors of p_n#, i.e., gcd(p, p_n#) = p. Since the totatives of m can be paired such that a + b = m, we need only determine if (p_n# - q) is not prime in order to count pairs (q, k).
a(n) < floor(A005867(n)/2).
a(n) <= A048862(n).
The totative pair (q,1) = (p_n# - 1, 1) is counted by a(n) for n in A057704, with (p_n# - 1) appearing in A057705.

			a(0) = 0 by definition. A002110(0) = 1; 1 is coprime to all numbers; the only possible totative pair is (1,1) and this does not include both a prime and a nonprime.
a(1) = 0 since, of the floor(A005867(1)/2) = 1 totative pair (1,1) of A002110(1) = 2, none include a both a prime and a nonprime.
a(2) = 1 since, the only totative pair (1,5) of A002110(1) = 6 includes both a prime and a nonprime.
a(3) = 1 since only (1,29) includes both a prime and a nonprime.
a(4) = 4 since (23,187), (41,169), (67,143), (89,121) include a both a prime and a nonprime.

C. K. Caldwell, The Prime Glossary, Primorial.
Eric Weisstein's World of Mathematics, Totative.

Cf. A000849, A002110, A005867, A048862, A048863, A057704, A057705, A116979, A117929.

Table[Function[P, Count[Prime@ Range[n + 1, PrimePi[P]], q_ /; ! PrimeQ[P - q]]]@ Product[Prime@ i, {i, n}], {n, 0, 9}] (* Michael De Vlieger, May 08 2017 *)

a(n) = (A000010(A002110(n)) - A048863(n)) - 2*A117929(A002110(n))
     = (A005867(n) - A048863(n)) - 2*A117929(A002110(n))
     = A048862(n) - 2*A117929(A002110(n)).

a(11) from Giovanni Resta, May 09 2017

cp's OEIS Frontend

A285905 a(n) = A275768(A002110(n)).

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A343119 Number of compositions (ordered partitions) of the n-th primorial into distinct parts.

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Maple

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A372730 Number of primes <= A005867(n).

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PARI

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A048980 Difference between number of nonprimes and primes in reduced residue system of primorial numbers.

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A065898 Which composite number is the product of first n primes (the n-th primorial number)?: a(n) = k such that A002808(k) = A002110(n), or 0 if A002110(n) is not composite.

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A066264 Number of composites < primorial(p) with all prime factors > p.

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A106558 a(n)=1+floor(sqrt(Pi/2)*p(n)#).

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A166678 a(n) = pi((sqrt(P(n))+1)^2) - pi(P(n)), where pi(n) = number of primes <= n and P(n) = n-th primorial.

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