A000849 -id:A000849 - cp's OEIS frontend

A007014 Largest prime <= Product prime(k).

2, 5, 29, 199, 2309, 30029, 510481, 9699667, 223092827, 6469693189, 200560490057, 7420738134751, 304250263527209, 13082761331669941, 614889782588491343, 32589158477190044657, 1922760350154212638963, 117288381359406970983181, 7858321551080267055878989

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

A057705 contains terms of a(n) such that A002110(n) - a(n) = 1. -Michael De Vlieger, May 15 2017

			From _Michael De Vlieger_, May 15 2017: (Start)
a(1) = 2 since A002110(1) = 2. 2 is prime thus the largest prime <= 2 = 2.
a(2) = 5 since A002110(2) = 6. 5 is the largest prime <= 6. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..350
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1994

Cf. A000849, A002110, A038710, A057705, A151799.

Array[Abs@ NextPrime[Product[Prime@ i, {i, #}], -1] &, 14] (* Michael De Vlieger, May 15 2017 *)

lista(n) = {prd = 1; for (i=1, n, prd *= prime(i); print1(precprime(prd), ", "););} \\ Michel Marcus, Jun 17 2013

a(n)=precprime(prod(i=1,n,prime(i))) \\ Charles R Greathouse IV, Jun 17 2013

From Michael De Vlieger, May 15 2017: (Start)
a(n) = prime(A000849(n)).
a(n) = A151799(A002110(n)). (End)

Corrected by Jud McCranie, Jan 03 2001

More terms from Michael De Vlieger, May 15 2017

A048862 Number of primes in the reduced residue system of n-th primorial number (=A002110(n)).

Original entry on oeis.org

0, 0, 1, 7, 42, 338, 3242, 42324, 646021, 12283522, 300369786, 8028642999, 259488750732, 9414916809082, 362597750396726, 15397728527812843, 742238179058722875, 40068968501510691877, 2251262473052300960808, 139566579945945392719394

Offset: 0

Labos Elemer

			For n = 3, the 3rd primorial is 30, phi(30) = 8, a(3) = 8-1 = 7 since 1 is nonprime. See A048597.
For n = 4, the 4th primorial is 210, the size of its reduced residue system (RRS) is 48 of which 42 are primes and 6 are either composite numbers or 1.

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000010 (phi), A000720, A000849, A001221, A002110, A007625, A048597, A048863.

a(n) = A000849(n) - n = A000720(A002110(n)) - A001221(A002110(n)).

a(0) prepended and extended by Max Alekseyev, Feb 22 2016

a(17) corrected and a(18)-a(19) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 03 2024

A048863 Number of nonprimes (1 and composites) in the reduced residue system of n-th primorial number (A002110).

Original entry on oeis.org

1, 1, 1, 1, 6, 142, 2518, 49836, 1012859, 24211838, 721500294, 22627459401, 844130935668, 34729870646918, 1491483322755274, 69890000837179157, 3692723747920861125, 217158823263305180123, 13182405032836651359192, 879055475442725460400606

Offset: 0

Labos Elemer

			For n = 3, the 3rd primorial is 30, phi(30) = 8, a(3) = 1 since 1 is nonprime. See A048597.
For n = 4, the 4th primorial is 210, the size of its reduced residue system (RRS) is 48 of which 6 are either composite numbers or 1: {1, 121, 143, 169, 187, 209}.

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000010 (phi), A000720, A002110, A005867, A007625, A048597, A048862.

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

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

a(n) = A000010(A002110(n)) - A000720(A002110(n)) + A001221(A002110(n)).

a(14)-a(15) from Max Alekseyev, Aug 21 2013
a(0) prepended, a(15) corrected, a(16)-a(17) computed from A000849 by Max Alekseyev, Feb 21 2016
a(18)-a(19) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 03 2024

A061568 Number of primes <= sum of first n primes.

Original entry on oeis.org

1, 3, 4, 7, 9, 13, 16, 21, 25, 31, 37, 45, 51, 60, 66, 75, 85, 95, 103, 115, 127, 138, 150, 162, 177, 191, 205, 219, 233, 250, 267, 283, 300, 319, 338, 360, 376, 400, 421, 441, 461, 481, 509, 531, 557, 578, 602, 630, 653, 684, 707, 737, 765, 793, 825, 853, 884

Offset: 1

Harvey P. Dale, May 18 2001

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

Cf. A000849, A000720, A007504.

PrimePi/@FoldList[Plus, 0, Prime[Range[50]]]//Rest

{ default(primelimit, 3682913); n=0; s=0; forprime (p=2, prime(1000), s+=p; write("b061568.txt", n++, " ", primepi(s)) ) } \\ Harry J. Smith, Jul 24 2009

a(n) = primepi(sum(k=1, n, prime(k))); \\ Michel Marcus, Jul 02 2018

a(n) = A000720(A007504(n)). - Michel Marcus, Jul 02 2018

A342996 The number of partitions of the n-th primorial.

Original entry on oeis.org

1, 2, 11, 5604, 9275102575355, 21565010821742923705373368869534441911701199887419

Offset: 0

Alois P. Heinz, Apr 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..7

Cf. A000041, A002110, A000849, A005867, A054640, A058254, A062447, A343147.

b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*ithprime(n)) end:
a:= n-> combinat[numbpart](b(n)):
seq(a(n), n=0..5);

b[n_] := b[n] = If[n == 0, 1, b[n - 1]*Prime[n]];
a[n_] := PartitionsP[b[n]];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Jul 07 2021, from Maple *)

a(n) = numbpart(prod(k=1, n, prime(k))); \\ Michel Marcus, Jul 07 2021

from sympy import primorial
from sympy.functions import partition
def A342996(n): return partition(primorial(n)) if n > 0 else 1 # Chai Wah Wu, Apr 03 2021

a(n) = A000041(A002110(n)).

A343147 The number of partitions of the n-th primorial into distinct parts.

Original entry on oeis.org

1, 1, 4, 296, 884987529, 41144767887910339859917073881177514

Offset: 0

Alois P. Heinz, Apr 06 2021

nonn

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

Cf. A000009, A002110, A000849, A005867, A054640, A058254, A062447, A342996.

b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*ithprime(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> g(b(n)):
seq(a(n), n=0..5);

$RecursionLimit = 2^13;
b[n_] := b[n] = If[n == 0, 1, b[n - 1]*Prime[n]];
g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[
     If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := g[b[n]];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

a(n) = A000009(A002110(n)).

A000882 Number of twin prime pairs <= product of first n primes.

Original entry on oeis.org

0, 1, 4, 15, 69, 468, 4636, 57453, 896062, 18463713, 425177757, 11997649372, 385088898632, 13280323588034, 509456736126003

Offset: 1

gandalf(AT)hrn.office.ssi.net (James D. Ausfahl)

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) [From _Donovan Johnson_, Mar 01 2010]

Cf. A002110, A000849.

More terms from David W. Wilson

a(10)-a(15) from Donovan Johnson, Mar 01 2010

A072236 Numbers of primes between successive primorials.

Original entry on oeis.org

1, 2, 7, 36, 297, 2905, 39083, 603698, 11637502, 288086265, 7728273214, 251460107734, 9155428058351, 353182833587645, 15035130777416118, 726840450530910033, 39326730322451969003, 2211193504550790268932, 137315317472893091758587

Offset: 0

Stephan Wagler (stephanwagler(AT)aol.com), Jul 05 2002

			There are 3 primes less than 6, 7 primes between 6 and 30 and 36 primes between 30 and 210.

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) [From _Donovan Johnson_, Apr 25 2010]

Cf. A002110, A000849.

Table[ PrimePi[ Product[Prime[i], {i, 1, n}]] - PrimePi[ Product[ Prime[i], {i, 1, n - 1}]], {n, 1, 12}]
Join[{1},Differences[PrimePi/@Rest[FoldList[Times,1,Prime[Range[12]]]]]] (* Harvey P. Dale, Mar 16 2012 *) (* Mathematica's implementation of PrimePi will not work for the 13th primorial because it's too large *)

a(n) = A000849(n+1) - A000849(n). - Amiram Eldar, Jun 11 2024

Edited by Robert G. Wilson v, Jul 08 2002
a(13)-a(14) from Donovan Johnson, Apr 25 2010
a(15)-a(18) calculated using the data at A000849 and added by Amiram Eldar, Jun 11 2024

A261215 Triangle read by rows: T(n, k) is the number of squarefree integers x such that x<=primorial(n) and omega(n) = k with 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 10, 7, 1, 1, 46, 62, 19, 1, 1, 343, 643, 359, 58, 1, 1, 3248, 7429, 5723, 1708, 152, 1, 1, 42331, 110373, 104928, 44365, 7926, 422, 1, 1, 646029, 1848729, 2024368, 1068664, 276833, 31108, 995, 1

Offset: 0

Michel Marcus, Aug 12 2015

			Triangle begins:
1;
1, 1;
1, 3, 1;
1, 10, 7, 1;
1, 46, 62, 19, 1;
1, 343, 643, 359, 58, 1;
...

Giovanni Resta, Table of n, a(n) for n = 0..77

Cf. A002110 (primorials), A000849 (1st column), A001221 (omega), A005117 (squarefrees), A092479 (similar triangle).

primo(n) = prod(i=1, n, prime(i));
tabl(nn) = {v = vector(1); na = 1; for (n=0, nn, nb = primo(n); for (i=na, primo(n), if (issquarefree(i), v[1+omega(i)]++);); print(v); v = concat(v, 0); na = nb + 1;);}

A283427 a(n) is the number of consecutive smallest prime totatives of primorial A002110(n).

Original entry on oeis.org

0, 1, 7, 26, 34, 55, 65, 91, 137, 152, 208, 251, 270, 315, 394, 471, 502, 591, 656, 685, 790, 864, 977, 1139, 1227, 1268, 1354, 1395, 1494, 1847, 1945, 2109, 2157, 2455, 2512, 2693, 2878, 3005, 3202, 3396, 3471, 3826, 3902, 4045, 4119, 4581, 5059, 5226, 5307

Offset: 1

Jamie Morken and Michael De Vlieger, May 15 2017

Let p_n# = A002110(n) be the n-th primorial, and let t be a totative of p_n#, i.e., gcd(t, p_n#) = 1. Let q be the smallest prime totative of p_n#. We know q must be p_(n+1) by the definition of "primorial" as the product of the smallest n primes. This is the starting point of the range of primes we are considering. The ending point is the smallest composite totative, which is a square semiprime. This semiprime in fact must be q^2, since q is the smallest prime totative of p_n#. Stated in terms of prime n, the range we are considering are primes p_(n+1) <= t <= prevprime((p_(n+1))^2). For the smallest primorials, q^2 > p_n# with n <= 3. Thus a(n) < A054272(n) for n <= 3.

			a(2) = pi(min(prime(3)^2, p_2#)) - 2 = pi(min(25,6)) - 2 = 3 - 2 = 1.
a(4) = pi(min(prime(5)^2, p_4#)) - 4 = pi(min(121,210)) - 4 = 30 - 4 = 26.

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

Cf. A000849, A054272, A283425.

Table[PrimePi[Min[Prime[n + 1]^2, Product[Prime@ i, {i, n}]]] - n, {n, 49}] (* Michael De Vlieger, May 16 2017 *)

a(n) = pi(min(prime(n+1)^2, Product_{k=1..n} ( prime(k) ) )) - n.

cp's OEIS Frontend

A007014 Largest prime <= Product prime(k).

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A048862 Number of primes in the reduced residue system of n-th primorial number (=A002110(n)).

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A048863 Number of nonprimes (1 and composites) in the reduced residue system of n-th primorial number (A002110).

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A061568 Number of primes <= sum of first n primes.

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A342996 The number of partitions of the n-th primorial.

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A343147 The number of partitions of the n-th primorial into distinct parts.

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Maple

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A000882 Number of twin prime pairs <= product of first n primes.

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A072236 Numbers of primes between successive primorials.

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