A054640 -id:A054640 - cp's OEIS frontend

A217757 Product_{i=0..n} (i! + 1).

2, 4, 12, 84, 2100, 254100, 183206100, 923541950100, 37238134969982100, 13513011656042074430100, 49036030210457135734021310100, 1957361459740805606124917565020990100, 937579272951542930363610919638075856505150100

Offset: 0

Jon Perry, Mar 23 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..42

Cf. A000142, A000178, A054640, A082480, A258325, A260231, A306729.

function factorial(n) {
var i,c=1;
for (i=2;i<=n;i++) c*=i;
return c;
}
a=2;
for (j=1;j<10;j++) {
a*=(factorial(j)+1);
document.write(a+", ");
}

a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(n!+1)) end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 20 2013

Table[Product[i!+1,{i,0,n}],{n,0,12}]  (* Geoffrey Critzer, May 04 2013 *)
Rest[FoldList[Times,1,Range[0,15]!+1]] (* Harvey P. Dale, May 28 2013 *)

a(n) ~ c * A000178(n), where c = A238695 = Product_{k>=0} (1 + 1/k!) = 7.364308272367257256372772509631... . - Vaclav Kotesovec, Jul 20 2015

A260613 Triangle read by rows: T(n, k) = coefficient of x^(n-k) in Product_{m=1..n} (x+prime(m)); 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 10, 31, 30, 1, 17, 101, 247, 210, 1, 28, 288, 1358, 2927, 2310, 1, 41, 652, 5102, 20581, 40361, 30030, 1, 58, 1349, 16186, 107315, 390238, 716167, 510510, 1, 77, 2451, 41817, 414849, 2429223, 8130689, 14117683, 9699690

Offset: 0

Matthew Campbell, Aug 10 2015

Up to signs and order of coefficients the same as A070918. Except for signs and the first column the same as A238146. - M. F. Hasler, Aug 13 2015

			The triangle starts:
Row 0: 1;
Row 1: 1, 2;  Coefficients of x + 2.
Row 2: 1, 5, 6;  Coefficients of (x+2)(x+3) = x^2 + 5x + 6.
Row 3: 1, 10, 31, 30; Coeff's of (x+2)(x+3)(x+5) = x^3 + 10x^2 + 31x + 30.
Row 5: 1, 17, 101, 247, 210;
Row 6: 1, 28, 288, 1358, 2927, 2310;
...

Alois P. Heinz, Rows n = 0..140, flattened (first 20 rows from Matthew Campbell)

Main diagonal gives A002110.
Row sums give A054640.
Cf. A000040.

T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(mul(x+ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019

row[n_] := CoefficientList[Product[x + Prime[m], {m, 1, n}] + O[x]^(n+1), x] // Reverse;
row /@ Range[0, 8] // Flatten (* Jean-François Alcover, Sep 16 2019 *)

tabl(nn) = {for (n=0, nn, polp = prod(k=1, n, x+prime(k)); forstep (k= n, 0, -1, print1(polcoeff(polp, k), ", ");); print(););} \\ Michel Marcus, Aug 10 2015

T(n, 1) = A007504(n) for n >= 1.

T(n, 2) = A024447(n) for n >= 2.

Corrected and edited by M. F. Hasler, Aug 13 2015

a(20) in b-file corrected by Andrew Howroyd, Dec 31 2017

A072986 Least k such that Product_{i=1..k} (prime(i) + 1) >= n*Product_{i=1..k} prime(i).

Original entry on oeis.org

1, 2, 6, 11, 24, 52, 113, 248, 553, 1245, 2828, 6481, 14963, 34770, 81253, 190836, 450202, 1066269, 2534269, 6042375, 14447465, 34632759, 83212840, 200360193, 483361096, 1168159015, 2827750519

Offset: 1

Benoit Cloitre, Aug 14 2002

Least k such that A054640(k) >= n*A002110(k). - Michel Marcus, Jan 09 2021

Cf. A000720, A001113, A002110, A054640, A072997.

a = b = k = 1; Do[ While[a = a*Prime[k]; b = b*(Prime[k] + 1); b < n*a, k++ ]; Print[k]; k++, {n, 1, 16}]

a(n)=if(n<0,0,s=1; while(prod(i=1,s, prime(i)+1)

Limit_{n->oo} a(n+1)/a(n) = e. - Robert Gerbicz, May 09 2008

a(n) = PrimePi(A072997(n)) = A000720(A072997(n)). - Amiram Eldar, Apr 18 2025

Edited and extended by Robert G. Wilson v, Aug 20 2002
a(17)-a(22) from Robert Gerbicz, May 09 2008
a(23)-a(27) from Amiram Eldar, Apr 18 2025

A078558 GCD of sigma(p#) and phi(p#) where p# = A002110(n) is the product of the first n primes.

Original entry on oeis.org

1, 2, 8, 48, 96, 1152, 9216, 1658880, 3317760, 92897280, 2786918400, 100329062400, 802632499200, 370816214630400, 741632429260800, 2966529717043200, 29665297170432000, 355983566045184000

Offset: 1

Labos Elemer, Dec 06 2002

nonn

			m=2,3,30,210 primorials are balanced numbers so these GCD() equals phi(): a(n)=1,2,8,48 (see A005867).

Harvey P. Dale, Table of n, a(n) for n = 1..400

Cf. A000203, A002110, A000005, A005867, A054640, A020492.

GCD[DivisorSigma[1,#],EulerPhi[#]]&/@FoldList[Times,Prime[Range[20]]] (* Harvey P. Dale, Feb 28 2016 *)

a(n)=gcd(prod(i=1,n,prime(i)-1),prod(i=1,n,prime(i)+1)) \\ Charles R Greathouse IV, Dec 09 2013

a(n) = gcd(A000203(A002110(n)), A000005(A002110(n))) = gcd(A005867(n), A054640(n)).

A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

Original entry on oeis.org

0, 1, 6, 42, 366, 4602, 66738, 1231314, 25136790, 612982650, 18612572370, 602072009070, 23079296834790, 976751205195990, 43281303292150770, 2090585319354906990, 113506497027753468870, 6842978980142398176930, 426187457118982899608730, 29098035465450244144376910, 2102916875063497845451016610, 156173789584825539524342644530

Offset: 0

Amarnath Murthy, Aug 22 2002

nonn

			a(3) = (2+1)*(3+1)*(5+1) - 2*3*5 = 72 - 30 = 42.

Antti Karttunen, Table of n, a(n) for n = 0..349 (terms 1..349 from Harvey P. Dale)
Index entries for sequences related to primorial numbers

Cf. A001065, A002110, A003415, A024451, A051674, A054640, A070826, A348507.

for n from 1 to 25 do a[n] := product(ithprime(i)+1,i=1..n)-product(ithprime(i),i=1..n): od:seq(a[j],j=1..25);

Module[{nn=20,p,pr,pr1},p=Prime[Range[nn]];pr=FoldList[Times,1,p];pr1= FoldList[Times,1,p+1];#[[2]]-#[[1]]&/@Rest[Thread[{pr,pr1}]]](* Harvey P. Dale, Feb 07 2015 *)

A074107(n) = (prod(i=1,n,1+prime(i))-prod(i=1,n,prime(i))); \\ Antti Karttunen, Nov 19 2024

From Antti Karttunen, Nov 19 2024: (Start)
a(n) = A348507(A002110(n)) = A054640(n) - A002110(n) = A001065(A002110(n)).
a(n) >= A024451(n), because A348507(n) >= A003415(n).
For n >= 1, a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n).
(End)

More terms from Sascha Kurz, Feb 01 2003

Term a(0)=0 prepended, data section further extended, and secondary definition added by Antti Karttunen, Nov 19 2024

A180617 Sum of divisors of the product of two consecutive primes.

Original entry on oeis.org

12, 24, 48, 96, 168, 252, 360, 480, 720, 960, 1216, 1596, 1848, 2112, 2592, 3240, 3720, 4216, 4896, 5328, 5920, 6720, 7560, 8820, 9996, 10608, 11232, 11880, 12540, 14592, 16896, 18216, 19320, 21000, 22800, 24016, 25912, 27552, 29232, 31320, 32760, 34944, 37248, 38412

Offset: 1

Thomas Kellar, Sep 12 2010

nonn

			a(1) = sigma(2*3) = 12, a(2) = sigma(3*5) = 24.

A distant relative of A054640.

Cf. A000203, A001043, A006094, A126199.

[(1+NthPrime(n))*(1+NthPrime(n+1)): n in [1..50]]; // Vincenzo Librandi, Feb 16 2015

DivisorSigma[1,#]&/@(Times@@@Partition[Prime[Range[50]],2,1]) (* Harvey P. Dale, Apr 04 2015 *)
Table[Prime[n]*Prime[n+1]+Prime[n]+Prime[n+1]+1,{n,1,30}] (* Metin Sariyar, Dec 08 2019 *)

for (n=1,10, i=prod(x=n,n+1,prime(x)); p=sigma(i); print1(p, ", "); )

a(n)=my(p=prime(n)); (p+1)*(nextprime(p+1)+1) \\ Charles R Greathouse IV, Feb 16 2015

a(n) = A000203(A006094(n)). - Omar E. Pol, Dec 08 2019
a(n) = A006094(n) + A001043(n) + 1. - Metin Sariyar, Dec 08 2019
a(n) = A126199(n) + 1 (after above formula). - Omar E. Pol, Dec 08 2019

More terms from Vincenzo Librandi, Feb 16 2015

Name simplified by Omar E. Pol, Dec 08 2019

A309802 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+2} (prime(i)*x-1).

Original entry on oeis.org

1, 10, 101, 1358, 20581, 390238, 8130689, 201123530, 6166988769, 201097530280, 7754625545261, 329758834067168, 14671637258193181, 711027519310719868, 38706187989054920001, 2338431642812927422310, 145908145906128304198449, 9976861293427674211625032

Offset: 0

Alexey V. Bazhin, Aug 17 2019

Alexey V. Bazhin, Table of n, a(n) for n = 0..300

Cf. A000040, A002110, A024451, A070918, A309803, A309804, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+2), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 18 2019

a(n) = [x^n] Product_{i=1..n+2} (prime(i)*x-1).
a(n) = abs(A070918(n+2,2)).
a(n) = abs(A238146(n+2,n)) for n>0.
a(n) = A260613(n+2,n).

A309803 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+3} (prime(i)*x-1).

Original entry on oeis.org

-1, -17, -288, -5102, -107315, -2429223, -64002818, -2057205252, -69940351581, -2788890538777, -122099137635118, -5580021752377242, -276932659619923555, -15388458479166668283, -946625238259888348698, -60082571176666116692888, -4171440414742758122621945

Offset: 0

Alexey V. Bazhin, Aug 17 2019

Alexey V. Bazhin, Table of n, a(n) for n = 0..300

Cf. A000040, A002110, A024451, A070918, A309802, A309804, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+3), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2019

a(n) = [x^n] Product_{i=1..n+3} (prime(i)*x-1).
a(n) = -abs(A070918(n+3,3)).
a(n) = -abs(A238146(n+3,n)) for n>0.
a(n) = -A260613(n+3,n).

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)).

cp's OEIS Frontend

A217757 Product_{i=0..n} (i! + 1).

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A260613 Triangle read by rows: T(n, k) = coefficient of x^(n-k) in Product_{m=1..n} (x+prime(m)); 0 <= k <= n, n >= 0.

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A072986 Least k such that Product_{i=1..k} (prime(i) + 1) >= n*Product_{i=1..k} prime(i).

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A078558 GCD of sigma(p#) and phi(p#) where p# = A002110(n) is the product of the first n primes.

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A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

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A180617 Sum of divisors of the product of two consecutive primes.

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A309802 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+2} (prime(i)*x-1).

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