A013939 -id:A013939 - cp's OEIS frontend

A060238 a(n) = det(M) where M is an n X n matrix with M[i,j] = lcm(i,j).

1, 1, -2, 12, -48, 960, 11520, -483840, 3870720, -69672960, -2786918400, 306561024000, 7357464576000, -1147764473856000, -96412215803904000, -11569465896468480000, 185111454343495680000, -50350315581430824960000, -1812611360931509698560000

Offset: 0

MCKAY john (mckay(AT)cs.concordia.ca), Mar 21 2001

sign

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 695, pp. 90, 297-298, Ellipses, Paris, 2004.
J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, p. 265, eq. 10.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..200
Index to sequences related to determinants.

Cf. A000142, A001088, A013939, A023900, A048803, A060239, A085542.

A060238:=n->n!*mul((1-ithprime(i))^floor(n/ithprime(i)), i=1..numtheory[pi](n)): seq(A060238(n), n=0..20); # Wesley Ivan Hurt, Aug 15 2016

A060238[n_]:=n!*Product[(1 - Prime[i])^Floor[n/Prime[i]], {i, PrimePi[n]}]; Array[A060238, 20] (* Enrique Pérez Herrero, Jun 08 2010 *)

a(n)=n!*prod(p=1,sqrtint(n),if(isprime(p),(1-p)^floor(n/p),1)) \\ Benoit Cloitre, Jan 31 2008

For n >= 2, a(n) = n! * Product_{j=2..n} Product_{p|j} (1-p) (where the second product is over all primes p that divide j) (cf. A023900). - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001
a(n) = n! * Product_{p<=n} (1-p)^floor(n/p) where the product runs through the primes. - Benoit Cloitre, Jan 31 2008
a(n) = A000142(n) * A085542(n). - Enrique Pérez Herrero, Jun 08 2010
a(n) = A001088(n) * A048803(n) * (-1)^A013939(n). - Amiram Eldar, Dec 19 2018
a(n) = Product_{k=1..n} (-1)^A001221(k) * A000010(k) * A007947(k) [De Koninck & Mercier]. - Bernard Schott, Dec 11 2020

a(0)=1 prepended by Alois P. Heinz, Jan 25 2023

A112966 Sum(mu(i)*omega(j): i+j=n), with mu=A008683 and omega=A001221.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 07 2005

			a(5)=mu(1)*omega(4)+mu(2)*omega(3)+mu(3)*omega(2)+mu(4)*omega(1)
= 1*1 - 1*1 - 1*1 + 0*0 = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A013939, A112965, A068341, A112962, A112963, A112964, A112968.

a112966 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001221_list
-- Reinhard Zumkeller, Feb 29 2012

A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

Original entry on oeis.org

0, 0, 1, 0, 0, -2, -2, -2, -2, -2, -6, -2, -4, -2, -7, -1, -5, 0, -7, -3, -9, 1, -11, 2, -7, 1, -12, 1, -11, 7, -8, -5, -8, -1, -18, 3, -10, 1, -13, 1, -7, 13, -12, -2, -13, 6, -16, 3, -11, 3, -15, -4, -16, 13, -15, -4, -15, 4, -17, 11, -14, 4, -13, 7, -12, 15, -17, -5, -15, 16, -13, 3, -12, 3, -20, 3, -27, 19, -20, -3, -11, 3

Offset: 1

Reinhard Zumkeller, Oct 07 2005

sign

			a(5) = mu(1)*Omega(4)+mu(2)*Omega(3)+mu(3)*Omega(2)+mu(4)*Omega(1) = 1*2 - 1*1 - 1*1 + 0*1 = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A013939, A112967, A068341, A112962, A112963, A112964, A112966.

a112968 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001222_list
-- Reinhard Zumkeller, Feb 29 2012

A112968[n_]:=Plus@@Table[MoebiusMu[i]*PrimeOmega[n-i],{i,1,n-1}]; Array[A112968,200] (* Enrique Pérez Herrero, Feb 28 2012 *)

Corrected by N. J. A. Sloane, Mar 01 2006

A329354 a(n) = Sum_{d|n} d*omega(d).

Original entry on oeis.org

0, 2, 3, 6, 5, 17, 7, 14, 12, 27, 11, 45, 13, 37, 38, 30, 17, 62, 19, 71, 52, 57, 23, 101, 30, 67, 39, 97, 29, 162, 31, 62, 80, 87, 82, 162, 37, 97, 94, 159, 41, 220, 43, 149, 137, 117, 47, 213, 56, 152, 122, 175, 53, 197, 126, 217, 136, 147, 59, 410, 61, 157, 187, 126, 148, 336, 67, 227, 164, 342, 71, 362, 73, 187, 213, 253, 172, 394, 79

Offset: 1

Antti Karttunen, Nov 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001221, A013939, A062799, A180253, A323599, A328260, A329375.

Table[Sum[d*PrimeNu[d], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 18 2021 *)

PARI
```
A329354(n) = sumdiv(n,d,omega(d)*d);
```

a(n) = Sum_{d|n} d*A001221(d).
a(n) = A180253(n) - A323599(n).
a(n) = A328260(n) + A329375(n).
a(n) = Sum_{d|n} (n/d) * sopf(d). - Wesley Ivan Hurt, May 24 2021
Dirichlet g.f.: primezeta(s-1) * zeta(s-1) * zeta(s). - Ilya Gutkovskiy, Aug 18 2021
Conjecture: Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (log(log(n)) + A077761) / 12. - Vaclav Kotesovec, Mar 03 2023

A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 8, 16, 32, 32, 32, 64, 64, 128, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 12288, 12288, 12288, 24576, 49152, 98304, 196608, 196608, 393216, 786432, 1572864, 1572864, 4718592, 4718592, 9437184, 18874368, 37748736

Offset: 1

Gus Wiseman, Jul 20 2022

nonn

Also partial products of A001221 without the first term 0, sum A013939.

For initial terms up to n = 29 we have a(n) = 2^A355538(n). The first non-power of 2 is a(30) = 12288.

			The a(n) choices for n = 2, 6, 10, 12, with prime(k) replaced by k:
  (1)  (12131)  (121314121)  (12131412151)
       (12132)  (121314123)  (12131412152)
                (121324121)  (12131412351)
                (121324123)  (12131412352)
                             (12132412151)
                             (12132412152)
                             (12132412351)
                             (12132412352)

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
The version for divisors instead of prime factors is A066843.
The integers themselves are the rows of A131818.
The version with multiplicity is A327486.
Using prime indices instead of 2..n gives A355741, for multisets A355744.
Counting sequences instead of multisets gives A355746.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
Cf. A000005, A000040, A000720, A002110, A013939, A076610, A355538, A355731, A355733, A355745, A355747.

Table[Times@@PrimeNu/@Range[2,m],{m,2,30}]

A060832 a(n) = Sum_{k>0} floor(n/k!).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 10, 11, 13, 14, 16, 17, 20, 21, 23, 24, 26, 27, 30, 31, 33, 34, 36, 37, 41, 42, 44, 45, 47, 48, 51, 52, 54, 55, 57, 58, 61, 62, 64, 65, 67, 68, 71, 72, 74, 75, 77, 78, 82, 83, 85, 86, 88, 89, 92, 93, 95, 96, 98, 99, 102, 103, 105, 106, 108, 109, 112, 113

Offset: 0

Henry Bottomley, May 01 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A013936, A013939, A038663, A055881.

[0] cat [&+[Floor(m/Factorial(k)):k in [1..m]]:m in [1..70]]; // Marius A. Burtea, Jul 11 2019

a(n)={my(s=0, d=1, f=1); while (n>=d, s+=n\d; f++; d*=f); s} \\ Harry J. Smith, Jul 12 2009

a(n) = round(sumpos(k=1, n\k!)); \\ Michel Marcus, Jan 24 2025

a(n) = a(n-1) + A055881(n).
a(n) = (e-1)*n + f(n) where f(n) < 0. - Benoit Cloitre, Jun 19 2002
f is unbounded in the negative direction. The assertion that f(n) < 0 is correct, since (e-1)*n = Sum_{k>=1} n/k! is term for term >= this sequence. - Franklin T. Adams-Watters, Nov 03 2005
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k!)/(1 - x^(k!)). - Ilya Gutkovskiy, Jul 11 2019

A338943 a(n) is the least number k such that the average number of distinct prime divisors of {1..k} is >= n.

Original entry on oeis.org

1, 6, 455, 8167302

Offset: 0

Ilya Gutkovskiy, Nov 17 2020

10^18 < a(4) < 10^19. - Daniel Suteu, Nov 17 2020

			a(2) = 455 because the average number of distinct prime divisors of {1..455} is >= 2.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 455
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A013939, A085829, A328331, A338891.

A368611 a(n) = Sum_{k=1..n} pi(k) * floor(n/k).

Original entry on oeis.org

0, 1, 3, 6, 9, 15, 19, 26, 32, 40, 45, 58, 64, 75, 86, 99, 106, 123, 131, 149, 163, 177, 186, 212, 224, 240, 255, 277, 287, 316, 327, 351, 369, 388, 406, 441, 453, 474, 494, 528, 541, 578, 592, 622, 651, 675, 690, 737, 756, 788, 812, 845, 861, 903, 927, 969, 995, 1022

Offset: 1

Wesley Ivan Hurt, Dec 31 2023

Cf. A000720 (pi), A013939, A046992, A368610, A368612, A368613.

Table[Sum[PrimePi[k] Floor[n/k], {k, n}], {n, 100}]

a(n) = A013939(n) + A368613(n).

A092494 a(n) = Sum_{p prime and p<=n} ceiling(n/p).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 10, 11, 12, 13, 16, 17, 20, 21, 23, 25, 27, 28, 31, 32, 34, 36, 39, 40, 42, 43, 45, 46, 49, 50, 54, 55, 56, 58, 60, 62, 65, 66, 68, 70, 73, 74, 78, 79, 81, 83, 86, 87, 89, 90, 92, 94, 97, 98, 100, 102, 104, 106, 109, 110, 114, 115, 117, 119, 120, 122

Offset: 1

Reinhard Zumkeller, Apr 05 2004

nonn

a(n) = A013939(n) + A048865(n).

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

Cf. A006590.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
p:= 0:
do
  p:= nextprime(p);
  if p > N then break fi;
  V[p]:= V[p]+1;
  for k from 2 to floor(N/p) do
    V[(k-1)*p+1 .. k*p]:= V[(k-1)*p+1 .. k*p] +~ k;
  od;
  if (k-1)*p+1<=N then V[(k-1)*p+1..N]:= V[(k-1)*p+1..N]+~ k fi
od:
convert(V,list); # Robert Israel, Jun 19 2019

a(n) = sum(k=1, n, isprime(k)*ceil(n/k)); \\ Michel Marcus, Jun 19 2019

A112965 a(n) = Sum_{i+j=n} omega(i)*omega(j), where omega = A001221.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 7, 8, 9, 10, 14, 14, 17, 18, 23, 24, 27, 26, 32, 32, 35, 36, 44, 42, 47, 48, 52, 50, 58, 54, 65, 62, 67, 66, 78, 70, 79, 78, 88, 84, 94, 88, 100, 100, 103, 100, 118, 106, 119, 114, 124, 116, 135, 122, 138, 134, 141, 136, 155, 142, 155, 154, 163, 156

Offset: 1

Reinhard Zumkeller, Oct 07 2005

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A013939, A112966, A112967.

Table[Sum[PrimeNu[i]*PrimeNu[n - i], {i, n - 1}], {n, 65}] (* Ivan Neretin, Jan 21 2017 *)

a(n) = sum(i=2, n-2, omega(i)*omega(n-i)); \\ Michel Marcus, Jan 22 2017

G.f.: (Sum_{k>=1} x^prime(k)/(1 - x^prime(k)))^2. - Ilya Gutkovskiy, Jan 31 2017

cp's OEIS Frontend

A060238 a(n) = det(M) where M is an n X n matrix with M[i,j] = lcm(i,j).

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A112966 Sum(mu(i)*omega(j): i+j=n), with mu=A008683 and omega=A001221.

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A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

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A329354 a(n) = Sum_{d|n} d*omega(d).

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A355537 Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

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Mathematica

A060832 a(n) = Sum_{k>0} floor(n/k!).

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A338943 a(n) is the least number k such that the average number of distinct prime divisors of {1..k} is >= n.

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A368611 a(n) = Sum_{k=1..n} pi(k) * floor(n/k).

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