A195207 -id:A195207 - cp's OEIS frontend

A195208 Number of odd divisors of !n.

1, 0, 1, 1, 3, 2, 4, 6, 8, 8, 10, 4, 6, 12, 8, 4, 24, 4, 8, 24, 32, 16, 24, 96, 32, 48, 24, 16, 384, 32, 64, 144, 16, 8, 36, 128, 16, 80, 24, 16, 96, 32, 16, 96, 4, 16, 448, 128, 64, 24, 192, 48, 192, 64, 16, 96, 768, 32, 96, 32, 64

Offset: 0

Michel Lagneau, Sep 13 2011

nonn

!k is a subfactorial number (A000166).

			a(7) = 6 because the divisors of  !7 = 1854 are  {1, 2, 3, 6, 9, 18, 103, 206, 309, 618, 927, 1854} with 6  odd divisors  1, 3, 9, 103, 309 and 927 => 6 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 0..81

Cf. A000005, A000166, A000265, A001227, A195207, A195209, A195210.

A166 := proc(n) option remember; (n-1)*(procname(n-1)+procname(n-2)); end:
A166(0):= 1: A166(1):= 0:
f:= proc(n) local v; v:= A166(n); numtheory:-tau(v/2^padic:-ordp(v,2)) end proc:
map(f, [$1..60]); # Robert Israel, Aug 01 2024

f[n_] := Block[{d = Divisors[Subfactorial[n]]}, Count[OddQ[d], True]]; Table[f[n], {n, 0, 60}]

a(n)=if(n<2,1-n,my(x='x,k=n!*polcoeff(exp(-x+x*O(x^n))/(1-x),n));numdiv(k>>valuation(k,2))) \\ Charles R Greathouse IV, Sep 13 2011

a(n) = A000005(A000265(A000166(n))). - Robert Israel, Aug 01 2024

a(n) = A001227(A000166(n)), for n != 1. - Amiram Eldar, Aug 02 2024

A195209 Sum of even divisors of !n.

Original entry on oeis.org

0, 0, 0, 2, 0, 72, 0, 2704, 0, 268128, 0, 17621496, 0, 4974281208, 0, 549790303712, 0, 245357774621280, 0, 67684833350266496, 0, 36434223374991414336, 0, 10781900341537812480000, 0, 14585451990785851336761600, 0, 4706112122106541811177554944, 0, 5607912482098966721098538956800, 0

Offset: 0

Michel Lagneau, Sep 13 2011

nonn

			a(7) = 2704 because the divisors of  !7 = 1854 are  {1, 2, 3, 6, 9, 18, 103, 206, 309, 618, 927, 1854}, and the sum of the 6  even divisors  2, 6, 18, 206, 618 and 1854 is 2704.

Amiram Eldar, Table of n, a(n) for n = 0..82

Cf. A000166, A146076, A195207, A195208, A195210.

Table[Total[Select[Divisors[Subfactorial[n]], EvenQ[ # ]&]], {n, 0, 25}]

From Amiram Eldar, Aug 02 2024: (Start)
a(n) = A146076(A000166(n)), for n != 1.
a(2*n) = 0. (End)

a(26)-a(30) from Amiram Eldar, Aug 02 2024

A195210 Sum of odd divisors of !n.

Original entry on oeis.org

1, 0, 1, 1, 13, 12, 324, 1352, 18368, 19152, 1994322, 8810748, 193690826, 829046868, 34563283272, 274895151856, 14554448767872, 8178592487376, 2493850026398472, 33842416675133248, 1037660717168380800, 6072370562498569056, 682597834324732589120, 5390950170768906240000

Offset: 0

Michel Lagneau, Sep 13 2011

nonn

!n is a subfactorial number (A000166).

Property of the sequence: for n > 4 the sum of odd divisors of !n seems even.

			a(7) = 1352 because the divisors of  !7 = 1854 are  {1, 2, 3, 6, 9, 18, 103, 206, 309, 618, 927, 1854}, and the sum of the 6  odd divisors  1, 3, 9, 103, 309 and 927 is 1352.

Amiram Eldar, Table of n, a(n) for n = 0..81

Cf. A000166, A000593, A195207, A195208, A195209.

Table[Total[Select[Divisors[Subfactorial[n]], OddQ[ # ]&]], {n, 0, 21}]

a(n) = A000593(A000166(n)), for n != 1. - Amiram Eldar, Aug 02 2024

a(22)-a(23) from Amiram Eldar, Aug 02 2024

A378157 The least prime dividing !n = A000166(n).

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 11, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 11, 2, 3, 2, 11, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 11, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2

Offset: 3

Amiram Eldar, Nov 18 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 3..10000

Cf. A000166, A020639, A152024, A195207, A195208, A195209, A195210, A378158, A378159.

lpf[n_] := Module[{p = 2}, While[! Divisible[n, p], p = NextPrime[p]]; p]; Array[lpf[Subfactorial[#]] &, 50, 3]

lpf(n) = {my(p = 2); while(n % p, p = nextprime(p+1)); p;}
lista(nmax) = {my(s = 1); for(n = 3, nmax, s = n * s + (-1)^n; print1(lpf(s), ", ")); }

a(n) = A020639(A000166(n)).

a(n) = min(A020639(n-1), A378159(n-2)) for n >= 2.

A378160 The number of distinct prime factors of !n = A000166(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 2, 4, 3, 3, 4, 3, 3, 4, 5, 5, 4, 7, 5, 6, 4, 5, 7, 6, 6, 7, 4, 4, 4, 8, 4, 6, 4, 5, 6, 6, 4, 7, 2, 4, 7, 8, 6, 5, 7, 6, 7, 7, 4, 6, 9, 6, 6, 6, 6, 6, 4, 4, 5, 4, 3, 6, 6, 6, 6, 6, 7, 7, 4, 8, 6, 5, 8, 6, 4, 4, 5, 8, 4, 7, 7, 8, 6

Offset: 2

Amiram Eldar, Nov 18 2024

nonn

Sean A. Irvine, Table of n, a(n) for n = 2..108

Cf. A000166, A001221, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378161, A378162.

Array[PrimeNu[Subfactorial[#]] &, 40, 2]

lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(omega(s), ", "));}

a(n) = A001221(A000166(n)).

a(n) >= A001221(n-1) + 1 for n >=5.

a(82)-a(88) from Jinyuan Wang, Nov 24 2024

A378161 The number of prime factors of !n = A000166(n), counted with multiplicity.

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Nov 18 2024

nonn

Sean A. Irvine, Table of n, a(n) for n = 2..108

Cf. A001222, A000166, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378160, A378162.

Array[PrimeOmega[Subfactorial[#]] &, 40, 2]

lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(bigomega(s), ", "));}

a(n) = A001222(A000166(n)).

a(n) >= A001222(n-1) + 1 for n >=4, with equality if and only if n is in A301423.

a(82)-a(86) from Jinyuan Wang, Nov 24 2024

cp's OEIS Frontend

A195208 Number of odd divisors of !n.

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A195209 Sum of even divisors of !n.

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A195210 Sum of odd divisors of !n.

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A378157 The least prime dividing !n = A000166(n).

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A378160 The number of distinct prime factors of !n = A000166(n).

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A378161 The number of prime factors of !n = A000166(n), counted with multiplicity.

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