A331286 -id:A331286 - cp's OEIS frontend

A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

1, 2, 4, 3, 8, 6, 16, 4, 9, 12, 32, 8, 64, 24, 18, 5, 128, 12, 256, 16, 36, 48, 512, 10, 27, 96, 16, 32, 1024, 24, 2048, 6, 72, 192, 54, 15, 4096, 384, 144, 20, 8192, 48, 16384, 64, 32, 768, 32768, 12, 81, 36, 288, 128, 65536, 20, 108, 40, 576, 1536, 131072, 30, 262144, 3072, 64, 7, 216, 96, 524288, 256, 1152, 72, 1048576, 18, 2097152, 6144, 48

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A000005, A000040, A000142, A002110, A010052, A034386, A052126, A056239, A108951, A329902, A329378, A329382, A329614, A329617, A331283, A331286 (odd part).
Cf. A329606 (rgs-transform), A329608, A331284 (ordinal transform).
Cf. A331285 (the position where for the first time some term has occurred n times in this sequence).

Block[{a}, a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Array[DivisorSigma[0, a@ #] &, 75]] (* Michael De Vlieger, Jan 24 2020, after Jean-François Alcover at A108951 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329605(n) = numdiv(A108951(n));

A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

A329605(n) = if(1==n,1,my(f=factor(n),e=0,d); forstep(i=#f~,1,-1, e += f[i,2]; d = (primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1]))); f[i,1] = (e+1); f[i,2] = d); factorback(f)); \\ Antti Karttunen, Jan 14 2020

a(n) = A000005(A108951(n)).
a(n) >= A329382(n) >= A329617(n) >= A329378(n).
A020639(a(n)) = A329614(n).
From Antti Karttunen, Jan 14 2020: (Start)
a(A052126(n)) = A329382(n).
a(A002110(n)) = A000142(1+n), for all n >= 0.
a(n) > A056239(n).
a(A329902(n)) = A002183(n).
A000265(a(n)) = A331286(n).
gcd(n,a(n)) = A331283(n).
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > ... > kx, then a(n) = (1+e(k1))^(k1-k2) * (1+e(k1)+e(k2))^(k2-k3) * ... * (1+e(k1)+e(k2)+...+e(kx))^kx.
A000035(a(n)) = A000035(A000005(n)) = A010052(n).
(End)

A212181 Largest odd divisor of tau(n): a(n) = A000265(A000005(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Matthew Vandermast, Jun 04 2012

Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172).
Not the same as the number of odd divisors of n (A001227(n)); see example.
Multiplicative because A000005 is multiplicative and A000265 is completely multiplicative. - Andrew Howroyd, Aug 01 2018
a(n) = 1 iff the number of divisors of n is a power of 2 (A036537). - Bernard Schott, Nov 04 2022

			48 has a total of 10 divisors (1, 2, 3, 4, 6, 8, 12, 16, 24 and 48). Since the largest odd divisor of 10 is 5, a(48) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000079, A000265, A036537, A108951, A212172, A295664, A331286 (applied to primorial inflation of n).

Table[Block[{nd=DivisorSigma[0, n]}, nd/2^IntegerExponent[nd, 2]], {n, 100}] (* Indranil Ghosh, Jul 19 2017, after PARI code *)

a(n) = my(nd = numdiv(n)); nd/2^valuation(nd, 2); \\ Michel Marcus, Jul 19 2017

from sympy import divisor_count, divisors
def a(n): return [i for i in divisors(divisor_count(n)) if i%2][-1]
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017

a(n) = A000265(A000005(n)).
From Antti Karttunen, Jan 14 2020: (Start)
a(n) = A000005(n) / A000079(A295664(n)).
a(A108951(n)) = A331286(n).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p odd prime} ((1 - 1/p)*(1 + Sum_{k>=1} a(k+1)/p^k)) = 2.076325817863586... . - Amiram Eldar, Oct 15 2022

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A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A212181 Largest odd divisor of tau(n): a(n) = A000265(A000005(n)).

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