A338159 -id:A338159 - cp's OEIS frontend

A338160 Number of ways to represent n as a product of the greatest number of distinct factors.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4

Offset: 1

Vladimir Letsko, Oct 14 2020

nonn

a(n) = A058060(n) for 1 < n < 60; a(60) = 3, A058060(60) = 1.
a(n) is the number of factorizations of n into A086435(n) distinct factors > 1.
a(n) depends only on the prime signature of n.

			a(72) = 3 because 72 = 2*3*12 = 2*4*9 = 3*4*6 and 72 cannot be represented as a product of 4 distinct factors each greater than 1 (adding the factor 1 to each product doesn't change anything).

Cf. A086435, A140207, A338159.

a(n)={my(d=divisors(n)); my(F(r,k)=if(r==1, [0,1], my(b=-1,c=0); for(k=2, k, if(r%d[k]==0, my([tb,tc]=self()(r/d[k], k-1)); if(tb>b, b=tb; c=0); if(tb==b, c+=tc))); [b+1, c])); F(n, #d)[2]} \\ Andrew Howroyd, Oct 14 2020

More terms from Andrew Howroyd, Oct 14 2020

A338261 The least number of the form 2^i*3^j (i, j >= 0) which can be represented as a product of the greatest number of distinct positive integers in exactly n ways.

Original entry on oeis.org

1, 12, 72, 96, 3456, 576, 1536, 55296, 864, 9216, 56623104, 6912, 1769472, 62208, 34359738368, 746496, 110592, 93312, 3145728, 82944, 15925248, 1327104, 32614907904, 995328, 1679616, 3538944, 42467328, 1207959552, 18874368, 382205952, 286654464, 22463437455746924544, 8707129344, 1855425871872, 13060694016, 14495514624, 2717908992, 270826551115776, 17915904, 226492416

Offset: 1

Dmitry Khomovsky, Oct 19 2020

nonn

The numbers 2^i*3^j and 2^j*3^i have the same number of ways to represent them as a product of the greatest number of distinct divisors. Therefore each term of the sequence is a number of the form 2^i*3^j for which i>=j>=0.

			a(5) = 2^7*3^3 = 3456 because 3456 = 1*2*3*4*6*24 = 1*2*3*4*8*18 = 1*2*3*4*9*16 = 1*2*3*6*8*12 = 1*2*4*6*8*9 and each number of the form 2^i*3^j (i, j >= 0) less than 3456 does not have 5 representations as a product of the greatest number of distinct positive integers.

Cf. A338159.

cp's OEIS Frontend

A338160 Number of ways to represent n as a product of the greatest number of distinct factors.

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