A072078 -id:A072078 - cp's OEIS frontend

A382487 The number of divisors of n whose largest prime factor is 3.

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

Offset: 1

Amiram Eldar, Mar 29 2025

The number of 3-smooth divisors of n that are not powers of 2.

The number of terms of A065119 that divide n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001511, A001651, A065119, A072078, A007949, A051064, A169611, A301461, A306771.

a[n_] := (IntegerExponent[n, 2] + 1) * IntegerExponent[n, 3]; Array[a, 100]

a(n) = (valuation(n, 2) + 1) * valuation(n, 3);

a(n) = A072078(n) - A001511(n).
a(n) = A001511(n) * A007949(n).
a(n) = 0 if and only if n is in A001651.
a(n) = 1 if and only if n is in A306771.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.
In general, the asymptotic mean of the number prime(k+1)-smooth divisors of n that are not prime(k)-smooth, for k >= 1, is (1/(prime(k+1)-1)) * Product_{i=1..k} (prime(i)/(prime(i)-1)).
Dirichlet g.f.: (zeta(s)/(1-1/2^s))*(1/(1-1/3^s) - 1).

A382492 a(n) is the least number that has exactly n 3-smooth divisors.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 72, 4096, 192, 144, 216, 65536, 288, 262144, 432, 576, 3072, 4194304, 864, 1296, 12288, 2304, 1728, 268435456, 2592, 1073741824, 3456, 9216, 196608, 5184, 6912, 68719476736, 786432, 36864, 10368, 1099511627776, 15552, 4398046511104

Offset: 1

Amiram Eldar, Mar 29 2025

The record values occur at A046022.

All the terms are in A003586 and A025487.

Amiram Eldar, Table of n, a(n) for n = 1..3322

Cf. A003586, A005179, A025487, A037143, A046022, A072078, A382493.

a[n_] := Min[Table[2^(d-1)*3^(n/d-1), {d, Divisors[n]}]]; Array[a, 60]

a(n) = vecmin(apply(d -> 2^(d-1)*3^(n/d-1), divisors(n)));

a(n) = Min_{d|n} (2^(d-1)*3^(n/d-1)).
a(n) = 2^A382493(n) * 3^(n/(A382493(n)+1)-1).
a(p) = 2^(p-1) for prime p.
a(n) = A005179(n) if n is in A037143.

A088244 Number of 3-smooth divisors of n!.

Original entry on oeis.org

1, 1, 2, 4, 8, 8, 15, 15, 24, 40, 45, 45, 66, 66, 72, 84, 112, 112, 153, 153, 171, 190, 200, 200, 253, 253, 264, 336, 364, 364, 405, 405, 480, 512, 528, 528, 630, 630, 648, 684, 741, 741, 800, 800, 840, 924, 946, 946, 1081, 1081, 1104, 1152, 1200, 1200

Offset: 0

Reinhard Zumkeller, Jan 23 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A003586.
Cf. A000142, A072078.
Cf. A011371, A054861.

a[n_] := ((1 + IntegerExponent[n!, 2])*(1 + IntegerExponent[n!, 3])); Array[a, 100, 0] (* Amiram Eldar, Aug 30 2019 *)

a(n) = A072078(A000142(n)).
a(n) = a(n-1) iff gcd(n,6) = 1.
a(n) = (A011371(n)+1)*(A054861(n)+1).

A382490 The number of infinitary 3-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 1, 4, 1, 4, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 8, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 8, 1, 4, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 2, 2, 2, 1, 8, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2

Offset: 1

Amiram Eldar, Mar 29 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A003586, A007310, A007814, A036537, A037445, A065331, A007949, A138302, A382488.

a[n_] := 2^(DigitCount[IntegerExponent[n, 2], 2, 1] + DigitCount[IntegerExponent[n, 3], 2, 1]); Array[a, 100]

a(n) = 1 << (hammingweight(valuation(n, 2)) + hammingweight(valuation(n, 3)));

Multiplicative with a(p^e) = 2^A000120(e) of p <= 3, and 1 otherwise.
a(n) = 2^(A000120(A007814(n)) + A000120(A007949(n))).
a(n) = A037445(A065331(n)).
a(n) = A037445(n) if and only if n is 3-smooth (A003586).
a(n) = A382488(n) if and only if n is an exponentially 2^n number (A138302).
a(n) = A072078(n) if and only if n is a product of a 5-rough number (A007310) and a 3-smooth number whose number of divisors is a power of 2 (i.e., in both A003586 and A036537).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/3) * Product_{k>=0} (1+1/2^(2^k-1))*(1+2/3^(2^k)) = 2.36739050930467832207... .

A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

Original entry on oeis.org

1, 5, 13, 71, 97, 1355, 793, 19163, 53473, 292355, 60073, 13102907, 535537, 78584915, 790859641, 3523099499, 43112257, 99646519235, 387682633, 2764285630427, 7604811750289, 7337148996275, 31385253913, 2226944658077771, 3656440886376673, 2341258386360995, 80539587570991081

Offset: 1

Amiram Eldar, Mar 29 2025

The denominator that corresponds to a(n) is 3*6^(n-1) = A169604(n-1) = A081341(n).

			Fractions begin with 1/3, 5/18, 13/108, 71/648, 97/3888, 1355/23328, 793/139968, 19163/839808, 53473/5038848, 292355/30233088, 60073/181398528, 13102907/1088391168, ...
a(1) = 1 since a(1)/A081341(1) = 1/3 is the asymptotic density of the numbers with a single 3-smooth divisor, 1, i.e., the numbers that are congruent to 1 or 5 mod 6 (A007310).
a(2) = 5 since a(2)/A081341(2) = 5/18 is the asymptotic density of the numbers with exactly two 3-smooth divisors, either 1 and 2 or 1 and 3, i.e., A171126.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007310, A072078, A081341 (denominators), A169604, A171126.

a[n_] := DivisorSum[n, 2^(n-#) * 3^(n-n/#) &]; Array[a, 30]

PARI
```
a(n) = sumdiv(n, d, 2^(n-d)*3^(n-n/d));
```

a(n) = Sum_{d|n} 2^(n-d) * 3^(n-n/d).
a(p) = 2^(p-1) + 3^(p-1).
Let f(n) = a(n)/A081341(n). Then:
f(n) = (1/3) * Sum_{d|n} (1/2)^(d-1) * (1/3)^(n/d-1).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} n * f(n) = 3 (the asymptotic mean of A072078).
Sum_{n>=1} n^2 * f(n) = 18, and therefore, the asymptotic variance of A072078 is 18 - 3^2 = 9, and its asymptotic standard deviation is 3.

A076302 Triangle T(n,k) = number of k-smooth divisors of n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 3, 3, 1, 1, 1, 1, 2, 1, 2, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 2, 1, 4, 4, 4, 4, 4, 4, 4, 1, 1, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 2, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Mar 14 2003

			Triangle begins:
                   1
                 1   2
               1   1   2
             1   3   3   3
           1   1   1   1   2
         1   2   4   4   4   4
       1   1   1   1   1   1   2
     1   4   4   4   4   4   4   4
   1   1   3   3   3   3   3   3   3

Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000005, A001511, A072078, A355583.

T[n_, k_] := Times@@(IntegerExponent[n, #]+1& /@ Select[Range[2, k], PrimeQ]);
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2021 *)

T(n,n) = A000005(n);
T(n,2) = A001511(n) for n>1.
T(n,3) = A072078(n) for n>2.
T(n,5) = A355583(n) for n>4.
Limit_{m->oo} (1/m) * Sum_{n=k..m} T(n,k) = 1/Product_{p prime <= k} (1 - 1/p). - Amiram Eldar, Apr 17 2025

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A382487 The number of divisors of n whose largest prime factor is 3.

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A382492 a(n) is the least number that has exactly n 3-smooth divisors.

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A088244 Number of 3-smooth divisors of n!.

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A382490 The number of infinitary 3-smooth divisors of n.

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A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

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A076302 Triangle T(n,k) = number of k-smooth divisors of n, read by rows.

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