A136568 -id:A136568 - cp's OEIS frontend

A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.

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

Offset: 1

Leroy Quet, Jan 07 2008

nonn

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 192, 2089, 21405, 215730, 2162136, 21636277, 216410510, 2164253043, 21642998932, ... . Apparently, the asymptotic mean of this sequence is 2.164... . - Amiram Eldar, Jun 30 2025

			120 = 2^3 * 3^1 * 5^1. The exponents of the prime factorization are therefore 3,1,1. The distinct values which equal these exponents are 1 and 3. So a(120) = 1+3 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Diana Mecum)
Index entries for sequences computed from exponents in factorization of n.

Cf. A066328, A071625, A088529, A136566, A136568, A181591, A181819.

Join[{0},Table[Total[Union[Transpose[FactorInteger[n]][[2]]]],{n,2,110}]] (* Harvey P. Dale, Jun 23 2013 *)

A136565(n) = vecsum(apply(primepi,factor(factorback(apply(e->prime(e),(factor(n)[,2]))))[,1])); \\ Antti Karttunen, Sep 06 2018

a(n) = A088529(n) = A181591(n) for n: 2 <= n < 24. - Reinhard Zumkeller, Nov 01 2010

a(n) = A066328(A181819(n)). - Antti Karttunen, Sep 06 2018

More terms from Diana L. Mecum, Jul 17 2008

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3780, 3800, 3816, 3960, 3996, 4056, 4116, 4200, 4248, 4312, 4392, 4428

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A163569 at n = 25: a(25) = 2520 = 2^3 * 3^2 * 5 * 7 is not a term of A163569.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 2, 3}.
The asymptotic densities of this sequence and A375074 are equal (0.0156712..., see A375074 for a formula), since the terms in A375074 that are not in this sequence (A375073) have a density 0.

Intersection of A375072 and A317090.
Equals A375074 \ A375073.
Subsequence of A046100 and A176297.
A163569 is a subsequence.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A136568, A215267.

Select[Range[4500], Union[FactorInteger[#][[;; , 2]]] == {1, 2, 3} &]

is(k) = Set(factor(k)[,2]) == [1, 2, 3];

A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

Original entry on oeis.org

24, 40, 54, 56, 88, 104, 120, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696, 702, 712, 728, 744, 750

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A360793 at n = 79: a(79) = 1080 = 2^3 * 3^3 * 5 is not a term of A360793.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 3}.
The asymptotic density of this sequence is ((zeta(6)/zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) - 1)/zeta(2) = 0.076359822332835689478... .

Equals A336591 \ (A005117 UNION A062838).
Subsequences: A065036, A360793.
Cf. A002117, A013661, A013664, A136568.

Select[Range[750], Union[FactorInteger[#][[;; , 2]]] == {1, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [1, 3];
```

cp's OEIS Frontend

A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.

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A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

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A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

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A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

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Mathematica

PARI