A038109 -id:A038109 - cp's OEIS frontend

A297075 Lexicographically earliest sequence of distinct positive numbers such that the prime factorizations of two consecutive terms never share a prime exponent >= 1.

1, 2, 4, 3, 8, 5, 9, 6, 16, 7, 25, 10, 27, 11, 32, 12, 64, 13, 36, 14, 49, 15, 72, 17, 81, 18, 125, 19, 100, 21, 108, 22, 121, 23, 128, 20, 216, 26, 144, 24, 169, 29, 196, 30, 200, 31, 225, 33, 243, 28, 256, 34, 288, 35, 289, 37, 324, 38, 343, 39, 361, 40, 400

Offset: 1

Rémy Sigrist, Dec 25 2017

nonn

For any n > 0, if a prime number p divides a(n) and a prime number q divides a(n+1), then the p-adic valuation of a(n) differs from the q-adic valuation of a(n+1).
Equivalently, for any n > 0, A297404(a(n)) AND A297404(a(n+1)) = 0 (where AND denotes the bitwise AND operator).
This sequence is a permutation of the natural numbers, with inverse A297403.
The curves visible in the logarithmic scatterplot of the first terms seems to be related to a(n) belonging to A038109 and to A052485 (see Links section).
Lexicographically earliest sequence of distinct numbers such that gcd(A181819(a(n)), A181819(a(n+1))) = 1. - Peter Munn, Oct 02 2023
From Peter Munn, Jan 25 2024: (Start)
The sequence bisections might be characterized as being monotonic with interruptions. The major interruptions are apparent from the coloring in the author's 15000 term logarithmic scatterplot -- they occur where the occurrence of terms belonging to A038109 switches between the bisections.
Other interruptions are too small to be seen in the scatterplot. Some relate to numbers that have both the square of a prime and cube of a prime as a unitary divisor (a subset of A038109).
Two such terms are a(4154) = 1350 and a(4156) = 1368, interrupting the even bisection's monotonicity after a(4152) = 1380. These 3 terms are each followed by a 4-full number (A036967): a(4153) = 1185921, a(4155) = 1229312, a(4157) = 1250000. Then we see an odd bisection interruption with a(4159) = 1191016.
(End)

			The first terms, alongside the corresponding sets of prime exponents, are:
  n       a(n)    Set of prime exponents of a(n)
  --      ----    ------------------------------
   1       1      {}
   2       2      {1}
   3       4      {2}
   4       3      {1}
   5       8      {3}
   6       5      {1}
   7       9      {2}
   8       6      {1, 1}
   9      16      {4}
  10       7      {1}
  11      25      {2}
  12      10      {1, 1}
  13      27      {3}
  14      11      {1}
  15      32      {5}
  16      12      {2, 1}
  17      64      {6}
  18      13      {1}
  19      36      {2, 2}
  20      14      {1, 1}

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 15000 terms
Rémy Sigrist, C++ program for A297075
Index entries for sequences that are permutations of the natural numbers

Cf. A001694 (numbers in odd bisection), A036967, A038109, A052485 (numbers in even bisection), A181819, A297403 (inverse), A297404.

Nest[Append[#, Block[{k = 3, m = FactorInteger[#[[-1]] ][[All, -1]]}, While[Nand[FreeQ[#, k], ! IntersectingQ[m, FactorInteger[k][[All, -1]]]], k++]; k]] &, {1, 2}, 61] (* Michael De Vlieger, Dec 29 2017 *)

A326815 Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 2 * p^(-s)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, -2, 0, 1, 1, 0, 1, 1, 1, -5, 1, 0, 1, 0, 1, 1, 1, -2, 0, 1, -2, 0, 1, 1, 1, -9, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 0, 1, 1, -5, 0, 0, 1, 0, 1, -2, 1, -2, 1, 1, 1, 0, 1, 1, 0, -14, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -5, -5, 1, 1, 0, 1

Offset: 1

Ilya Gutkovskiy, Oct 19 2019

Inverse Moebius transform applied twice to A076479 (unitary Moebius function).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001221, A005117 (positions of 1's), A007425, A008683, A038109 (positions of 0's), A046951, A076479, A080956, A326814.

Table[Sum[(-1)^PrimeNu[n/d] DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := (e + 1)*(2 - e)/2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

A326815(n) = sumdiv(n,d,((-1)^omega(n/d))*numdiv(d)); \\ Antti Karttunen, Nov 17 2019

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

a(n) = Sum_{d|n} (-1)^omega(n/d) * tau(d), where omega = A001221 and tau = A000005.
a(n) = Sum_{d|n} tau_3(n/d) * mu(d) * 2^omega(d), where tau_3 = A007425 and mu = A008683.
Multiplicative with a(p^e) = (e+1)*(2-e)/2 = A080956(e). - Amiram Eldar, Oct 26 2020

A362841 Numbers with at least one 5 in their prime signature.

Original entry on oeis.org

32, 96, 160, 224, 243, 288, 352, 416, 480, 486, 544, 608, 672, 736, 800, 864, 928, 972, 992, 1056, 1120, 1184, 1215, 1248, 1312, 1376, 1440, 1504, 1568, 1632, 1696, 1701, 1760, 1824, 1888, 1944, 1952, 2016, 2080, 2144, 2208, 2272, 2336, 2400, 2430, 2464, 2528, 2592, 2656, 2673, 2720, 2784, 2848, 2912, 2976

Offset: 1

R. J. Mathar, May 05 2023

nonn

Contains all odd multiples of 2^5: Each second term of A174312 is in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^5 + 1/p^6) = 0.01863624892... . - Amiram Eldar, May 05 2023

			Contains 2^5, 2^5*3, 2^5*5, 2^5*7, 3^5, 2^5*3^2, 2^5*11, 2^5*13, 2^5*3*5, 2*3^5, etc.

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

Cf. A038109 (at least one 2), A176297 (at least one 3), A050997 (subsequence), A178740 (subsequence), A179646 (subsequence), A179667 (subsequence), A179671 (subsequence), A174312.

Select[Range[3000], MemberQ[FactorInteger[#][[;;, 2]], 5] &] (* Amiram Eldar, May 05 2023 *)

A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245

Offset: 1

Amiram Eldar, Jul 28 2024

Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .
Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .
Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.
Each term is a product of two coprime numbers: an exponentially odd number (A268335) and a square of a squarefree number (A062503) that is larger than 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = A065466 - A065463 = 0.2432910611445097832029... .

			4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
144 = 2^4 * 3^2 is not a term because it has the exponent 4 in its prime factorization which is even and larger than 2.

Subsequence of A013929, A038109 and A304365.
A062503 \ {1} is a subsequence.
Cf. A065463, A065466, A268335, A335275, A350388, A375033.

q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]

is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}

A375033(a(n)) = 2.

A375847 The maximum exponent in the prime factorization of the largest unitary cubefree divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Aug 31 2024

Cf. A004709, A036966, A038109, A051903, A330596, A337050, A360539.

Cf. A007424 (analogous with the largest cubefree divisor, for n >= 2).

a[n_] := Max[Join[{0}, Select[FactorInteger[n][[;; , 2]], # <= 2 &]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = select(x -> x <= 2, factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

a(n) = A051903(A360539(n)).
a(n) = 0 if and only if n is cubefull (A036966).
a(n) = 1 if and only if n is in A337050 \ A036966.
a(n) = 2 if and only if n is in A038109.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - A330596 = 1.25146474031763643535... .

cp's OEIS Frontend

A297075 Lexicographically earliest sequence of distinct positive numbers such that the prime factorizations of two consecutive terms never share a prime exponent >= 1.

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A326815 Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 2 * p^(-s)).

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A362841 Numbers with at least one 5 in their prime signature.

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A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

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A375847 The maximum exponent in the prime factorization of the largest unitary cubefree divisor of n.

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