A052485 -id:A052485 - cp's OEIS frontend

A304776 A weakening function. a(n) = n / A007947(n)^(A051904(n) - 1) where A007947 is squarefree kernel and A051904 is minimum prime exponent.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 12, 73, 74, 75, 76, 77, 78, 79, 80, 3, 82, 83

Offset: 1

Gus Wiseman, May 18 2018

nonn

This function takes powerful numbers (A001694) to weak numbers (A052485) and leaves weak numbers unchanged.

First differs from A052410 at a(72) = 12, A052410(72) = 72.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001597, A001694, A005117, A007947, A013929, A047966, A051903, A051904, A052485, A062759, A066636, A066638, A072774, A304768, A354090.

spr[n_]:=Module[{f,m},f=FactorInteger[n];m=Min[Last/@f];n/Times@@First/@f^(m-1)];
Array[spr,100]

A007947(n) = factorback(factorint(n)[, 1]);
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A304776(n) = (n/(A007947(n)^(A051904(n)-1))); \\ Antti Karttunen, May 19 2022

a(n) = if(n == 1, 1, my(f = factor(n), p = f[, 1], e = f[, 2]); n / vecprod(p)^(vecmin(e) - 1)); \\ Amiram Eldar, Sep 12 2024

a(n) = n / A354090(n). - Antti Karttunen, May 19 2022

Sum_{k=1..n} a(k) ~ n^2 / 2. - Amiram Eldar, Sep 12 2024

Data section extended up to a(83) by Antti Karttunen, May 19 2022

A352493 Number of non-constant integer partitions of n into prime parts with prime multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 3, 0, 1, 4, 5, 3, 1, 3, 5, 7, 3, 5, 6, 8, 8, 11, 7, 6, 8, 15, 14, 14, 10, 15, 17, 21, 18, 23, 20, 28, 25, 31, 27, 35, 32, 33, 37, 46, 41, 50, 45, 58, 56, 63, 59, 78, 69, 76, 81, 85, 80, 103, 107, 111, 114, 127

Offset: 0

Gus Wiseman, Mar 24 2022

nonn

			The a(n) partitions for selected n (B = 11):
n = 10    16       19        20         25          28
   ---------------------------------------------------------------
    3322  5533     55333     7733       77722       BB33
          55222    55522     77222      5533333     BB222
          3322222  3333322   553322     5553322     775522
                   33322222  5522222    55333222    55533322
                             332222222  55522222    772222222
                                        333333322   3322222222222
                                        3333322222

Constant partitions are counted by A001221, ranked by A000961.
Non-constant partitions are counted by A144300, ranked A024619.
The constant version is A230595, ranked by A352519.
This is the non-constant case of A351982, ranked by A346068.
These partitions are ranked by A352518.
A000040 lists the primes.
A000607 counts partitions into primes, ranked by A076610.
A001597 lists perfect powers, complement A007916.
A038499 counts partitions of prime length.
A053810 lists primes to primes.
A055923 counts partitions with prime multiplicities, ranked by A056166.
A257994 counts prime indices that are themselves prime.
A339218 counts powerful partitions into prime parts, ranked by A352492.
Cf. A000005, A007690, A031368, A035444, A052485, A056239, A066208, A089723, A114639, A320628, A330945.

Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&And@@PrimeQ/@#&& And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A386223 Nonsquarefree weak numbers k that are products of primorials.

Original entry on oeis.org

12, 24, 48, 60, 96, 120, 180, 192, 240, 360, 384, 420, 480, 720, 768, 840, 960, 1080, 1260, 1440, 1536, 1680, 1920, 2160, 2520, 2880, 3072, 3360, 3840, 4320, 4620, 5040, 5760, 6144, 6300, 6480, 6720, 7560, 7680, 8640, 9240, 10080, 11520, 12288, 12600, 12960, 13440

Offset: 1

Michael De Vlieger, Jul 15 2025

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    12   2^2 * 3
 2    24   2^3 * 3
 3    48   2^4 * 3
 4    60   2^2 * 3 * 5
 5    96   2^5 * 3
 6   120   2^3 * 3 * 5
 7   180   2^2 * 3^2 * 5
 8   192   2^6 * 3
 9   240   2^4 * 3 * 5
10   360   2^3 * 3^2 * 5
11   384   2^7 * 3
12   420   2^2 * 3 * 5 * 7

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A126706, A286708, A332785, A364930, A380543.

(* Load May 19 2018 function f at A025487, then run the following: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Subset of A380543.
Intersection of A025487 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.
The union of this sequence and A364930 is A126706.

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

Original entry on oeis.org

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 *)

A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 11 2018

nonn

All terms are squarefree numbers. First differs from A304328 at a(500) = 1, A304328(500) = 4.

			f maps 500 -> 4 -> 1 -> 1, so a(500) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001597, A001694, A005117, A007916, A052485, A052486, A059404, A091050, A160400, A203025, A294068, A303707, A304327, A304328.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
op[n_]:=n/Last[Select[Divisors[n],!radQ[#]&]];
Table[FixedPoint[op,n],{n,200}]

a(n)={while(1, my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); if(m==1, return(n)); n/=m)} \\ Andrew Howroyd, Aug 26 2018

A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67

Offset: 1

Gus Wiseman, May 18 2018

nonn

Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.

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

Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.

acj[n_]:=Module[{f,m},f=FactorInteger[n];m=Max[Last/@f];Times@@Table[p[[1]]^(m-p[[2]]+1),{p,f}]];
Array[acj,100]

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[,2]) + 1); prod(i = 1, #f~, f[i,1]^e) / n);} \\ Amiram Eldar, Feb 12 2023

If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).

A331802 Integers having no representation as sum of two nonsquarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 19, 23

Offset: 1

Bernard Schott, Feb 23 2020

This sequence is finite with 14 terms and 23 is the largest term (see Prime Curios link); a proof can be found in comments of A331801.

			With the two smallest nonsquarefree numbers 4 and 8, it is not possible to get 1, 2, 3, 4, 5, 6, 7, 9, 10 and 11 as sum of two nonsquarefree numbers.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 23

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331801 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

max = 25; Complement[Range[max], Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &]] (* Amiram Eldar, Feb 24 2020 *)

A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

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

Cf. A001222, A001694 (positions of nonzero terms), A005361, A007427, A008683, A008836, A028242, A052485 (positions of 0's), A062838 (positions of 1's), A112526, A252505, A322483, A332685, A332713.

Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]
A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
f[p_, e_] := 3*Floor[e/2] - e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(2*s)^2 * zeta(3*s) / zeta(6*s).
a(n) = Sum_{d|n} mu(lcm(d, n/d)/d).
a(n) = Sum_{d|n} (-1)^bigomega(n/d) * A005361(d).
a(n) = Sum_{d|n} A010052(n/d) * A112526(d).
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)*log(n)/(2*zeta(3)) + ((2*gamma - 1)*zeta(3/2) + 3*zeta'(3/2)/2 - 3*zeta(3/2)*zeta'(3)/zeta(3)) * sqrt(n)/zeta(3) + 6*zeta(2/3)^2 * n^(1/3)/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 21 2020
Multiplicative with a(p^e) = A028242(e). - Amiram Eldar, Nov 30 2020

A374785 Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.

Original entry on oeis.org

223092870, 281291010, 300690390, 6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330

Offset: 1

Amiram Eldar, Jul 20 2024

nonn

Numbers k such that A374783(k)/A374784(k) > 2.
The least odd term is A070826(43) = 5.154... * 10^74, and the least term that is coprime to 6 is Product_{k=3..219} prime(k) = 1.0459... * 10^571.
The least nonsquarefree (A013929) term is a(613) = 148802944290 = 2 * 3 * 5 * 7 * 11 * 13 * 17 *19 * 23^2 * 29.
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, A374783/(k)/A374784(k) < Product_{p prime} (1 + 1/(2*p)) = 1.242534... (A366586).

			223092870 is a term since A374783(223092870)/A374784(223092870) = 666225/330752 = 2.014... > 2.

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

Subsequence of A052485.
Cf. A001221, A001694, A013929, A034683, A070826, A366586, A374783, A374784.
Similar sequences: A245214, A374788.

f[p_, e_] := 1 + 1/(2*p^e); r[1] = 1; r[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^8], s[#] > 2 &]

is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])) > 2;}

A001221(a(n)) >= 9.

A386224 Nonsquarefree weak numbers k that are not products of primorials, whose squarefree kernel is a primorial.

Original entry on oeis.org

18, 54, 90, 150, 162, 270, 300, 450, 486, 540, 600, 630, 750, 810, 1050, 1200, 1350, 1458, 1470, 1500, 1620, 1890, 2100, 2250, 2400, 2430, 2940, 3000, 3150, 3240, 3750, 3780, 4050, 4200, 4374, 4410, 4800, 4860, 5250, 5670, 5880, 6000, 6750, 6930, 7290, 7350, 7500

Offset: 1

Michael De Vlieger, Jul 15 2025

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    18   2 * 3^2
 2    54   2 * 3^3
 3    90   2 * 3^2 * 5
 4   150   2 * 3 * 5^2
 5   162   2 * 3^4
 6   270   2 * 3^3 * 5
 7   300   2^2 * 3 * 5^2
 8   450   2 * 3^2 * 5^2
 9   486   2 * 3^5
10   540   2^2 * 3^3 * 5
11   600   2^3 * 3 * 5^2
12   630   2 * 3^2 * 5 * 7

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A056808, A126706, A286708, A332785, A369420, A380543, A386223.

(* Load May 19 2018 function f at A025487, then run the following: *)
fQ[x_] :=
 Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &,
   Nest[Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &,
     If[x == 1, {0},
       Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@
         Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ x], 2] ] == x;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Select[Union@ Flatten@ f[6][[3 ;; -1, 2 ;; -1]], Nor[Divisible[#, rad[#]^2], fQ[#]] &]

{a(n)} = A380543 \ A386223.
Intersection of A056808 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694, and A056808 = A055932 \ A025487.
The union of this sequence and A369420 is A126706.

cp's OEIS Frontend

A304776 A weakening function. a(n) = n / A007947(n)^(A051904(n) - 1) where A007947 is squarefree kernel and A051904 is minimum prime exponent.

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A352493 Number of non-constant integer partitions of n into prime parts with prime multiplicities.

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A386223 Nonsquarefree weak numbers k that are products of primorials.

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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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A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

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A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

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A331802 Integers having no representation as sum of two nonsquarefree numbers.

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A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

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