A008479 -id:A008479 - cp's OEIS frontend

A372972 Numbers k such that A372720(k) is negative.

162, 250, 324, 384, 486, 648, 686, 768, 972, 1152, 1250, 1296, 1372, 1458, 1536, 1728, 1875, 1944, 2058, 2250, 2304, 2430, 2500, 2560, 2592, 2662, 2738, 2916, 3000, 3072, 3362, 3402, 3456, 3698, 3750, 3840, 3888, 3993, 4050, 4116, 4374, 4394, 4418, 4500, 4608

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
For s > 1, an infinite number of k in R are such that g(k) is negative. For example, with s = 6, all terms k > 864 in A033845 are in this sequence.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 162 = 2*3^4, since tau(162) - f(162)
     = (1+1)*(4+1) - card(A369609(162))
     = 10 - 12 = -2.
a(2) = 250 = 2*5^3, since tau(250) - f(250)
     = (1+1)*(3+1) - card(A369609(250))
     = 8 - 9 = -1.
a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
     = (2+1)*(4+1) - card(A369609(324))
     = 15 - 16 = -1, etc.

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

Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 5000}], ?(# < 0 &)][[All, 1]]

A373738 a(1) = 1, a(n) = floor((1/omega(n)!) * Product_{p | n} 1 + (log n)/(log p)), where omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 7, 2, 5, 4, 5, 2, 9, 2, 7, 4, 6, 2, 10, 3, 6, 4, 7, 2, 12, 2, 6, 5, 6, 4, 13, 2, 6, 5, 10, 2, 13, 2, 8, 7, 7, 2, 14, 3, 11, 5, 8, 2, 15, 4, 10, 5, 7, 2, 19, 2, 7, 7, 7, 4, 15, 2, 8, 5, 13, 2, 17, 2, 7, 9, 8, 4, 16, 2, 13, 5, 8, 2

Offset: 1

Michael De Vlieger, Jul 16 2024

nonn

This sequence is the integer part of the omega(n)-rank content of an omega(n)-rank orthogonal simplex S(n) with axes measuring 1 + (log n)/(log p) for all primes p | n.
Let R(n) be the arrangement of row n of A162306(n) according to the order of exponents of distinct prime factors p | n. Then A010846(n) is the content of a rank omega(n) Hauy construction where the numbers are placed in omega(n) dimensional cubes, while S(n) is the corresponding simplex.
Conjecture: A010846(k) - a(k) approaches 0 as k increases toward infinity, for k with omega(k) > 1 that have the same squarefree kernel r. Therefore, the difference is most significant for composite squarefree k.
Observation: A008479(n) < a(n) <= A010846(n).

			Let b = A010846.
a(6) = 4 since the floor of the area of a right triangle with axial edge lengths {1+log_p 6 : p | 6} = {3.58496..., 2.63093...}, a(6) = floor(9.43178.../2) = 4. b(6) = 5.
a(10) = 5 since the floor of the area of a right triangle with axial edge lengths {1+log_p 12 : p | 12} = {4.32193..., 2.43068...}, a(10) = floor(10.5052.../2) = 5. b(10) = 6.
a(30) = 12 since the floor of the volume of a trirectangular tetrahedron with axial edge lengths {1+log_p 30 : p | 30} = {5.90689..., 4.0959..., 3.11328...}, a(30) = floor(75.3229.../6) = 12. b(30) = 18.
a(210) = 34 since the floor of the content of a 4-simplex with a vertex with orthogonal edges at origin and axial edge lengths {1+log_p 210 : p | 210} = {8.71425..., 5.86715..., 4.32234..., 3.74787...}, a(210) = floor(828.248.../24) = 12. b(210) = 68, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Comparison of S(900) in red with R(900) in blue
Eric Weisstein's World of Mathematics, Simplex.

Cf. A001221, A008479, A010846, A162306, A246655.

{1}~Join~Table[Floor[(1/PrimeNu[n]!)*Times @@ Map[Log[#, n] + 1 &, FactorInteger[n][[All, 1]] ] ], {n, 2, 82}]

a(n) = A010846(n) = A008479(n) + 1 = 2 for n such that omega(n) = 1.

A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

Original entry on oeis.org

2, 3, 2, 4, 2, 5, 3, 2, 2, 6, 4, 2, 7, 3, 2, 2, 2, 8, 3, 2, 5, 2, 3, 3, 2, 9, 4, 2, 6, 3, 10, 5, 2, 2, 4, 2, 3, 2, 4, 3, 2, 11, 3, 2, 5, 3, 2, 2, 7, 12, 2, 4, 2, 2, 2, 4, 6, 3, 2, 4, 13, 6, 3, 8, 2, 2, 4, 2, 14, 2, 7, 5, 2, 3, 3, 2, 7, 5, 2, 3, 3, 9, 5, 2, 2, 4

Offset: 1

Michael De Vlieger, Sep 21 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).
The set R(k) is a list of numbers beginning with the empty product 1 and including all m such that p | m implies p | n. For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.
Then k*{R(k)} is the list of numbers beginning with k, followed by nonsquarefree k*m such that rad(k*m) = k.
The number k is composite and the only squarefree term in k*{R(k)} and appears in A120944; the rest of the list is in A126706.

			a(1) = 2 since rad(b(1)) = rad(12) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 12 is the 2nd term.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 3rd term in k*{R(6)}.
a(3) = 2 since rad(b(3)) = rad(20) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, ...}, 20 is the 2nd term.
a(4) = 4 since rad(b(4)) = rad(24) = 6, and 24 is the 4th term in k*{R(6)}.
a(5) = 2 since rad(b(5)) = rad(28) = 14, and in the sequence k*{R(14)} = 14*{A003591} = {14, 28, 56, 98, 112, ...}, 28 is the 2nd term, etc.

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

Cf. A007947, A126706, A365783, A365792.

nn = 270;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

a(n) = A008479(A126706(n)).

a(n) > 1 for all n.

A365793 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A286708(n).

Original entry on oeis.org

5, 8, 6, 10, 11, 6, 8, 14, 5, 15, 16, 8, 11, 18, 5, 7, 12, 20, 21, 8, 7, 11, 14, 23, 18, 9, 24, 15, 6, 9, 25, 8, 5, 26, 8, 9, 13, 8, 6, 14, 18, 29, 19, 26, 11, 30, 19, 12, 8, 31, 10, 20, 32, 6, 32, 11, 16, 10, 33, 5, 10, 17, 22, 6, 8, 8, 13, 35, 28, 36, 8, 14

Offset: 1

Michael De Vlieger, Sep 22 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).

			a(1) = 5 since rad(b(1)) = rad(36) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 36 is the 5th term.
a(2) = 8 since rad(b(2)) = rad(72) = 6, and 72 is the 8th term in k*{R(6)}.
a(3) = 6 since rad(b(3)) = rad(100) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, 100, ...}, 100 is the 6th term, etc.

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

Cf. A007947, A286708, A365786, A365792.

nn = 4000;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[
  Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &],
  AllTrue[FactorInteger[#][[All, -1]], # > 1 &] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]]][[1]] &, Length[t]]

a(n) = A008479(A286708(n)).

a(n) > 1 for all n.

A371630 Numbers k that set records in A372720.

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 120, 210, 420, 840, 1260, 1680, 2520, 4620, 9240, 13860, 18480, 27720, 32760, 55440, 65520, 102960, 110880, 120120, 180180, 240240, 360360, 556920, 720720, 1081080, 1441440, 1884960, 2162160, 2827440, 2882880, 3063060, 3603600, 4084080, 6126120

Offset: 1

Michael De Vlieger, Jun 04 2024

In other words, numbers k that set records for d(k) - f(k), where d = A000005 and f = A008479.
Largest primorial in this sequence is A002110(4) = 210.
The primorials A002110(0..4) are the only squarefree numbers in this sequence.
{a(n)} \ A002110(0..4) is contained in A126706.
Not a subset of A060735; a(13) = 2520 is not in A060735. Though common for small n, the set of a(n) in A060735 is likely finite; the restriction is connected with the finite number of primorials in the sequence.
Not a subset of A025487 or A055932; a(19) = 32760 is the smallest term without a primorial kernel.
The prime shape of a(n) appears to feature exponents m of prime power factors p^m | a(n) that are nonincreasing as pi(p) increases.

			Table of a(n) and A371634(n) = b(n) for n = 1..20. Asterisks in the a(n) column denote squarefree terms while "+" denotes numbers not in A055932 (i.e., in A080259).
   n     a(n)  A067255(a(n))            d(n)-f(n) = b(n)
  ------------------------------------------------------
   1       1*  1                          1 -  1 =   0
   2       2*  2                          2 -  1 =   1
   3       6*  2 * 3                      4 -  1 =   3
   4      12   2^2 * 3                    6 -  2 =   4
   5      30*  2 * 3 * 5                  8 -  1 =   7
   6      60   2^2 * 3 * 5               12 -  2 =  10
   7     120   2^3 * 3 * 5               16 -  4 =  12
   8     210*  2 * 3 * 5 * 7             16 -  1 =  15
   9     420   2^2 * 3 * 5 * 7           24 -  2 =  22
  10     840   2^3 * 3 * 5 * 7           32 -  4 =  28
  11    1260   2^2 * 3^2 * 5 * 7         36 -  6 =  30
  12    1680   2^4 * 3 * 5 * 7           40 -  8 =  32
  13    2520   2^3 * 3^2 * 5 * 7         48 - 11 =  37
  14    4620   2^2 * 3 * 5 * 7 * 11      48 -  2 =  46
  15    9240   2^3 * 3 * 5 * 7 * 11      64 -  4 =  60
  16   13860   2^2 * 3^2 * 5 * 7 * 11    72 -  6 =  66
  17   18480   2^4 * 3 * 5 * 7 * 11      80 -  8 =  72
  18   27720   2^3 * 3^2 * 5 * 7 * 11    96 - 12 =  84
  19   32760+  2^3 * 3^2 * 5 * 7 * 13    96 - 11 =  85
  20   55440   2^4 * 3^2 * 5 * 7 * 11   120 - 20 = 100

Michael S. Branicky, Table of n, a(n) for n = 1..67 (terms 1..56 from Michael De Vlieger)
Michael De Vlieger, Prime power decomposition of A371630(n), n = 1..56.

Cf. A000005, A002110, A008479, A055932, A060735, A080259, A126706, A371634, A372720.

A371634 Records in A372720.

Original entry on oeis.org

0, 1, 3, 4, 7, 10, 12, 15, 22, 28, 30, 32, 37, 46, 60, 66, 72, 84, 85, 100, 101, 102, 111, 124, 138, 152, 180, 181, 219, 226, 252, 253, 271, 272, 277, 282, 291, 312, 372, 373, 458, 480, 481, 538, 539, 587, 588, 608, 644, 645, 681, 682, 683, 685, 687, 756, 759, 760

Offset: 1

Michael De Vlieger, Jun 04 2024

Michael S. Branicky, Table of n, a(n) for n = 1..67

Cf. A000005, A008479, A371630, A372720.

a(57) and beyond from Michael S. Branicky, Jun 14 2024

A372864 Numbers k such that A372720(k) = 0.

Original entry on oeis.org

1, 500, 578, 722, 750, 1058, 1500, 1682, 1922, 2646, 2744, 3430, 3645, 4800, 5202, 5346, 5476, 5488, 5625, 6318, 6400, 6724, 7168, 7396, 8000, 8836, 10092, 10976, 11236, 11532, 11979, 12005, 13068, 13924, 14450, 14884, 15309, 16810, 16875, 16896, 18050, 18225

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since f(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since f(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
Apart from terms in this sequence, all the rest of the terms k in R are such that g(k) is negative.
There are no 3-smooth numbers k > 1 in this sequence, however there are 3 terms {500, 6400, 8000} in A033846 (with s = rad(k) = 10). For s = 2*3*23, there are 6 terms {19044, 25392, 38088, 70656, 536544, 953856}.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 1 since tau(1) - f(1) = 1 - 1 = 0.
a(2) = 500 = 2^2 * 5*3, since tau(500) - f(500)
     = (2+1)*(3+1) - card({10,20,40,50,80,100,160,200,250,320,400,500})
     = 12 - 12 = 0.
a(3) = 578 = 2*17^2, since tau(578) - f(578)
     = (1+1)*(2+1) - card({34,68,136,272,544,578})
     = 6 - 6 = 0, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..2000

Cf. A000005, A001221, A007947, A008479, A010846, A372720, A372972.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 20000}], ?(# == 0 &)][[All, 1]]

A381498 a(n) = sum of numbers k <= n that have the same squarefree kernel as n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 36, 19, 30, 21, 22, 23, 60, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 96, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 144, 56, 120, 51, 78, 53, 198, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66

Offset: 1

Michael De Vlieger, Mar 03 2025

nonn

Analogous to A244974(n) = sum of row n of A162306; row n of A369609 is a proper subset of A162306.

			 n  a(n)  Factor(a(n))  Row n of A369609
----------------------------------------
 4    6   2 * 3         {2, 4}
 8   14   2 * 7         {2, 4, 8}
 9   12   2^2 * 3       {3, 9}
12   18   2 * 3^2       {6, 12}
16   30   2 * 3 * 5     {2, 4, 8, 16}
18   36   2^2 * 3^2     {6, 12, 18}
20   30   2 * 3 * 5     {10, 20}
24   60   2^2 * 3 * 5   {6, 12, 18, 24}
25   30   2 * 3 * 5     {5, 25}
27   39   3 * 13        {3, 9, 27}
28   42   2 * 3 * 7     {14, 28}
32   62   2 * 31        {2, 4, 8, 16, 32}
36   96   2^5 * 3       {6, 12, 18, 24, 36}

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing a(n) for prime n in red, squarefree composite n in green, proper prime powers n in gold, powerful n that are not prime powers in magenta, and other numbers in blue.

Cf. A007947, A008479, A244974, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Total@ Select[Range[n], rad[#] == r &], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]);
a(n) = my(r=rad(n)); sum(k=1, n, if(rad(k)==r, k)); \\ Michel Marcus, Mar 03 2025

a(n) = sum of row n of A369609.
For squarefree k, a(k) = k.
For prime power p^m, a(p^m) = Sum_{i=1..m} p^i.

A140661 Number of pairs (b,c) with the same prime factors, 1<=b<=c<=n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 24, 25, 28, 29, 31, 32, 33, 34, 38, 40, 41, 44, 46, 47, 48, 49, 54, 55, 56, 57, 62, 63, 64, 65, 68, 69, 70, 71, 73, 75, 76, 77, 83, 85, 89, 90, 92, 93, 100, 101, 104, 105, 106, 107, 109, 110, 111, 113, 119, 120, 121, 122

Offset: 1

R. J. Mathar, Jul 11 2008

If pairs are restricted to b

			a(16)=24 counts the 16 pairs (b,b) with 1<=b<=16 plus the 8 pairs (2,4), (2,8), (2,16), (4,8), (4,16), (8,16), (3,9), (6,12).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Erdos and T. Motzkin, Density of pairs with same prime factors, Am. Math. Month. vol 97 no 10 (1990) p 937, problem 5735.

Partial sums of A008479.

samepf(m,n)=my(g=gcd(m,n),t=g); m/=g; while((t=gcd(t,m))>1, m/=t); if(m!=1, return(0)); t=g; while((t=gcd(t,n))>1, n/=t); n==1
a(n)=sum(b=1,n, sum(c=b,n, samepf(b,c))) \\ Charles R Greathouse IV, Jan 09 2018

A357241 a(n) is the number of j in the range 1 <= j <= n such that j / rad(j) = n / rad(n).

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 6, 1, 1, 7, 8, 2, 9, 10, 11, 1, 12, 2, 13, 3, 14, 15, 16, 2, 1, 17, 1, 4, 18, 19, 20, 1, 21, 22, 23, 1, 24, 25, 26, 3, 27, 28, 29, 5, 3, 30, 31, 2, 1, 2, 32, 6, 33, 2, 34, 4, 35, 36, 37, 7, 38, 39, 4, 1, 40, 41, 42, 8, 43, 44, 45, 1, 46, 47, 3, 9, 48, 49, 50, 3

Offset: 1

Ilya Gutkovskiy, Sep 19 2022

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A001694 (positions of 1's), A003557, A008479.

Table[Length[Select[Range[n], #/Last[Select[Divisors[#], SquareFreeQ]] == n/Last[Select[Divisors[n], SquareFreeQ]] &]], {n, 1, 80}]

f(n) = n/factorback(factor(n)[, 1]); \\ A003557
a(n) = my(x=f(n)); sum(j=1, n, f(j) == x); \\ Michel Marcus, Sep 20 2022

a(n) = |{j <= n : A003557(j) = A003557(n)}|.

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cp's OEIS Frontend

A372972 Numbers k such that A372720(k) is negative.

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A373738 a(1) = 1, a(n) = floor((1/omega(n)!) * Product_{p | n} 1 + (log n)/(log p)), where omega = A001221.

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A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

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A365793 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A286708(n).

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A371630 Numbers k that set records in A372720.

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A371634 Records in A372720.

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A372864 Numbers k such that A372720(k) = 0.

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A381498 a(n) = sum of numbers k <= n that have the same squarefree kernel as n.

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