A372720 -id:A372720 - 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]]

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]]

A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

Original entry on oeis.org

162, 250, 686, 1875, 7203, 2662, 4394, 750, 3993, 578, 12005, 722, 6591, 2058, 1058, 14739, 73205, 20577, 1682, 1922, 142805, 5346, 36501, 3430, 2738, 102487, 6318, 3362, 417605, 3698, 73167, 199927, 89373, 4418, 651605, 5202, 25725, 5618, 13310, 151959, 6498

Offset: 1

Michael De Vlieger, Jun 24 2024

Numbers k whose position i in S(n) is such that tau(k) <= i, i.e., that A372720(k) is not positive.
For k = p^m, m > 0, in S(p), p prime, tau(p^m) > A008479(p^m) since tau(p^m) = m + 1 and A008479(p^m) = m. Therefore we consider only composite squarefree q in this sequence.
a(n) is in A126706.
Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.

			a(1) = 162 since the 12th term in S(6) = A033845 = {6, 12, 18, 24, 36, 48, 54, ..., 162, ...} is the smallest k = S(6, i) such that tau(S(6, i)) <= i: tau(162) = 10 while i = 12.
a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.
a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.
Table of first and some notable terms:
       n        q     i         a(n) a(n)/q  A372720(a(n))
  --------------------------------------------------------
       1        6    12         162   3^3         -2
       2       10     9         250   5^2         -1
       3       14    10         686   7^2         -2
       4       15    11        1875   5^3         -1
       5       21    13        7203   7^3         -3
       6       22    12        2662   11^2        -4
       7       26    13        4394   13^2        -5
       8       30    16         750   5^2          0
      82      210    51       26250   5^3        -11
    1061     2310    99      635250   5^2 * 11    -3
   15013    30030   222    25375350   5 * 13^2   -30
  268015   510510   338   679488810   11^3       -18

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..268015 in blue, and A120944(n)^3 in red.
Michael De Vlieger, Diagram of S(6) = A033845 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function relates to A372720(k), with positive values from largest in light yellow grading to A372720(k) = 1 in orange, and negative values with smallest absolute value in dark blue to greatest in light blue. a(1) = 162 appears at right.
Michael De Vlieger, Diagram of S(30) = A143207 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function is as above, but with A372720(k) = 0 in red. a(2) = 750 appears at left in red.

Cf. A000005, A008479, A120944, A126706, A372720.

(* First, load function f from A162306 *)
Table[k = 1; s = f[n, n^3]; While[DivisorSigma[0, n*s[[k]]] - k > 0, k++]; s[[k]], {n, Select[Range[6, 120], And[SquareFreeQ[#], CompositeQ[#]] &]}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A371630 Numbers k that set records in A372720.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A371634 Records in A372720.

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica