A369609 -id:A369609 - cp's OEIS frontend

A372720 a(n) = A000005(n) - A008479(n).

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

Offset: 1

Michael De Vlieger, May 13 2024

sign

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

			Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
  n  b(n) c(n) a(n)
 ------------------
  1    1    1    0
  2    2    1    1
  3    2    1    1
  4    3    2    1
  5    2    1    1
  6    4    1    3
  7    2    1    1
  8    4    3    1
  9    3    2    1
 10    4    1    3
 11    2    1    1
 12    6    2    4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

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

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.
a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).
a(n) = A183093(n) - A355432(n).

A381094 Triangle read by rows where row n contains k < n that are neither coprime to n nor have the same squarefree kernel as n, or 0 if there are no such k.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 4, 0, 6, 6, 2, 4, 5, 6, 8, 0, 2, 3, 4, 8, 9, 10, 0, 2, 4, 6, 7, 8, 10, 12, 3, 5, 6, 9, 10, 12, 6, 10, 12, 14, 0, 2, 3, 4, 8, 9, 10, 14, 15, 16, 0, 2, 4, 5, 6, 8, 12, 14, 15, 16, 18, 3, 6, 7, 9, 12, 14, 15, 18, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 20

Offset: 1

Michael De Vlieger, Feb 14 2025

Let rad(k) = A007947(k), the squarefree kernel of k.
Let T(n) be row n of this sequence and let S(n) be row n of A133995.
T(n) contains numbers k < n such that k and n share at least one prime factor p, but not all distinct prime p | n.
T(n) is a superset of S(n), since S(n) does not contain any divisor d | n, while T(n) allows d | n such that rad(d) != rad(n).

			Table begins:
   n   row n
  ---------------------------
   1:  0;
   2:  0;
   3:  0;
   4:  0;
   5:  0;
   6:  2, 3, 4;
   7:  0;
   8:  6;
   9:  6;
  10:  2, 4, 5, 6, 8;
  11:  0;
  12:  2, 3, 4, 8, 9, 10;
  13:  0;
  14:  2, 4, 6, 7, 8, 10, 12;
  15:  3, 5, 6, 9, 10, 12;
  16:  6, 10, 12, 14;
From _Michael De Vlieger_, Mar 03 2025: (Start)
Row 10 is the union of {2, 4, 6, 8, 10} and {5, 10} without 10.
Row 12 is the union of {2, 4, 6, 8, 10, 12} and {3, 6, 9, 12} without {6, 12}.
Row 30 is the union of {2, 4, ..., 30}, {3, 6, ..., 30}, and {5, 10, ..., 30} without 30.
Row 84 is the union of {2, 4, ..., 84}, {3, 6, ..., 84}, and {7, 14, ..., 84} without {42, 84}, etc. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11873 (rows n = 1..250, flattened)

Cf. A007947, A121998, A133995, A369609, A381096.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

T(n) = { k < n : 1 < gcd(k,n), rad(k) != rad(n) }.
T(n) = S(n) \ { k : k | n, rad(k) = rad(n) }.
For prime p, T(p) = {}, but we write 0 to signify the empty set.
T(4) = 0, since k < 4 is either coprime to 4 or rad(k) = 2.
Let U(n) be row n of A121998 and let R(n) be row n of A369609. T(n) = U(n) \ R(n). - Michael De Vlieger, Mar 03 2025

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

Original entry on oeis.org

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]]

A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 9, 0, 6, 6, 25, 0, 36, 0, 49, 45, 42, 0, 81, 0, 100, 84, 121, 0, 144, 45, 169, 96, 196, 0, 315, 0, 210, 198, 289, 175, 354, 0, 361, 273, 430, 0, 609, 0, 484, 435, 529, 0, 648, 140, 655, 459, 676, 0, 801, 385, 826, 570, 841, 0, 1260, 0, 961, 798

Offset: 1

Michael De Vlieger, Mar 02 2025

nonn

Analogous to A066760(n), the sum of row n of A133995, and A381499(n), sum of row n of A272619.

			Table of n and a(n) for select n, showing prime power decomposition of both and row n of A381094:
   n   Factor(n) a(n)  Factor(a(n))  Row n of A381094
  -------------------------------------------------------------------
   6   2 * 3       9   3^2           {2,3,4}
   8   2^3         6   2 * 3         {6}
   9   3^2         6   2 * 3         {6}
  10   2 * 5      25   5^2           {2,4,5,6,8}
  12   2^2 * 3    36   2^2 * 3^2     {2,3,4,8,9,10}
  14   2 * 7      49   7^2           {2,4,6,7,8,10,12}
  15   3 * 5      45   3^2 * 5       {3,5,6,9,10,12}
  16   2^4        42   2 * 3 * 7     {6,10,12,14}
  18   2 * 3^2    81   3^4           {2,3,4,8,9,10,14,15,16}
  20   2^2 * 5   100   2^2 * 5^2     {2,4,5,6,8,12,14,15,16,18}
  21   3 * 7      84   2^2 * 3 * 7   {3,6,7,9,12,14,15,18}
  22   2 * 11    121   11^2          {2,4,6,8,10,11,12,14,16,18,20}
  24   2^3 * 3   144   2^4 * 3^2     {2,3,4,8,9,10,14,15,16,20,21,22}
a(6) = (2+4) + (3) = 9,
a(n) = 6 for n in {8, 9} since 6 is the only number less than n that shares a factor with n but does not have the same squarefree kernel as n.
a(10) = (2+4+6+8) + (5) = 25.
a(12) = (2+4+8+10) + (3+9) = 36.
a(14) = (2+4+6+8+10+12) + (7) = 49.
a(15) = (3+6+9+12) + (5+10) = 45.
a(16) = (6+10+12+14) = 42, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 6..2^14, ignoring a(n) = 0, showing a(n) for prime power n in gold, a(n) for squarefree n in green, otherwise blue.

Cf. A007947, A038566, A066760, A067392, A121998, A369609, A381094, A381096, A381498, A381499.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Total@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

a(n) is the sum of row n of A381094.
a(n) = 0 for prime n and n = 4.
a(n) = A067392(n) - A381498(n).

A376505 Number of m <= n such that rad(m) | n that are neither squarefree nor prime powers, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 5, 0, 1, 1, 0, 0, 5, 0, 3, 0, 1, 0, 6, 0, 2, 0, 0, 0, 11, 0, 0, 1, 0, 0, 7, 0, 1, 0, 5, 0, 7, 0, 0, 2, 1, 0, 8, 0, 4, 0, 0, 0, 11, 0, 0, 0

Offset: 1

Michael De Vlieger, Sep 28 2024

			a(2) = a(4) = a(p^k) = 0 since numbers m <= p^k such that rad(m) | p^k are all divisors that are prime powers p^j, j = 0..k.
a(k) = 0 for k < 12 since 12 is the smallest number that is neither squarefree nor prime powers.
a(12) = 1 since m = 12 is such that 12 <= 12 and rad(12) | 12.
a(18) = 2 since both k = 12 and k = 18 are such that rad(k) | 18.
a(30) = 4 since row 30 of A162306 has 4 numbers that are neither squarefree nor prime powers: {1, 2, 3, 4, 5, 6, 8, 9, 10, [12], 15, 16, [18], [20], [24], 25, 27, 30}, indicated by brackets. (The bracketed numbers happen to be the first 4 terms of A126706.)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagram of row 1440 of A162306 showing 4 squarefree composites in green, 3 primes in red, the empty product in gray, 17 perfect powers of primes in yellow, and 72 numbers that are neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers.

Cf. A000005, A000961, A001221, A010846, A126706, A162306, A376504, A361373 (intersection of A246655 and row n of A162306), A376504 (intersection of A120944 and row n of A162306).

(* Load "theta" program from this A369609/a369609.txt">link in A369609 *)
{0}~Join~Table[theta[n] - Total@ Map[Floor@ Log[#, n] &, #1] - 2^#2 + #2 & @@ {#, Length[#]} &@ FactorInteger[n][[All, 1]], {n, 2, 120}]

rad(n) = factorback(factorint(n)[, 1]);
a(n) = sum(m=1, n, if (!issquarefree(m) && !isprimepower(m), ((n % rad(m))==0))); \\ Michel Marcus, Sep 29 2024

a(n) = A010846(n) - (Sum_{p|n} floor(log n / log p)) - 2^omega(n) + omega(n), where omega = A001221.
a(n) = A010846(n) - A361373(n) - A376504(n) + 1.
a(n) = 0 for n = p^k, where p is prime and k >= 0, i.e., n in A000961.
Intersection of A126706 and row n of A162306.

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.

A382926 Irregular table where row n lists numbers k in row n of A162306 for which there exists a prime p | n such that k*p > n.

Original entry on oeis.org

2, 3, 4, 5, 3, 4, 6, 7, 8, 9, 4, 5, 8, 10, 11, 6, 8, 9, 12, 13, 4, 7, 8, 14, 5, 9, 15, 16, 17, 8, 9, 12, 16, 18, 19, 5, 8, 10, 16, 20, 7, 9, 21, 4, 8, 11, 16, 22, 23, 9, 12, 16, 18, 24, 25, 4, 8, 13, 16, 26, 27, 7, 8, 14, 16, 28, 29, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30

Offset: 2

Michael De Vlieger, Apr 28 2025

The number n appears in each row. For n in A024619, for all p|n, p^floor(log_p n) is in row n. Thus, the number of terms in row n for n in A024619 is at least 1+omega(n), where omega = A001221 is the number of distinct prime factors of n.

			Let s(n) = A382964(n).
Table of select rows:
 n  s(n)    row n of this sequence
--------------------------------------------------------
 6    3     3,  4,  6;
10    4     4,  5,  8, 10;
12    4     6,  8,  9, 12;
14    4     4,  7,  8, 14;
15    3     5,  9, 15;
18    5     8,  9, 12, 16, 18;
20    5     5,  8, 10, 16, 20;
21    3     7,  9, 21;
22    5     4,  8, 11, 16, 22;
24    5     9, 12, 16, 18, 24;
26    5     4,  8, 13, 16, 26;
28    5     7,  8, 14, 16, 28;
30   12     8,  9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30.
In the examples below, we place terms in row n in brackets [] among other terms in row n of A162306, presented in order of row n of A275280.
Row p^m for m > 0 and prime p is {p^m}, since multiplying p^m by p exceeds p^m.
Row 10 = {4, 5, 8, 10}, since numbers k such that rad(k) | 10 contains these numbers, furthermore, we have the following: 2 or 5 times 8 exceeds 10, 5*4 > 10, 2 or 5 times 10 exceeds 10, and 5*5 > 10.
      1    2   [4]  [8]
     [5] [10]
Row 24 = {9, 12, 16, 18, 24}, since numbers k such that rad(k) | 24 contains these numbers, furthermore, we have the following: 2 or 3 times 16 exceeds 24, 2 or 3 times 24 exceeds 24, 3*12 > 24, 2 or 3 times 18 exceeds 24, and 3*9 > 24.
      1    2    4    8  [16]
      3    6  [12] [24]
     [9] [18]

Michael De Vlieger, Table of n, a(n) for n = 2..11938 (rows n = 2..1000, flattened)

Cf. A000961, A007947, A024619, A162306, A275280, A382964 (row lengths).

(* First, run the "regs" function from A369609, then: *)
Table[Select[regs[n], Function[k, AnyTrue[FactorInteger[n][[All, 1]], #*k > n &]]], {n, 2, 30}] // Flatten

For n in A000961, row n is {n}.

A382964 Number of k <= n such that rad(k) divides n and g * k > n where g is the largest prime factor of n and rad = A007947.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 4, 3, 1, 1, 5, 1, 5, 3, 5, 1, 5, 1, 5, 1, 5, 1, 12, 1, 1, 4, 6, 3, 6, 1, 6, 4, 6, 1, 14, 1, 6, 4, 6, 1, 6, 1, 6, 4, 6, 1, 6, 3, 6, 4, 6, 1, 16, 1, 6, 4, 1, 3, 17, 1, 7, 4, 13, 1, 7, 1, 7, 4, 7, 3, 18, 1, 7, 1, 7, 1, 19, 3

Offset: 1

Michael De Vlieger, Apr 28 2025

nonn

			Table of select n, a(n), and numbers k in row n of A382926:
 n  a(n)    row n of A382926
--------------------------------------------------------
 6    3     3, 4, 6;
10    4     4, 5, 8, 10;
12    4     6, 8, 9, 12;
14    4     4, 7, 8, 14;
15    3     5, 9, 15;
18    5     8, 9, 12, 16, 18;
20    5     5, 8, 10, 16, 20;
21    3     7, 9, 21;
22    5     4, 8, 11, 16, 22;
24    5     9, 12, 16, 18, 24;
26    5     4, 8, 13, 16, 26;
28    5     7, 8, 14, 16, 28;
30   12     8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30.

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

Cf. A007947, A010846, A024619, A162306, A246655, A382926.

(* First, run the "regs" function from A369609, then: *)
Table[Count[regs[n], _?(Function[k, AllTrue[FactorInteger[n][[All, 1]], #*k > n &]])], {n, 105}]

a(n) = 1 for n in A246655.

1+A001221(n) <= a(n) < A010846(n) for n in A024619.

A381112 a(1) = 1, let q = greatest prime in S(n) = {p; p = A053669(a(i)); 1 <= i <= n-1}. Then for n > 1, a(n) is the smallest number not yet in the sequence such that: (i) q|a(n), and (ii) p a prime and p^k|a(n) implies p in S(n) and k <= cardinality of p in S(n).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 20, 30, 7, 14, 21, 28, 35, 42, 56, 63, 70, 84, 105, 112, 126, 140, 168, 175, 189, 210, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 132, 154, 165, 176, 198, 220, 231, 264, 275, 297, 308, 330, 352, 385, 396, 440, 462, 495, 528, 539, 550, 594

Offset: 1

David James Sycamore, Feb 14 2025

nonn

Sequence contains all primes (in order), but no perfect prime powers. All primorial numbers are terms but no multiples m*A002110(k) are terms, where m = 2,3,4,...,prime(k+1)-1. Each prime p results in a trajectory (see graph) of p-smooth numbers corresponding to rows of a table; see Example.

Let P(i) = A002110(i). For n > 178 and a(n) = P(i)/prime(i-1), a(n+1) = prime(i-1)^2 * prime(i). See tables in Example. - Michael De Vlieger, Mar 06 2025

			The smallest nondivisor prime of a(1) = 1 is 2, so S(2) = {2} and so a(2) = 2. Likewise the smallest nondivisor prime of a(2) = 2 is 3, so S(3) = {2,3} and a(3) = 3. At this point we have S(4) = {2,3,2} and the next term must have greatest prime factor = q = 3 and other prime factors must be in S(4). Therefore a(4) = 2*3 = 6.
Sequence expressed in tabular form, rows (after first) starting with a prime and ending with a primorial:
  1;
  2;
  3,6;
  5,10,15,20,30;
  7,14,21,28,35,42,56,63,70,84,105,112,126,140,168,175,189,210;
  11,22,33,44,55,66,...
Row (n+1) commencing p = prime(n) contains all p-smooth numbers m; p <=m <= A002110(n) except for any perfect powers p^k, (k>1) of p which lie in that range.

Michael De Vlieger, Table of n, a(n) for n = 1..6382
Michael De Vlieger, Log log scatterplot of a(n), n = 1..6382.
Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..4096, 24X vertical exaggeration, with a color function representing m = 1 in black, m = 2 in red, ..., largest m in the dataset in magenta. The colored bar below the image is red if a(n) is prime, gold if a(n) is a proper prime power, green for squarefree composite a(n), and blue or purple if a(n) is neither squarefree nor prime power, where purple additionally represents powerful a(n) that are not prime powers.
Michael De Vlieger, Notes on the mechanics of this sequence, regarding the prime decomposition of a(n) and the cardinalities of primes p in the sequence S(1..n), where S(n) = A053669(a(n)).

Cf. A000040, A002110, A006530, A053669.

(* First, load the program regs from the "fast Mathematica programs" in the links at A369609, then: *)
nn = 8; p = q = 1; c[] := False; m[] := 0;
f[x_] := f[x] = Block[{qx = 2}, While[Divisible[x, qx], qx = NextPrime[qx]]; qx];
{1}~Join~Reap[Do[
  Map[If[! c[#],
    If[PrimeQ[#],
      Sow[#]; c[#] = True; m[f[#]]++,
        If[And[AllTrue[FactorInteger[#], #2 <= m[#1] & @@ # &],  Divisible[#, p]],
          Sow[#]; c[#] = True; m[f[#]]++] ] ] &, regs[q, q]];
  q *= Prime[i]; p = Prime[i]; u = 1, {i, nn}] ][[-1, 1]] (* Michael De Vlieger, Feb 16 2025 *)

A383180 Irregular table T(n,k) = A010846(A019565(2^n + k)).

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 5, 18, 2, 6, 5, 19, 5, 20, 16, 68, 2, 7, 6, 22, 5, 21, 18, 77, 5, 22, 17, 79, 16, 74, 60, 283, 2, 7, 6, 23, 5, 23, 18, 80, 5, 22, 18, 82, 16, 78, 62, 295, 5, 24, 19, 87, 16, 82, 64, 315, 15, 80, 62, 316, 55, 290, 226, 1161

Offset: 0

Michael De Vlieger, May 09 2025

			Triangle begins:
  0: 1;
  1: 2;
  2: 2, 5;
  3: 2, 6, 5, 18;
  4: 2, 6, 5, 19, 5, 20, 16, 68;
  5: 2, 7, 6, 22, 5, 21, 18, 77, 5, 22, 17, 79, 16, 74, 60, 283;
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows n = 0..14, flattened)
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^14.
Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..14, with a color function showing smaller numbers in reds and oranges, larger numbers in blues and magentas.

Cf. A005117, A010846, A019565, A363061.

(* Load the "theta" program at the Mathematica link in A369609, then: *)
f[x_] := Times @@ Prime@ Position[Reverse@ IntegerDigits[x, 2], 1][[All, 1]]; Table[theta[f[2^n + k] ], {n, 0, 7}, {k, 0, 2^n - 1}]

T(0,0) = 1.
T(n,0) = 2.
T(n,2^(n-1)-1) = A363061(n).

cp's OEIS Frontend

A372720 a(n) = A000005(n) - A008479(n).

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A381094 Triangle read by rows where row n contains k < n that are neither coprime to n nor have the same squarefree kernel as n, or 0 if there are no such k.

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

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A381497 a(n) = sum of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n), where rad = A007947.

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A376505 Number of m <= n such that rad(m) | n that are neither squarefree nor prime powers, where rad = A007947.

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

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A382926 Irregular table where row n lists numbers k in row n of A162306 for which there exists a prime p | n such that k*p > n.

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A382964 Number of k <= n such that rad(k) divides n and g * k > n where g is the largest prime factor of n and rad = A007947.