A361373 -id:A361373 - cp's OEIS frontend

A380870 a(n) = A381798(n) - A361373(n) - 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 2, 0, 1, 4, 6, 0, 0, 0, 8, 0, 1, 0, 1, 0, 0, 3, 3, 7, 2, 0, 13, 0, 1, 0, 4, 0, 7, 6, 6, 0, 1, 0, 15, 14, 8, 0, 13, 3, 0, 15, 23, 0, 1, 0, 0, 5, 0, 5, 7, 0, 3, 9, 12, 0, 2, 0, 30, 18, 14, 10, 6, 0, 3, 0, 14, 0

Offset: 1

Michael De Vlieger, Apr 08 2025

nonn

a(n) = cardinality of the intersection of A024619 and row n of A381799.
Let S(n,p) = {p^m : p | n, m = 1..floor(log_p n)}. Therefore S(10,2) = {1,2,4,8} and S(30,3) = {1,3,9,27}. Then U({S(n,p) : p|n}) = row n of A377485.
Let T(n,p) = {p^m (mod n) : p | n} the set of prime divisor power residues r (mod n) == p^m, p | n. Thus T(10,2) = {1,2,4,8,6} and T(30,3) = {1,3,9,27,21}. Then U({T(n,p) : p|n}) = row n of A381799.

			Table of n, a(n), and H(n) = intersection of row n of A381799 with A024619.
 n   facs(n)   a(n)  H(n)
--------------------------------------------
 6   2 * 3       0   -
10   2 * 5       1   {6}
12   2^2 * 3     0   -
14   2 * 7       0   -
15   3 * 5       3   {6, 10, 12}
18   2 * 3^2     2   {10, 14}
20   2^2 * 5     1   {12}
21   3 * 7       4   {6, 12, 15, 18}
22   2 * 11      6   {6, 10, 12, 14, 18, 20}
24   2^3 * 3     0   -
30   2 * 3 * 5   1   {21}
.
a(6) = 0 since Q(6) = R(6) = {1,2,3,4}, i.e., all terms in row 6 of A381799 are powers of primes.
a(10) = 1 since Q(10) = {1,2,4,5,8} but R(10) = {1,2,4,5,6,8}; the latter set contains 1 term (i.e., 6) that is not a member of the former set.
a(14) = 0 since R(14) = {1,2,4,7,8} are all powers of primes.
a(15) = 3 since R(15) = {1,3,5,6,9,10,12} has 3 terms {6,10,12} that are not powers of primes.
a(18) = 2 since R(18) = {1,2,3,4,8,9,10,14,16} has 2 terms {6,10} that are not powers of primes, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Cf. A000961, A024619, A361373, A377485, A381750, A381798, A381799.

f[x_, p_] := Block[{m = 2, r, c},
  Which[
    PrimePowerQ[x],
    Join[{0}, #1^Range[0, #2 - 1]] & @@ FactorInteger[x][[1]],
    PowerMod[p, m, x] == p, {1, p},
    True, c[_] := False;
    c[1] = c[p] = True; {1, p}~Join~
      Reap[While[r = PowerMod[p, m, x]; ! c[r], Sow[r];
        c[r] = True; m++] ][[-1, 1]] ] ]
Table[Count[Union@ Flatten@ Map[f[n, #] &, FactorInteger[n][[All, 1]] ], _?(And[# > 1, ! PrimePowerQ[#]] &)], {n, 120}]

Let Q(n) = {1} joined to row n > 1 of A377485 and let R(n) = row n of A381799.
a(n) = card(U(Q(n) \ R(n))).
a(p^m) = 0 for prime power p^m, m >= 0.
a(n) = 0 for n in A381750.

A384442 Smallest k such that A361373(k) = n.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 40, 36, 30, 60, 102, 84, 132, 150, 264, 210, 540, 330, 420, 660, 630, 840, 1050, 2100, 2340, 2520, 3150, 2310, 2730, 4290, 4620, 6930, 9240, 15960, 16170, 17850, 18480, 20790, 34650, 62370, 68250, 30030, 62790, 60060, 78540, 90090, 117810

Offset: 0

Michael De Vlieger, Jun 12 2025

nonn

For n > 1, a(n) is composite, since A361373(p) = 1 for prime p.

For n = 0..2, a(n) = 2^n. For n > 2, a(n) is in A024619.

			Table of n, a(n) for n = 1..12, showing row a(n) of A377485.
          log n/log p
 n  a(n)  p_1 p_2 p_3  row n of A377485
-------------------------------------------------------------------------
 1:   2   1            {p}
 2:   4   2            {p, p^2}
 3:   6   2   1        {p, q, p^2}
 4:  10   3   1        {p, p^2, q, p^3}
 5:  12   3   2        {p, q, p^2, p^3, q^2}
 6:  18   4   2        {p, q, p^2, p^3, q^2, p^4}
 7:  40   5   2        {p, p^2, q, p^3, p^4, q^2, p^5}
 8:  36   5   3        {p, q, p^2, p^3, q^2, p^4, q^3, p^5}
 9:  30   4   3   2    {p, q, p^2, r, p^3, q^2, p^4, r^2, q^3}
10:  60   5   3   2    {p, q, p^2, r, p^3, q^2, p^4, r^2, q^3, p^5}
11: 102   6   4   1    {p, q, p^2, p^3, q^2, p^4, r, q^3, p^5, p^6, q^4}
12:  84   6   4   2    {p, q, p^2, r, p^3, q^2, p^4, q^3, p^5, r^2, p^6, q^4}

Cf. A361373, A377845.

nn = 30030; t[_] := 0; u = 1; Do[(If[t[#] == 0, t[#] = n]; If[# == u, While[t[u] != 0, u++]]) &[Total@ Map[Floor@ Log[#, n] &, FactorInteger[n][[All, 1]] ] ], {n, 2, nn}]; {1}~Join~Array[t, u - 1]

A377485 Irregular triangle where row n lists powers p^k of primes p | n such that p^k <= n and k > 0.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 2, 3, 4, 7, 2, 4, 8, 3, 9, 2, 4, 5, 8, 11, 2, 3, 4, 8, 9, 13, 2, 4, 7, 8, 3, 5, 9, 2, 4, 8, 16, 17, 2, 3, 4, 8, 9, 16, 19, 2, 4, 5, 8, 16, 3, 7, 9, 2, 4, 8, 11, 16, 23, 2, 3, 4, 8, 9, 16, 5, 25, 2, 4, 8, 13, 16, 3, 9, 27, 2, 4, 7, 8, 16, 29

Offset: 1

Michael De Vlieger, Oct 29 2024

Row 1 is {1} by convention, since 1 is the empty product.

			Table of the first 12 rows:
   n   row n
  -------------------
   1:  1;
   2:  2;
   3:  3:
   4:  2, 4;
   5:  5;
   6:  2, 3, 4;
   7:  7;
   8:  2, 4, 8;
   9:  3, 9;
  10:  2, 4, 5, 8;
  11: 11;
  12:  2, 3, 4, 8, 9;

Michael De Vlieger, Table of n, a(n) for n = 1..12747 (rows n = 1..1500, flattened.)

Cf. A162306, A246655.

{{1}}~Join~Table[Union[Join @@ Map[#^Range[Floor@ Log[#, n]] &, FactorInteger[n][[All, 1]] ] ], {n, 2, 30}]

Row n is { p^k : p | n, k = 1..floor(log n/log p) }, i.e., intersection of A246655 and row n of A162306.
Row p = {p} for prime p.
Row p^k = { p^j : j = 1..k } for prime p and k > 0.
A361373(n) = length of row n for n > 1.

A376504 Number of divisors of n that are both composite and squarefree.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 4, 0, 1, 1, 0, 1, 4, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, 0, 1, 0, 4, 1, 1, 1

Offset: 1

Michael De Vlieger, Sep 25 2024

Also number of composite and squarefree m <= n such that rad(m) | n, i.e., in row n of A162306, where rad = A007947.
This sequence is distinct from A327517; A327517(210) != a(210).
Record setters are primorials, a(6) = 1, a(30) = 4, a(210) = 11, etc., since primorials P(n) = A002110(n) are the smallest instance of omega(n) = A001221(n).

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, A000295, A000961, A001221, A002110, A007947, A034444, A120944, A162306, A327517, A361373 (number of prime powers in row n of A162306), A374514 (number of divisors of n that are neither squarefree nor prime powers).

Mathematica
```
Array[2^# - # - 1 &@ PrimeNu[#] &, 120]
```

a(n) = 2^omega(n) - omega(n) - 1 = A034444(n) - A001221(n) - 1.
a(n) = 0 for n = p^m, where p is prime and m >= 0, i.e., n in A000961.
a(n) = A000295(omega(n)) = A000295(A001221(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.

A384936 a(n) = Sum_{k=1..n} floor( log(A002110(n)) / log(prime(k)) ).

Original entry on oeis.org

0, 1, 3, 9, 16, 28, 42, 57, 76, 97, 121, 148, 177, 208, 242, 279, 316, 359, 401, 446, 493, 545, 596, 651, 708, 767, 829, 893, 958, 1026, 1096, 1170, 1246, 1319, 1400, 1484, 1567, 1657, 1742, 1834, 1923, 2021, 2119, 2218, 2316, 2419, 2526, 2635, 2745, 2857, 2972

Offset: 0

Michael De Vlieger, Jun 12 2025

A384442(a(n)) = A002110(n) for n <= 8; does it hold for n > 5?

			Table of n, a(n) for n = 0..10, listing terms in row n of A287010:
      Terms in row n of A287010 corresponding
      to the primes listed in the header
 n\k   2   3   5   7  11  13  17  19  23  29   a(n)
---------------------------------------------------
 0:    0   .   .   .   .   .   .   .   .   .     0
 1:    1   .   .   .   .   .   .   .   .   .     1
 2:    2   1   .   .   .   .   .   .   .   .     3
 3:    4   3   2   .   .   .   .   .   .   .     9
 4:    7   4   3   2   .   .   .   .   .   .    16
 5:   11   7   4   3   3   .   .   .   .   .    28
 6:   14   9   6   5   4   4   .   .   .   .    42
 7:   18  11   8   6   5   5   4   .   .   .    57
 8:   23  14   9   8   6   6   5   5   .   .    76
 9:   27  17  11   9   8   7   6   6   6   .    97
10:   32  20  14  11   9   8   7   7   7   6   121

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A002110, A287010, A361373, A377485, A384442.

P = 2; s = {2}; {0}~Join~Reap[Do[Sow@ Total@ Map[Floor@ Log[#, P] &, s]; (AppendTo[s, #]; P *= #) &[Prime[k]], {k, 2, 51}] ][[-1, 1]]

a(n) = my(v=primes(n), pp=vecprod(v)); sum(k=1, n, log(pp)\log(v[k])); \\ Michel Marcus, Jun 14 2025

a(n) = A361373(A002110(n)).

Row sums of A287010.

cp's OEIS Frontend

A380870 a(n) = A381798(n) - A361373(n) - 1.

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A384442 Smallest k such that A361373(k) = n.

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A377485 Irregular triangle where row n lists powers p^k of primes p | n such that p^k <= n and k > 0.

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A376504 Number of divisors of n that are both composite and squarefree.

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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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A384936 a(n) = Sum_{k=1..n} floor( log(A002110(n)) / log(prime(k)) ).

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