A183093 -id:A183093 - cp's OEIS frontend

A183096 a(n) = number of divisors of n that are not perfect powers.

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 7, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183093: a(72) = 7, A183093(72) = 6.

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001597, A091050, A183093, A183095.

Cf. A001620, A072102.

ppQ[n_] := GCD @@ FactorInteger[n][[;;, 2]] > 1; ppQ[1] = True; a[n_] := DivisorSum[n, 1 &, !ppQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

A091050(n) = (1+ sumdiv(n, d, ispower(d)>1)); \\ This function from Michel Marcus, Sep 21 2014
A183096(n) = (numdiv(n) - A091050(n)); \\ Antti Karttunen, Nov 23 2017

a(n) = A000005(n) - A091050(n).
a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - A072102 - 2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 29 2025

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

Original entry on oeis.org

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).

A183095 a(n) = number of divisors d of n that are either 1 or of the form Product_(i) (p_i^e_i) where e_i = 1 for at least one i.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

a(n) = number of non-powerful divisors d of n where powerful numbers are numbers of the form A001694(m) for m >= 1.

			For n = 12, set of such divisors is {1, 2, 3, 6, 12}; a(12) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001694, A183093, A183094.

Cf. A001620, A082695.

nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1;a[n_] := DivisorSum[n, 1 &, nonpowQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

a(n) = sumdiv(n, d, d == 1 || vecmin(factor(d)[, 2]) == 1); \\ Amiram Eldar, Jan 30 2025

a(n) = A000005(n) - A183094(n) = A183093(n) + 1.
a(1) = 1, a(p) = 2, a(pq) = 4, a(pq...z) = 2^k, a(p^k) = 2, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(2)*zeta(3)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Name corrected by Amiram Eldar, Jan 30 2025

A380819 Triangle read by rows where row n lists "weak" divisors d | n (i.e., d in A052485) such that rad(d)^2 does not divide d, where rad = A007947.

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 6, 7, 2, 3, 2, 5, 10, 11, 2, 3, 6, 12, 13, 2, 7, 14, 3, 5, 15, 2, 17, 2, 3, 6, 18, 19, 2, 5, 10, 20, 3, 7, 21, 2, 11, 22, 23, 2, 3, 6, 12, 24, 5, 2, 13, 26, 3, 2, 7, 14, 28, 29, 2, 3, 5, 6, 10, 15, 30, 31, 2, 3, 11, 33, 2, 17, 34, 5, 7, 35, 2, 3, 6, 12, 18

Offset: 2

Michael De Vlieger, Feb 13 2025

Intersection of row n of A027750 and A052485 for n > 1.

			D(2) = {1, 2}; of these, only 2 is weak.
D(4) = {1, 2, 4}; of these, only 2 is weak.
D(6) = {1, 2, 3, 6}; of these, {2, 3, 6} are weak.
D(10) = {1, 2, 5, 10}; of these, {2, 5, 10} are weak.
D(12) = {1, 2, 3, 4, 6, 12}; of these, {2, 3, 6, 12} are weak.
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {2, 3, 6, 12, 18} are weak, etc.
Table begins:
   n:  row n
  ----------------
   2:  2;
   3:  3;
   4:  2;
   5:  5;
   6:  2, 3, 6;
   7:  7;
   8:  2;
   9:  3;
  10:  2, 5, 10;
  11:  11;
  12:  2, 3, 6, 12;
  13:  13;
  14:  2, 7, 14;
  15:  3, 5, 15;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11751 (rows n = 2..2000, flattened).

Cf. A000005, A001694, A005361, A007947, A027750, A052485, A183093, A379545, A380672.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], ! Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten

Row 1 is empty since d = 1 is powerful (i.e., in A001694).
Let P(n) = row n of A027748 for n > 1. P(n) is a subset of row n.
Length of row n = A183093(n) = tau(n) = tau(n/rad(n)).
For prime p and m > 0, row p^m = {p}, since d = 1 and p = p^j, j > 1 are powerful.
Let D(n) = row n of A027750. For squarefree composite n, row n = D(n) \ {1}, since d | n, d > 1, are squarefree for squarefree n.

A380672 Triangle read by rows where row n lists divisors d | n such that rad(d) != rad(n), where rad = A007947.

Original entry on oeis.org

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

Offset: 2

Michael De Vlieger, Feb 13 2025

Row n lists terms in row n of A027750 that do not have the same squarefree kernel as does n.

			D(6) = {1, 2, 3, 6}; of these, {1, 2, 3} are such that rad(d) != rad(6).
D(10) = {1, 2, 5, 10}; of these, {1, 2, 5} are such that rad(d) != rad(10).
D(12) = {1, 2, 3, 4, 6, 12}; of these, {1, 2, 3, 4} are such that rad(d) != rad(12).
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {1, 2, 3, 4, 9} are such that rad(d) != rad(36), etc.
Table begins:
   n:  row n
  ---------------
   2:  1;
   3:  1;
   4:  1;
   5:  1;
   6:  1, 2, 3;
   7:  1;
   8:  1;
   9:  1;
  10:  1, 2, 5;
  11:  1;
  12:  1, 2, 3, 4;
  13:  1;
  14:  1, 2, 7;
  15:  1, 3, 5;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11751 (rows n = 2..2000, flattened).

Cf. A000005, A005361, A007947, A027750, A183093, A284318, A380819.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Divisors[n], rad[#] != r &], {n, 2, 40}] // Flatten

Row 1 is empty since rad(1) | rad(1).
The first term of row n is 1 for all n > 1.
n is not in row n since rad(n) = rad(n).
Length of row n = A183093(n) = tau(n) - tau(n/rad(n)).
Let S(n) = row n of A284318 and let D(n) = row n of A027750. Then row n of this sequence is D(n) \ S(n).
For prime p and m > 0, row p^m = {1}, since d | p^m, d > 1, are such that rad(d) = p.
For squarefree composite n, row n = D(n) \ {n} with length 2^(omega(k)-1).

cp's OEIS Frontend

A183096 a(n) = number of divisors of n that are not perfect powers.

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

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A183095 a(n) = number of divisors d of n that are either 1 or of the form Product_(i) (p_i^e_i) where e_i = 1 for at least one i.

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A380819 Triangle read by rows where row n lists "weak" divisors d | n (i.e., d in A052485) such that rad(d)^2 does not divide d, where rad = A007947.

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A380672 Triangle read by rows where row n lists divisors d | n such that rad(d) != rad(n), where rad = A007947.

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