cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A326987 Number of nonpowers of 2 dividing n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 3, 1, 2, 3, 0, 1, 4, 1, 3, 3, 2, 1, 4, 2, 2, 3, 3, 1, 6, 1, 0, 3, 2, 3, 6, 1, 2, 3, 4, 1, 6, 1, 3, 5, 2, 1, 5, 2, 4, 3, 3, 1, 6, 3, 4, 3, 2, 1, 9, 1, 2, 5, 0, 3, 6, 1, 3, 3, 6, 1, 8, 1, 2, 5, 3, 3, 6, 1, 5, 4, 2, 1, 9, 3, 2, 3, 4, 1, 10, 3, 3, 3, 2, 3, 6, 1, 4, 5, 6
Offset: 1

Views

Author

Omar E. Pol, Aug 18 2019

Keywords

Comments

In other words: a(n) is the number of divisors of n that are not powers of 2.
a(n) is also the number of odd divisors > 1 of n, multiplied by the number of divisors of n that are powers of 2.
a(n) = 0 iff n is a power of 2.
a(n) = 1 iff n is an odd prime.
From Bernard Schott, Sep 12 2019: (Start)
a(n) = 2 iff n is an even semiprime >= 6 or n is a square of prime >= 9 (Aug 26 2019).
a(n) = 3 iff n is an odd squarefree semiprime, or n is an odd prime multiplied by 4, or n is a cube of odd prime (End).

Examples

			For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. There are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18], so a(18) = 4. On the other hand, there are two odd divisors > 1 of 18, they are [3, 9], and there are two divisors of 18 that are powers of 2, they are [1, 2], then we have that 2*2 = 4, so a(18) = 4.

Links

Crossrefs

Cf. A000005, A000079, A001227, A001248, A001511, A001620, A057716, A065091, A069283, A100484, A326988 (sum), A326989.

Programs

  • Magma
    sol:=[];  m:=1;  for n in [1..100] do v:=Set(Divisors(n)) diff {2^k:k in [0..Floor(Log(2,n))]};  sol[m]:=#v; m:=m+1; end for; sol; // Marius A. Burtea, Aug 24 2019
  • Maple
    a:= n-> numtheory[tau](n)-padic[ordp](2*n, 2):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 24 2019
  • Mathematica
    a[n_] := DivisorSigma[0, n] - IntegerExponent[n, 2] - 1; Array[a, 100] (* Amiram Eldar, Aug 31 2019 *)
  • PARI
    ispp2(n) = (n==1) || (isprimepower(n, &p) && (p==2));
a(n) = sumdiv(n, d, ispp2(d) == 0); \\ Michel Marcus, Aug 26 2019
  • Python
    from sympy import divisor_count
def A326987(n): return divisor_count(n)-(n&-n).bit_length() # Chai Wah Wu, Jul 13 2022

Formula

a(n) = A000005(n) - A001511(n).
a(n) = (A001227(n) - 1)*A001511(n).
a(n) = A069283(n)*A001511(n).
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

A326989 Triangle read by rows in which row n lists the nonpowers of 2 dividing n, or 0 if n is a power of 2.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 6, 7, 0, 3, 9, 5, 10, 11, 3, 6, 12, 13, 7, 14, 3, 5, 15, 0, 17, 3, 6, 9, 18, 19, 5, 10, 20, 3, 7, 21, 11, 22, 23, 3, 6, 12, 24, 5, 25, 13, 26, 3, 9, 27, 7, 14, 28, 29, 3, 5, 6, 10, 15, 30, 31, 0, 3, 11, 33, 17, 34, 5, 7, 35, 3, 6, 9, 12, 18, 36, 37, 19, 38, 3, 13, 39, 5, 10, 20, 40, 41
Offset: 1

Views

Author

Omar E. Pol, Aug 24 2019

Keywords

Comments

Row n has length A326987(n) if n is not a power of 2, otherwise row n has length 1.

Examples

			Triangle begins:
   0;
   0;
   3;
   0;
   5;
   3,  6;
   7;
   0;
   3,  9;
   5, 10;
  11;
   3,  6, 12;
  13;
   7, 14;
   3,  5, 15;
   0;
  17;
   3,  6,  9, 18;
  ...
For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. There are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18], the same as the 18th row of triangle.

Crossrefs

Row sums give A326988.
Cf. A000079, A027750, A057716, A326987.

A326990 Sum of odd divisors of n that are greater than 1.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 23, 0, 17, 12, 19, 5, 31, 11, 23, 3, 30, 13, 39, 7, 29, 23, 31, 0, 47, 17, 47, 12, 37, 19, 55, 5, 41, 31, 43, 11, 77, 23, 47, 3, 56, 30, 71, 13, 53, 39, 71, 7, 79, 29, 59, 23, 61, 31, 103, 0, 83, 47, 67, 17, 95, 47, 71, 12, 73, 37, 123, 19, 95, 55, 79, 5
Offset: 1

Views

Author

Omar E. Pol, Aug 24 2019

Keywords

Comments

Also sum of nonpowers of 2 dividing n, divided the sum of powers of 2 dividing n.
a(n) = 0 iff n is a power of 2.
a(n) = n iff n is an odd prime.
First differs from A284233 at a(15).

Examples

			For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. The sum of odd divisors of 18 that are greater than 1 is 3 + 9 = 12, so a(18) = 12. On the other hand, there are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18], and the sum of them is 3 + 6 + 9, 18 = 36. Also there are two divisors of 18 that are powers of 2, they are [1, 2], and the sum of them is 1 + 2 = 3. Then we have that 36/3 = 12, so a(18) = 12.

Links

Crossrefs

Cf. A000079, A000203, A000593, A038712, A057716, A065091, A069283 (Number), A284233, A326988, A326989.

Programs

  • Magma
    sol:=[]; m:=1; for n in [1..80] do v:=[d:d in Divisors(n)|d gt 1 and IsOdd(d)]; if #v ne 0 then sol[m]:=&+v; m:=m+1; else sol[m]:=0; m:=m+1; end if; end for; sol; // Marius A. Burtea, Aug 24 2019
  • Mathematica
    Table[Total[Select[Rest[Divisors[n]],OddQ]],{n,80}] (* Harvey P. Dale, Jan 11 2025 *)

Formula

a(n) = A000593(n) - 1.
a(n) = (A000203(n) - A038712(n))/A038712(n).
a(n) = A326988(n)/A038712(n).

A327327 Partial sums of the sum of nonpowers of 2 dividing n.

Original entry on oeis.org

0, 0, 3, 3, 8, 17, 24, 24, 36, 51, 62, 83, 96, 117, 140, 140, 157, 193, 212, 247, 278, 311, 334, 379, 409, 448, 487, 536, 565, 634, 665, 665, 712, 763, 810, 894, 931, 988, 1043, 1118, 1159, 1252, 1295, 1372, 1449, 1518, 1565, 1658, 1714, 1804, 1875, 1966, 2019, 2136, 2207, 2312, 2391, 2478, 2537, 2698
Offset: 1

Views

Author

Omar E. Pol, Sep 14 2019

Keywords

Comments

a(n) can be represented with a diagram since the symmetric diagram of A024916(n) is greater than or equal to the diagram of A080277(n). The difference between both diagrams is a representation of a(n). For more information about the symmetric diagram of A024916 see A236104 and A237593.

Examples

			The divisors of 6 are 1, 2, 3, 6. But 1 and 2 are powers of 2, so we only add up 3, 6 to get 9, and add that to the running total of 8 to get a(6) = 17.

Crossrefs

Partial sums of A326988.
Cf. A000203, A024916, A038712, A080277, A236104, A237593, A327326.

Programs

  • Mathematica
    Accumulate[Table[DivisorSigma[1, n] - Denominator[DivisorSigma[1, 2n]/DivisorSigma[1, n]], {n, 100}]] (* Alonso del Arte, Nov 18 2019, based on Wesley Ivan Hurt's program for A326988 *)

Formula

a(n) = A024916(n) - A080277(n).
a(n) = a(n-1) when n is a power of 2.
Showing 1-4 of 4 results.

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