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 1-5 of 5 results.

A081061 Union of 3-smooth numbers and prime powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 54, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 04 2003

Keywords

Comments

A081060(m)=1 iff m=a(k) for some k.
Complement of A081062.

Links

Crossrefs

Cf. A003586, A000961, A081060, A081062, A081063.

Programs

  • Mathematica
    smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
Select[Range[1000], PrimePowerQ[#] || smooth3Q[#]&] (* Jean-François Alcover, Oct 14 2021 *)
  • Python
    from sympy import integer_log, primepi, integer_nthroot
def A081061(n):
    def f(x): return int(n+x-1+(a:=x.bit_length())+(b:=integer_log(x,3)[0])-sum((x//3**i).bit_length() for i in range(b+1))-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, a)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 16 2024

A306999 Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21^e for e >= 0.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 28, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, 66, 69, 70, 72, 75, 77, 78, 84, 87, 90, 91, 93, 96, 98, 99, 102, 105, 108, 111, 112, 114, 117, 119, 120, 123, 126, 129, 132, 133, 135, 138, 140, 141, 144, 150, 153, 154, 156, 159, 161
Offset: 1

Views

Author

Michael De Vlieger, Aug 22 2019

Keywords

Comments

Complement of the union of A003594 and A160545.
Analogous to A081062 and A105115 regarding terms 1 and 2 of A120944, respectively. This sequence applies to A120944(5) = 21.

Examples

			6 is in the sequence since gcd(6, 21) = 3 and 6 does not divide 21^e with integer e >= 0.
5 is not in the sequence since it is coprime to 21.
3 is not in the sequence since 3 | 21.
9 is not in the sequence since 9 | 21^2.

Links

Crossrefs

Cf. A003594, A081062, A105115, A120944, A160545.

Programs

  • Maple
    filter:= proc(n) local g;
  g:= igcd(n,21);
  if g = 1 or g = n then return false fi;
  3^padic:-ordp(n,3)*7^padic:-ordp(n,7) < n
end proc:
select(filter, [$1..200]); # Robert Israel, Aug 28 2019
  • Mathematica
    With[{nn = 161, k = 21}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A316991 Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 22, 24, 26, 30, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 100, 102, 104, 105, 106, 108, 110, 114, 116, 118, 119, 120, 122, 124, 126, 130, 132
Offset: 1

Views

Author

Michael De Vlieger, Aug 02 2018

Keywords

Comments

Complement of A000027 and union of A003591 and A162699.
Analogous to A081062 and A105115 that apply to A120944(1) and A120944(2), respectively.
This sequence applies to A120944(3).

Examples

			6 is in the sequence since gcd(6, 14) = 2 and 6 does not divide 14^e with integer e >= 0.
2 is not in the sequence since 2 | 14.
4 is not in the sequence since 4 | 14^2.
3 and 5 are not in the sequence since they are coprime to 14.

Links

Crossrefs

Cf. A003591, A081062, A105115, A120944, A162699.

Programs

  • Mathematica
    With[{nn = 132, k = 14}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A307589 Numbers m such that 1 < gcd(m, 35) < m and m does not divide 35^e for e >= 0.

Original entry on oeis.org

10, 14, 15, 20, 21, 28, 30, 40, 42, 45, 50, 55, 56, 60, 63, 65, 70, 75, 77, 80, 84, 85, 90, 91, 95, 98, 100, 105, 110, 112, 115, 119, 120, 126, 130, 133, 135, 140, 145, 147, 150, 154, 155, 160, 161, 165, 168, 170, 180, 182, 185, 189, 190, 195, 196, 200, 203, 205
Offset: 1

Views

Author

Michael De Vlieger, Aug 22 2019

Keywords

Comments

Complement of the union of A003595 and A235933.
Analogous to A081062 and A105115 for terms 1 and 2 of A120944. This sequence applies to A120944(6) = 35.

Examples

			10 is in the sequence since gcd(10, 35) = 5 and 10 does not divide 35^e with integer e >= 0.
2 is not in the sequence since 2 is coprime to 35.
7 is not in the sequence since 7 | 35.
25 is not in the sequence since 25 | 35^2.

Links

Crossrefs

Cf. A003595, A081062, A105115, A120944, A235933, A306999.

Programs

  • Mathematica
    With[{nn = 205, k = 35}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]
  • PARI
    is(n)=gcd(n,35)>1 && n/5^valuation(n,5)/7^valuation(n,7)>1 \\ Charles R Greathouse IV, Sep 07 2022

Formula

a(n) = 35n/11 + O(log^2 n). - Charles R Greathouse IV, Sep 07 2022

A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 24, 30, 33, 35, 36, 39, 40, 42, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 78, 80, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 126, 129, 130, 132, 138, 140, 141, 144, 145, 147, 150
Offset: 1

Views

Author

Michael De Vlieger, Aug 02 2018

Keywords

Comments

Complement of A000027 and union of A003593 and A229829.
Analogous to A081062 and A105115 that apply to A120944(1)=6 and A120944(2)=10, respectively.
This sequence applies to term A120944(4)=15.

Examples

			6 is in the sequence since gcd(6, 15) = 3 and 6 does not divide 15^e with integer e >= 0.
2 and 4 are not in the sequence since they are coprime to 15.
3 and 5 are not in the sequence since they are divisors of 15.
9 is not in the sequence since 9 | 15^2.

Links

Crossrefs

Cf. A003593, A081062, A105115, A120944, A229829, A316991.

Programs

  • Mathematica
    With[{nn = 150, k = 15}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]
Showing 1-5 of 5 results.

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