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.

A091419 Squares arising in A089653.

Original entry on oeis.org

1, 1, 4, 1, 9, 1, 16, 9, 4, 25, 1, 36, 9, 9, 25, 1, 64, 9, 81, 9, 100, 25, 121, 4, 1, 144, 9, 81, 169, 196, 9, 25, 225, 121, 9, 1, 256, 16, 289, 9, 64, 81, 25, 169, 361, 9, 400, 25, 441, 225, 484, 121, 9, 1, 529, 576, 289, 4, 625, 9, 81, 25, 676, 169, 361, 144, 729, 784, 9
Offset: 1

Views

Author

Ray Chandler, Jan 05 2004

Keywords

Links

Crossrefs

A091420 gives square roots.
Cf. A007947.

Formula

a(n) = A007947(A089653(n)) - 1. - Amiram Eldar, Sep 29 2019

A089632 1 + product of prime factors of n is a perfect square.

Original entry on oeis.org

3, 9, 15, 27, 35, 45, 75, 81, 135, 143, 175, 195, 225, 243, 245, 255, 323, 375, 399, 405, 483, 585, 675, 729, 765, 875, 899, 975, 1023, 1125, 1155, 1197, 1215, 1225, 1275, 1295, 1443, 1449, 1573, 1599, 1715, 1755, 1763, 1859, 1875, 2025, 2187, 2295, 2535
Offset: 1

Views

Author

Joseph L. Pe, Jan 04 2004

Keywords

Comments

From Robert Israel, Apr 14 2019: (Start)
Numbers k such that A076618(k) is a square.
All terms are odd.
Squarefree terms are k^2-1 for k in A067874.
(End)

Examples

			The prime factors of 35 are 5 and 7 and 5 * 7 + 1 = 36 is a square; so 35 belongs to the sequence.

Links

Crossrefs

Cf. A089653. A091278 gives squares, A091279 gives square roots.
Cf. A067874, A076618.

Programs

  • Maple
    filter:= n -> issqr(1+convert(numtheory:-factorset(n),`*`)):
select(filter, [$1..10000]); # Robert Israel, Apr 14 2019
  • Mathematica
    ppf[n_] := Apply[Times, Transpose[FactorInteger[n]][[1]]]; Select[Range[2, 10^3], IntegerQ[Sqrt[ppf[ # ] + 1]] &]
  • PARI
    isok(n) =  my(f=factor(n)); issquare(1+prod(k=1, #f~, f[k,1])); \\ Michel Marcus, Apr 15 2019

Extensions

More terms from Ray Chandler, Jan 05 2004

A091420 Square roots of A091419.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 3, 2, 5, 1, 6, 3, 3, 5, 1, 8, 3, 9, 3, 10, 5, 11, 2, 1, 12, 3, 9, 13, 14, 3, 5, 15, 11, 3, 1, 16, 4, 17, 3, 8, 9, 5, 13, 19, 3, 20, 5, 21, 15, 22, 11, 3, 1, 23, 24, 17, 2, 25, 3, 9, 5, 26, 13, 19, 12, 27, 28, 3, 5, 29, 8, 13, 21, 30, 15, 31, 11, 3, 1, 23, 33, 34, 17, 35
Offset: 1

Views

Author

Ray Chandler, Jan 05 2004

Keywords

Links

Crossrefs

Cf. A089653, A091419.

Formula

a(n) = sqrt(A091419(n)). - Amiram Eldar, Sep 29 2019

A363084 Numbers k such that sqrt(A007947(k) - A007913(k)) is an integer m > 0.

Original entry on oeis.org

4, 16, 18, 25, 64, 72, 100, 162, 180, 256, 288, 289, 294, 400, 507, 625, 648, 676, 720, 722, 1024, 1152, 1176, 1210, 1369, 1458, 1600, 1620, 2178, 2205, 2500, 2548, 2592, 2646, 2704, 2880, 2888, 3150, 4096, 4225, 4500, 4563, 4608, 4704, 4840, 5202, 5832, 5887
Offset: 1

Views

Author

Michael De Vlieger, Sep 05 2023

Keywords

Comments

Let core(k) = A007913(k) and rad(k) = A007947(k).
Squarefree numbers k imply rad(k) - core(k) = k - k = 0.
Perfect squares k^2 such that rad(k) = m^2+1 and k > 1 imply rad(k^2) - core(k^2) = (m^2+1) - 1 = m^2, with integers k, m.
Generally, if there exists a minimal d such that d | k, k/d = m^2, and rad(k) - d = m^2, then k is in the sequence.
Subsets of this sequence include the sets of squares k^2 such that k is in A002496, A003592, and A089653, since A089653 contains both A002496 and A003592.

Examples

			a(1) = 4 since rad(4) = 1+1; rad(4) - core(4) = 2 - 1 = 1, a nonzero square.
a(2) = 18 since 18/2 = 9, and rad(18) - core(18) = 6 - 2 = 4, a nonzero square, etc.

Links

Crossrefs

Cf. A002496, A003592, A007913, A007947, A089653, A365517.

Programs

  • Mathematica
    Select[Range[6000], And[IntegerQ[#], # > 0] &[Sqrt[Times @@ FactorInteger[#][[All, 1]] - (Sqrt[#] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2)] ] &]
  • PARI
    isok(k) = my(s=factorback(factorint(k)[, 1])-core(k)); (s>0) && issquare(s); \\ Michel Marcus, Sep 18 2023
  • Python
    from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import factorint
def A363084_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        a, b = 1, 1
        for p, e in factorint(k).items():
            if e&1:
                a *= p
            else:
                b *= p
        if b>1 and is_square(a*(b-1)):
            yield k
A363084_list = list(islice(A363084_gen(),30)) # Chai Wah Wu, Sep 19 2023
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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