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.

A154893 Numbers whose number of proper divisors is not a prime number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 48, 53, 59, 61, 64, 67, 71, 73, 79, 80, 81, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167, 168, 173, 176, 179, 181, 191, 193, 196, 197, 199
Offset: 1

Views

Author

Omar E. Pol, Jan 25 2009

Keywords

Comments

Complement of A063806.

Links

Crossrefs

Cf. A032741, A063806.

Programs

  • Mathematica
    Select[Range[300], ! PrimeQ[DivisorSigma[0, # ] - 1] &] (* Stefan Steinerberger, Jan 31 2009 *)
  • PARI
    isok(n) = !isprime(numdiv(n)-1); \\ Michel Marcus, Apr 27 2018

Extensions

More terms from Stefan Steinerberger, Jan 31 2009

A223456 Composite numbers whose number of proper divisors has a prime number of proper divisors.

Original entry on oeis.org

16, 36, 48, 64, 80, 81, 100, 112, 120, 144, 162, 168, 176, 196, 208, 210, 216, 225, 256, 264, 270, 272, 280, 304, 312, 324, 330, 368, 378, 384, 390, 400, 405, 408, 440, 441, 456, 462, 464, 484, 496, 510, 512, 520, 546, 552, 567, 570, 576, 592, 594, 616, 625
Offset: 1

Views

Author

Christopher J. Hanson, Jul 19 2013

Keywords

Examples

			a(1) = 16, which has 4 proper divisors (1, 2, 4, 8). 4 has 2 proper divisors, 2 is prime. 2 steps were needed.

Links

Crossrefs

Cf. A000040, A063806, A032741.

Programs

  • Haskell
    a223456 n = a223456_list !! (n-1)
a223456_list = filter ((== 1 ) . a010051 . a032741 . a032741) a002808_list
-- Reinhard Zumkeller, Sep 22 2013
  • Maple
    isA223456 := proc(n)
        local npd ;
        if not isprime(n) and n >=4 then
                npd := A032741(n) ;
                if isprime( A032741(npd)) then
                        true;
                else
                        false;
                end if ;
        else
                false;
        end if;
end proc:
for n from 16 to 630 do
        if isA223456(n) then
                printf("%d,",n) ;
        end if;
end do: # R. J. Mathar, Sep 18 2013
  • Mathematica
    Select[Range[1000], PrimeQ[DivisorSigma[0, DivisorSigma[0, #] - 1] - 1] &] (* Alonso del Arte, Jul 21 2013 *)

Formula

{ n: n in A002808 and A032741(A032741(n)) in A000040}.
(1 - A010051(a(n))) * A010051(a032741(a032741(a(n)))) = 1. - Reinhard Zumkeller, Sep 22 2013

A223457 Composite numbers whose number of proper divisors has a number of proper divisors which has a prime number of proper divisors.

Original entry on oeis.org

44100, 46656, 57600, 65536, 108900, 112896, 152100, 213444, 260100, 278784, 298116, 313600, 324900, 331776, 389376, 476100, 509796, 592900, 636804, 656100, 665856, 736164, 756900, 774400, 828100, 831744, 864900, 933156, 1000000, 1081600, 1218816, 1232100
Offset: 1

Views

Author

Christopher J. Hanson, Jul 19 2013

Keywords

Examples

			a(1) = 44100, which has 80 divisors.  80 has 9 divisors. 9 has 2 divisors, 2 is prime.  3 steps were needed.

Crossrefs

Cf. A000040, A063806, A032741, A223456.

Programs

  • Mathematica
    d3Q[n_]:=PrimeQ[Nest[DivisorSigma[0,#]-1&,n,3]]; Select[Range[13*10^5],d3Q] (* Harvey P. Dale, Apr 21 2016 *)

Formula

{n in A002808 : A032741(A032741(A032741(n))) is prime}.

A255429 Numbers with a prime number of nontrivial divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 196
Offset: 1

Views

Author

Rory Glover, Feb 22 2015

Keywords

Comments

Empirically, numbers in this sequence seem to have few divisors.
This sequence appears to be the union of A130763 and the squares of A225649. - Kellen Myers, Apr 21 2015

Crossrefs

Cf. A063806, A070824, A130763, A225649.

Programs

  • Magma
    [n: n in [1..200] | IsPrime(NumberOfDivisors(n)-2)]; // Vincenzo Librandi, Apr 21 2015
  • Mathematica
    seq[n_] := Select[Range[n], PrimeQ[DivisorSigma[0, #] - 2] &] (* Kellen Myers, Apr 21 2015 *)
  • PARI
    isok(m) = isprime(numdiv(m)-1); \\ Michel Marcus, Jan 13 2023

Formula

{n: A070824(n) in A000040}.

Extensions

Terms fixed by Kellen Myers, Apr 21 2015
Name corrected by Michel Marcus, Jan 13 2023
Showing 1-4 of 4 results.

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