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 1-3 of 3 results.

A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 12, 1, 16, 1, 12, 1, 42, 1, 1, 15, 20, 13, 14, 1, 22, 17, 14, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 30, 17, 16, 23, 32, 1, 60, 1, 34, 17, 1, 19, 78, 1, 22, 27, 74, 1, 18, 1, 40, 29, 24, 19, 90, 1, 22, 1, 44
Offset: 1

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Author

N. J. A. Sloane

Keywords

Examples

			Unitary divisors of 12 are 1, 3, 4, 12. a(12) = 1 + 3 + 4 = 8.

Links

Crossrefs

Cf. A034444, A034448, A077610, A306633.
Cf. A063936 (squares > 1).
Cf. A063919 (essentially the same sequence).

Programs

  • Haskell
    a034460 = sum . init . a077610_row  -- Reinhard Zumkeller, Aug 15 2012
  • Maple
    A034460 := proc(n)
    A034448(n)-n ;
end proc:
seq(A034460(n),n=1..40) ; # R. J. Mathar, Nov 10 2014
  • Mathematica
    usigma[n_] := Sum[ If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := usigma[n] - n; Table[ a[n], {n, 1, 82}] (* Jean-François Alcover, May 15 2012 *)
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 03 2022 *)
  • PARI
    a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n \\ Charles R Greathouse IV, Aug 01 2016

Formula

a(n) = Sum_{k = 1..A034444(n)-1} A077610(n,k). - Reinhard Zumkeller, Aug 15 2012
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/2 = 0.1842163888... . - Amiram Eldar, Feb 22 2024

A063937 Sum of unitary divisors of n is a square > 1.

Original entry on oeis.org

3, 8, 22, 24, 66, 70, 76, 94, 115, 119, 170, 210, 214, 217, 228, 252, 265, 282, 310, 316, 322, 345, 357, 382, 385, 490, 497, 510, 517, 522, 527, 580, 612, 642, 651, 679, 710, 716, 742, 745, 782, 795, 801, 833, 862, 889, 920, 930, 935, 948, 952, 966, 970
Offset: 1

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Author

Felice Russo, Aug 31 2001

Keywords

Comments

A unitary divisor of n is a divisor d of n such that gcd(d, n/d) = 1.

Examples

			The unitary divisors of 3 are 1,3 and then 3 + 1 = 4 is a square.

Links

Crossrefs

Cf. A034448, A063936.

Programs

  • Haskell
    import Data.List (findIndices)
a063937 n = a063937_list !! (n-1)
a063937_list = map (+ 2) $
               findIndices ((== 1) . a010052) $ tail a034448_list
-- Reinhard Zumkeller, Aug 15 2012
  • Mathematica
    udQ[n_]:=Module[{totdivs=Total[Sort[Flatten[Outer[Times,Sequence@@({1,#}&/@Power@@@FactorInteger[n])]]]]},totdivs>1&&IntegerQ[Sqrt[totdivs]]]; Select[Range[1000],udQ] (* Harvey P. Dale, Apr 22 2012, using program from Eric Weisstein at https://mathworld.wolfram.com/UnitaryDivisor.html *)
  • PARI
    us(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
{ n=0; for (m=1, 10^9, u=us(m); if (issquare(u) && u > 1, write("b063937.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 03 2009

Formula

a(n) > 1 and A010052(A034448(a(n))) = 1. - Reinhard Zumkeller, Aug 15 2012

A333258 Numbers k that are not powers of primes such that the sum of proper unitary divisors of k is a cube.

Original entry on oeis.org

10, 12, 69, 122, 133, 153, 236, 363, 504, 752, 844, 992, 1001, 1018, 1243, 1685, 1819, 1940, 1994, 2295, 2323, 2619, 2871, 2900, 3184, 3403, 3483, 3641, 3763, 3981, 3984, 4024, 5482, 6471, 6892, 7128, 7925, 7928, 8186, 8856, 9077, 9352, 9641, 9664, 10113, 10404
Offset: 1

Views

Author

Amiram Eldar, Mar 13 2020

Keywords

Comments

Powers of primes are excluded since they are trivial terms: their sum of proper unitary divisors is 1 (except for 1 whose sum of proper unitary divisors is 0) .

Examples

			10 is a term since A034460(10) = 8 = 2^3.

Links

Crossrefs

The unitary version of A048698.
Cf. A000961, A024619, A034448, A034460, A063936, A329239.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); us[n_] := usigma[n] - n; Select[Range[10000], PrimeNu[#] > 1 && IntegerQ @ Surd[us [#], 3] &]
Showing 1-3 of 3 results.

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

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