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.

A187793 Sum of the deficient divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 10, 14, 24, 24, 31, 18, 15, 20, 22, 32, 36, 24, 18, 31, 42, 40, 28, 30, 36, 32, 63, 48, 54, 48, 19, 38, 60, 56, 30, 42, 48, 44, 84, 78, 72, 48, 34, 57, 93, 72, 98, 54, 42, 72, 36, 80, 90, 60, 40, 62, 96, 104, 127, 84, 72, 68, 126, 96, 74, 72, 27
Offset: 1

Views

Author

Timothy L. Tiffin, Jan 06 2013

Keywords

Comments

Sum of divisors d of n with sigma(d) < 2*d.
a(n) = sigma(n) when n is itself also deficient.
Also, a(n) agrees with the terms in A117553 except when n is a multiple (k > 1) of either a perfect number or a primitive abundant number.
Notice that a(1) = 1. The remaining fixed points are given by A125310. - Timothy L. Tiffin, Jun 23 2016
a(A028982(n)) is an odd integer. Also, if n is an odd abundant number that is not a perfect square and n has an odd number of abundant divisors (e.g., 945 has one abundant divisor and 4725 has three abundant divisors), then a(n) will also be odd: a(945) = 975 and a(4725) = 2675. - Timothy L. Tiffin, Jul 18 2016

Examples

			a(12) = 10 because the divisors of 12 are 1, 2, 3, 4, 6, 12; of these, 1, 2, 3, 4 are deficient, and they add up to 10.

Links

Crossrefs

Cf. A000203, A005100, A028982, A080226, A117553, A125310, A125499, A187794, A187795, A247328, A274338, A274339, A274340, A274380, A274549, A274829, A294886, A294934.

Programs

  • Maple
    A187793 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) < 2*d then
            a := a+d ;
        end if ;
    end do:
    a ;
end proc:# R. J. Mathar, May 08 2019
  • Mathematica
    Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &], {n, 72}] (* Michael De Vlieger, Jul 18 2016 *)
  • PARI
    a(n)=sumdiv(n,d,if(sigma(d,-1)<2,d,0)) \\ Charles R Greathouse IV, Jan 07 2013

Formula

From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294934(d)*d.
a(n) = A294886(n) + (A294934(n)*n).
a(n) + A187794(n) + A187795(n) = A000203(n).
(End)

Extensions

a(54) corrected by Charles R Greathouse IV, Jan 07 2013

A125746 a(n) = smallest divisor d of n such that n <= {sum of d and all smaller divisors of n}.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 14, 15, 16, 17, 9, 19, 10, 21, 22, 23, 8, 25, 26, 27, 14, 29, 15, 31, 32, 33, 34, 35, 12, 37, 38, 39, 20, 41, 21, 43, 44, 45, 46, 47, 16, 49, 50, 51, 52, 53, 27, 55, 28, 57, 58, 59, 20, 61, 62, 63, 64, 65, 33, 67, 68, 69, 35, 71, 24, 73
Offset: 1

Views

Author

Leroy Quet, Dec 05 2006

Keywords

Comments

Original name of this sequence: a(n) is the smallest positive integer such that (sum{k|n, 1<=k<=a(n)} k) is >= n.

Examples

			The divisors of 12 are 1,2,3,4,6,12. 1+2+3+4 = 10, which is smaller than 12; but 1+2+3+4+6 = 16, which is >= 12. So a(12) = 6.

Links

Crossrefs

Cf. A125747, A117553, A300826 (= n/a(n)).

Programs

  • Mathematica
    f[n_] := Block[{k = 1, d = Divisors[n]},While[Sum[d[[i]], {i, k}] < n, k++ ];d[[k]]];Table[f[n], {n, 76}] (* Ray Chandler, Dec 06 2006 *)
  • PARI
    A125746(n) = { my(k=0,s=0); fordiv(n,d, k++; s += d; if(s>=n,return(d))); }; \\ Antti Karttunen, Mar 21 2018

Extensions

Extended by Ray Chandler, Dec 06 2006
Name changed by Antti Karttunen, Mar 21 2018

A117552 Largest partial sum of the increasingly ordered divisors of n, not exceeding n.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 10, 1, 10, 9, 15, 1, 12, 1, 12, 11, 14, 1, 24, 6, 16, 13, 28, 1, 27, 1, 31, 15, 20, 13, 25, 1, 22, 17, 30, 1, 33, 1, 40, 33, 26, 1, 36, 8, 43, 21, 46, 1, 39, 17, 36, 23, 32, 1, 58, 1, 34, 41, 63, 19, 45, 1, 58, 27, 39, 1, 63, 1, 40, 49, 64, 19, 51, 1, 66
Offset: 1

Views

Author

Leroy Quet, Mar 28 2006

Keywords

Examples

			a(12)=10 because the increasingly ordered divisors of 12 are 1,2,3,4,6 and 12, with partial sums 1,3,6,10,16 and 28; the largest partial sum not exceeding 12 is 10.

Links

Crossrefs

Cf. A117553, A109883, A377247 (corresponding largest divisor index).

Programs

  • Maple
    with(numtheory): a:=proc(n) local div,j: if n=1 then 1 else div:=divisors(n): for j from 1 by 1 while sum(div[i],i=1..j)<=n do sum(div[k],k=1..j) od: fi: end: seq(a(n),n=1..90); # Emeric Deutsch, Apr 01 2006
  • Mathematica
    Table[Last@ TakeWhile[Accumulate@ Divisors@ n, # <= n &], {n, 80}] (* Michael De Vlieger, Oct 30 2017 *)
  • PARI
    A117552(n) = { my(divs=divisors(n), s=0); for(i=1,#divs,if((s+divs[i])>n,return(s),s+=divs[i])); s; }; \\ Antti Karttunen, Oct 30 2017

Formula

a(n) = n - A109883(n). - Ridouane Oudra, Jan 25 2024

Extensions

More terms from Emeric Deutsch, Apr 01 2006

A125747 a(n) is the smallest positive integer such that (Sum_{t(k)|n, 1 <= k <= a(n)} t(k)) >= n, where t(k) is the k-th positive divisor of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 7, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 7, 4, 7, 4, 4, 2, 10, 2, 4, 6, 7, 4, 7, 2, 6, 4, 7, 2, 10, 2, 4, 6, 6, 4, 7, 2, 9, 5, 4, 2, 10, 4, 4, 4, 7, 2, 10, 4, 6, 4, 4, 4, 10, 2, 6, 6, 8, 2, 7, 2
Offset: 1

Views

Author

Leroy Quet, Dec 05 2006

Keywords

Comments

a(n) = the least number of divisors of n, taken in increasing order as 1, A020639(n), A292269(n), etc. needed so that their sum is >= n. - Antti Karttunen, Mar 21 2018

Examples

			The divisors of 12 are 1,2,3,4,6,12. 1+2+3+4 = 10, which is smaller than 12; but 1+2+3+4+6 = 16, which is >= 12. 6 is the 5th divisor of 12, so a(12) = 5.

Links

Crossrefs

Cf. A020639, A292269, A125746, A117553.

Programs

  • Mathematica
    f[n_] := Block[{k = 1, d = Divisors[n]},While[Sum[d[[i]], {i, k}] < n, k++ ];k];Table[f[n], {n, 105}] (* Ray Chandler, Dec 06 2006 *)
  • PARI
    A125747(n) = { my(k=0,s=0); fordiv(n,d, k++; s += d; if(s>=n,return(k))); }; \\ Antti Karttunen, Mar 21 2018

Extensions

Extended by Ray Chandler, Dec 06 2006
Showing 1-4 of 4 results.

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