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 1-7 of 7 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

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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

A125310 Numbers n such that n = sum of deficient proper divisors of n.

Original entry on oeis.org

6, 28, 90, 496, 8128, 33550336, 8589869056
Offset: 1

Views

Author

Joseph L. Pe, Mar 19 2008

Keywords

Comments

Since any proper divisor of a perfect number is deficient, all perfect numbers are (trivially) included in the sequence.
Hence the interesting terms of the sequence are its non-perfect terms, which I call "deficiently perfect". 90 is the only such term < 10^8. Are there any more?
If a(n) were defined to be those numbers that are equal to the sum of their deficient divisors, then the sequence would begin with 1. So, up to 10^10, the only non-perfect numbers in that sequence would be 1 (a deficient number) and 90 (an abundant number). - Timothy L. Tiffin, Jan 08 2013
a(8) > 10^10. - Giovanni Resta, Jan 08 2013
These "deficiently perfect" numbers are pseudoperfect (A005835) and are a proper multiple of a nondeficient number (and hence abundant).

Examples

			90 has deficient proper divisors 1, 2, 3, 5, 9, 10, 15, 45, which sum to 90. Hence 90 is a term of the sequence.

Links

Crossrefs

Cf. A005100, A198470, A198471.
Subsequence of A005835. Fixed points of A294886. Cf. also A294900.

Programs

  • Mathematica
    sigdef[n_] := Module[{d, l, ct, i}, d = Drop[Divisors[n],-1]; l = Length[d]; ct = 0; For[i = 1, i <= l, i++, If[DivisorSigma[1, d[[i]]] < 2 d[[i]], ct = ct + d[[i]]]]; ct]; l = {}; For[i = 1, i <= 10^8, i++, If[sigdef[i] == i, l = Append[l, i]]]; l
  • PARI
    is(n)=sumdiv(n,d,(sigma(d,-1)<2 && dCharles R Greathouse IV, Jan 17 2013

A294888 Sum of nonabundant proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 24, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 25, 1, 22, 17, 30, 1, 54, 1, 40, 33, 26, 1, 40, 8, 43, 21, 46, 1, 48, 17, 64, 23, 32, 1, 46, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 33, 1, 40, 49, 64, 19, 90, 1, 46, 40, 44, 1, 86, 23, 46, 33, 92, 1, 96, 21
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Comments

Sum of divisors of n smaller than n that are nonabundant numbers (in A263837).

Examples

			Proper divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45. Of these 1, 2, 3, 5, 6, 9, 10, 15 and 45 are in A263837, thus a(90) = 1+2+3+5+6+9+10+15+45 = 96.

Links

Crossrefs

Cf. A001065, A263837, A294928, A294935, A294886, A294887, A294889.
Cf. A294900 (fixed points).

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Boole[# < n && DivisorSigma[1, #] <= 2#] * #&];
Array[a, 100] (* Jean-François Alcover, Nov 17 2017 *)
  • PARI
    A294888(n) = sumdiv(n, d, (d

Formula

a(n) = Sum_{d|n, dA294935(d)*d.
a(n) + A294889(n) = A001065(n).

A294889 Sum of abundant proper divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 62, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 0, 0, 0, 0, 0, 0, 0, 60, 0, 0, 0, 54, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 84, 0, 0, 0, 20, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Comments

Sum of divisors of n smaller than n that are abundant numbers (in A005101).

Examples

			The proper divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. Of these 12, 20 and 30 are in A005101, thus a(60) = 12+20+30 = 62.

Links

Crossrefs

Cf. A001065, A005101, A187795, A294929, A294937, A294886, A294887, A294888.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, # < n && DivisorSigma[1, #] > 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)
  • PARI
    A294889(n) = sumdiv(n, d, (d(2*d))*d);

Formula

a(n) = Sum_{d|n, dA294937(d)*d.
a(n) = A187795(n) - (A294937(n)*n).
a(n) + A294888(n) = A001065(n).

A294887 Sum of nondeficient proper divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 36, 0, 0, 0, 20, 0, 6, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 24, 0, 28, 0, 0, 0, 68, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 96, 0, 0, 0, 0, 0, 6, 0, 60, 0, 0, 0, 88, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 90, 0, 0, 0, 20, 0, 6, 0, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Comments

Sum of divisors n smaller than n that are nondeficient numbers (in A023196).

Examples

			Proper divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, and 45. Of these 6, 18 and 30 are in A023196, thus a(90) = 6+18+30 = 54.

Links

Crossrefs

Cf. A001065, A023196, A294927, A294936, A294886, A294888, A294889.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, # < n && DivisorSigma[1, #] >= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)
  • PARI
    A294887(n) = sumdiv(n, d, (d=(2*d))*d);

Formula

a(n) = Sum_{d|n, dA294936(d)*d. - Typo in A-number corrected by Antti Karttunen, Apr 04 2022
a(n) + A294886(n) = A001065(n).

A294926 Number of proper divisors of n that are deficient (A005100).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 4, 1, 4, 1, 5, 3, 3, 1, 5, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 5, 1, 3, 3, 6, 1, 6, 1, 5, 5, 3, 1, 6, 2, 5, 3, 5, 1, 5, 3, 6, 3, 3, 1, 7, 1, 3, 5, 6, 3, 6, 1, 5, 3, 7, 1, 6, 1, 3, 5, 5, 3, 6, 1, 7, 4, 3, 1, 7, 3, 3, 3, 7, 1, 8, 3, 5, 3, 3, 3, 7, 1, 5, 5, 7, 1, 6, 1, 7, 7
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Links

Crossrefs

Cf. A032741, A080226, A294886, A294927, A294928, A294929, A294934.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] < 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)
  • PARI
    A294926(n) = sumdiv(n, d, (dAntti Karttunen, Nov 14 2017

Formula

a(n) = Sum_{d|n, dA294934(d).
a(n) = A080226(n) - A294934(n).
a(n) + A294927(n) = A032741(n).

A296074 Sum of deficiencies of the proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 6, 1, 5, 1, 8, 7, 4, 1, 9, 1, 9, 9, 12, 1, 2, 5, 14, 8, 13, 1, 16, 1, 5, 13, 18, 11, 3, 1, 20, 15, 8, 1, 24, 1, 21, 18, 24, 1, -9, 7, 27, 19, 25, 1, 20, 15, 14, 21, 30, 1, -1, 1, 32, 24, 6, 17, 40, 1, 33, 25, 40, 1, -27, 1, 38, 32, 37, 17, 48, 1, -1, 22, 42, 1, 9, 21, 44, 31, 26, 1, 18, 19, 45, 33, 48, 23, -36, 1, 53, 36, 33
Offset: 1

Views

Author

Antti Karttunen, Dec 04 2017

Keywords

Examples

			For n = 6, whose proper divisors are 1, 2, 3, their deficiencies are 1, 1, 2, thus a(6) = 1+1+2 = 4.
For n = 12, whose proper divisors are 1, 2, 3, 4, 6, their deficiencies are 1, 1, 2, 1, 0, thus a(12) = 1+1+2+1+0 = 5.

Links

Crossrefs

Cf. A033879.
Cf. also A294886, A294887, A294888, A294889, A293438 (product of).

Programs

  • Mathematica
    f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2; a[1] = 0; a[n_] := Module[{f = FactorInteger[n]}, 3 * Times @@ f1 @@@ f - Times @@ f2 @@@ f - 2*n]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)
  • PARI
    A033879(n) = ((2*n)-sigma(n));
A296074(n) = sumdiv(n,d,(dA033879(d));

Formula

a(n) = Sum_{d|n, dA033879(d).
a(n) = A296075(n) - A033879(n).
Sum_{k=1..n} a(k) ~ (Pi^2/4 - Pi^4/72 - 1) * n^2. - Amiram Eldar, Dec 04 2023
Showing 1-7 of 7 results.

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