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.

A294900 Numbers k such that k = sum of nonabundant proper divisors of k (A294888).

Original entry on oeis.org

6, 24, 28, 126, 496, 8128, 5594428, 33550336, 8589869056, 17589794838, 35439846824, 49380301744
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Comments

Naturally, all the terms of A000396, including 137438691328, are in this sequence. - Antti Karttunen, Dec 01 2017
Thus, if there are infinitely many Mersenne primes, then this sequence is also, by definition of even perfect numbers, infinite. - Iain Fox, Dec 02 2017
All non-perfect terms are abundant. Proof: Assume d is a deficient number in this sequence. Because multiples of abundant numbers are abundant, d cannot have an abundant divisor, thus all its divisors are nonabundant. Since d is in this sequence, the sum of its proper divisors, which are all nonabundant, must equal d. However, if this were true, then d would be perfect. Therefore, this sequence contains no deficient numbers. - Iain Fox, Dec 07 2017
Questions from Iain Fox, Dec 07 2017: (Start)
Are there an infinite number of abundant terms?
Are all abundant terms in this sequence even?
(End)
No other terms up to 10^10. - Iain Fox, Dec 07 2017
a(13) > 6*10^10. - Giovanni Resta, Dec 11 2017
In comparison, the numbers which are the sum of their abundant proper divisors seems to be scarcer: up to 6*10^10 only 19514300 and 16333377500 have this property. - Giovanni Resta, Dec 11 2017
From Iain Fox, Dec 11 2017: (Start)
The first abundant term without a perfect divisor is 35439846824.
This term and any other abundant terms without perfect divisors are also terms in A125310.
(End)

Links

Crossrefs

Fixed points of A294888.
Subsequence of A005835; A000396 is a subsequence.
Cf. A125310.

Programs

  • PARI
    isok(n) = sumdiv(n, d, if ((dMichel Marcus, Nov 17 2017
  • PARI
    normalize(f)=f=select(v->v[2],f~)~;if(vecmax(matsize(f)),f,factor(1));
is(n,f=factor(n))=
{
my(p=Mat(f[,1]),g,s);
forvec(v=apply(k->[0,k],f[,2]~),
g=normalize(concat(p,v~));
if(sigma(g,-1)<=2,
s+=factorback(g)
);
);
s==if(sigma(f,-1)>2,n,2*n);
}
forfactored(n=6,10^9, if(is(n[1],n[2]), print1(n[1]", "))) \\ Charles R Greathouse IV, Dec 08 2017

Extensions

a(9) from Iain Fox, Dec 07 2017
a(10)-a(12) from Giovanni Resta, Dec 11 2017

A294886 Sum of deficient proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 22, 11, 14, 1, 18, 6, 16, 13, 28, 1, 36, 1, 31, 15, 20, 13, 19, 1, 22, 17, 30, 1, 48, 1, 40, 33, 26, 1, 34, 8, 43, 21, 46, 1, 42, 17, 36, 23, 32, 1, 40, 1, 34, 41, 63, 19, 72, 1, 58, 27, 74, 1, 27, 1, 40, 49, 64, 19, 84, 1, 46, 40, 44, 1, 52, 23, 46, 33, 92, 1, 90, 21
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Comments

Sum of divisors of n smaller than n that are deficient numbers (in A005100).

Examples

			Proper divisors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45. Of these 1, 2, 3, 5, 9, 10, 15 and 45 are in A005100, thus a(90) = 1+2+3+5+9+10+15+45 = 90.

Links

Crossrefs

Cf. A001065, A005100, A001065, A187793, A294926, A294934, A294887, A294888, A294889.
Cf. A125310 (fixed points).

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, # < n && DivisorSigma[1, #] < 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)
  • PARI
    A294886(n) = sumdiv(n, d, (d

Formula

a(n) = Sum_{d|n, dA294934(d)*d.
a(n) = A187793(n) - (A294934(n)*n).
a(n) + A294887(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).

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

A294928 Number of proper divisors of n that are nonabundant (A263837).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 6, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 6, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 7, 2, 5, 3, 5, 1, 6, 3, 7, 3, 3, 1, 8, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 7, 1, 3, 5, 5, 3, 7, 1, 7, 4, 3, 1, 9, 3, 3, 3, 7, 1, 9, 3, 5, 3, 3, 3, 8, 1, 5, 5, 7, 1, 7, 1, 7, 7
Offset: 1

Views

Author

Antti Karttunen, Nov 14 2017

Keywords

Examples

			The seven proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. All except 12 are nonabundant (in A263837), thus a(24) = 6.

Links

Crossrefs

Cf. A263837, A294888, A294926, A294927, A294929, A294935.
Differs from A032741 for the first time at n=24.
Cf. also A294901.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] <= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)
  • PARI
    A294928(n) = sumdiv(n, d, (d

Formula

a(n) = Sum_{d|n, dA294935(d).
a(n) + A294929(n) = A032741(n).

A364858 a(n) = Sum_{d|n, d < n, d in S} d, where S is the set defined in A118372.

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, 64, 8, 43, 21, 46, 1, 48, 17, 64, 23, 32, 1, 46, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 93, 1
Offset: 1

Views

Author

Amiram Eldar, Aug 11 2023

Keywords

Comments

First differs from A294888 at n = 48.

Links

Crossrefs

Cf. A001065, A118372, A181487, A294888.

Programs

  • Mathematica
    seq[nmax_] := Module[{s = {1}, a = {0}, sum}, Do[sum = Total[Select[Divisors[n], MemberQ[s, #] &]]; If[sum <= n, AppendTo[s, n]]; AppendTo[a, sum], {n, 2, nmax}]; a]; seq[100]
  • PARI
    lista(nmax) = {my(c = 0, s); print1(0, ", "); for(n=2, nmax, s = sumdiv(n, d, !bittest(c, d)*d) - n; if(s > n, c+=1<M. F. Hasler at A181487

Formula

a(n) <= A001065(n).
a(n) <= n if and only if n is in the set S.
a(n) = n if and only if n is S-perfect (A118372).
a(n) > n if and only if n is S-abundant (A181487).
Showing 1-7 of 7 results.

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