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.

A387405 a(n) is the least k that is a multiple of d=A240991(n), with abundancy ratio sigma(k)/k equal to (sigma(d)+1)/d), or -1 if no such k exists.

Original entry on oeis.org

18, 54, 196, 1521, 486, 1372, 15376, 24025, 4374, 1032256, 67228, 39366, 476656, 257049, 744775, 27237961, 354294, 3341637, 3294172, 23088025, 870899121, 131096512, 3188646, 4990433449, 18837288001, 28697814, 43647148561, 458066416, 161414428, 131785698529, 8362054027, 274810802176, 564736653, 508339054441, 258280326
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2025

Keywords

Links

Crossrefs

Cf. A006530, A240991, A253560, A387406.

Programs

  • PARI
    A387405(n) = { my(d=A240991(n), r=(sigma(d)+1)/d); forstep(k=d,oo,d,if(sigma(k)/k==r, return(k))); }

Formula

a(n) = A253560(A240991(n)) = A006530(A240991(n)) * A240991(n). - Conjectured, most likely true, especially if A240991 is a subsequence of A387406.

A240923 a(n) = numerator(sigma(n)/n) - sigma(denominator(sigma(n)/n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 3, 0, 4, 2, 0, 0, 1, 0, 3, 0, 6, 0, 2, 0, 7, 0, 1, 0, 6, 0, 0, 4, 9, 0, 0, 0, 10, 0, 2, 0, 8, 0, 9, 2, 12, 0, 3, 0, 0, 6, 7, 0, 7, 0, 7, 0, 15, 0, 8, 0, 16, 0, 0, 0, 12, 0, 9, 8, 24, 0, 5, 0, 19, 0, 15, 0, 14, 0, 3, 0, 21, 0
Offset: 1

Views

Author

Michel Marcus, Aug 03 2014

Keywords

Comments

a(n) is the integer t, such that if sigma(n)/n is written in its reduced form, nk/dk = A017665(n)/A017666(n), then we have (sigma(dk)+t)/dk.
It appears that a(n) is never negative.
a(n) = 0 if and only if n is in A014567 (n and sigma(n) are relatively prime).
That a(n) >= 0 is easily seen from the formula a(n) = sigma(n)/A009194(n) - sigma(n/A009194(n)), where A009194(n) = gcd(n,sigma(n)). - Antti Karttunen, Aug 30 2025

Examples

			For n=10, sigma(10)/10 = 18/10 = 9/5 = (sigma(5) + 3)/5, hence a(10)=3.

Links

Crossrefs

Cf. A000203, A009194, A017665, A017666, A240990, A272008.
Cf. A014567 (positions of 0's), A240991 (positions of 1's).

Programs

  • Haskell
    import Data.Ratio ((%), numerator, denominator)
a240923 n = numerator sq - a000203 (denominator sq)
   where sq = a000203 n % n
-- Reinhard Zumkeller, Aug 05 2014
  • Maple
    with(numtheory): A240923:=n->numer(sigma(n)/n) - sigma(denom(sigma(n)/n)): seq(A240923(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2014
  • Mathematica
    Table[Numerator[DivisorSigma[1, n]/n] - DivisorSigma[1, Denominator[ DivisorSigma[1, n]/n]], {n, 100}] (* Wesley Ivan Hurt, Aug 06 2014 *)
  • PARI
    a(n) = my(ab = sigma(n)/n); numerator(ab) - sigma(denominator(ab));
  • PARI
    A240923(n) = { my(s=sigma(n), g=gcd(s,n)); ((s/g) - sigma(n/g)); }; \\ Antti Karttunen, Aug 30 2025
  • Python
    from gmpy2 import mpq
from sympy import divisors
map(lambda x: x.numerator-sum(divisors(x.denominator)),[mpq(sum(divisors(n)),n) for n in range(1,10**5)]) # Chai Wah Wu, Aug 05 2014

Formula

a(n) = A017665(n) - sigma(A017666(n)) = sigma(n)/A009194(n) - sigma(n/A009194(n)). - Antti Karttunen, Aug 30 2025

A387406 Numbers k such that sigma(A253560(k)) / A253560(k) is equal to (sigma(k)+1) / k, where A253560(k) = k multiplied by its largest prime factor.

Original entry on oeis.org

6, 18, 28, 54, 117, 162, 196, 486, 496, 775, 1372, 1458, 1521, 4374, 8128, 9604, 13122, 15376, 19773, 24025, 39366, 67228, 88723, 118098, 257049, 354294, 470596, 476656, 744775, 796797, 1032256, 1062882, 2896363, 3188646, 3294172, 3341637, 6725201, 9565938, 12326221, 14776336, 23059204, 23088025, 25774633, 27237961
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2025

Keywords

Comments

Terms k for which sigma(k/A053585(k)) = A006530(k). This further entails that A001221(k) = 2 [See A023194].

Links

Crossrefs

Cf. A000203, A001221, A006530, A023194, A053585, A253560, A387405.
Subsequences: A000396 (even terms only), A240991 (conjectured, if true, then A000396 has only even terms).

Programs

  • Mathematica
    fk[k_]:=k*FactorInteger[k][[-1,1]];Select[Range[10^6],DivisorSigma[1,fk[#]]/fk[#]==(DivisorSigma[1,#]+1)/#&] (* James C. McMahon, Aug 31 2025 *)
  • PARI
    A253560(n) = if (n==1, 1, n*vecmax(factor(n)[, 1]));
isA387406(n) = { my(x=A253560(n)); ((sigma(x)/x) == ((sigma(n)+1)/n)); };
Showing 1-3 of 3 results.

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

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