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.

A097019 a(n) = sigma(A097018(n))/prime(n) = A000203(A097018(n))/A000040(n).

Original entry on oeis.org

2, 1, 3, 1, 4, 1, 4, 2, 6, 6, 1, 2, 4, 6, 6, 4, 6, 3, 6, 4, 7, 2, 4, 23, 2, 10, 6, 6, 6, 6, 1, 4, 4, 2, 10, 12, 2, 6, 10, 4, 10, 8, 22, 8, 4, 2, 2, 8, 4, 2, 16, 6, 14, 12, 12, 4, 6, 2, 12, 4, 6, 4, 1, 10, 6, 6, 2, 2, 6, 8, 10, 6, 2, 6, 2, 4, 6, 6, 22, 7, 16, 12, 4, 8, 2, 6, 6, 6, 12, 6, 4, 10, 12, 10, 2
Offset: 1

Views

Author

Labos Elemer, Aug 23 2004

Keywords

Examples

			For n = 11: the quotient of the least number whose sigma is divisible by the 11th prime is A097018(11) = 16, sigma(16) = 31, and prime(11) = 31, so, a(11) = 31/31 = 1.

Links

Crossrefs

Cf. A000040, A000203, A097018.

Programs

  • PARI
    list(len) = {my(v = vector(len), p = primes(len), k = 1, c = 0, s); while(c < len, s = sigma(k); for(i = 1, len, if(v[i] == 0 && !(s % p[i]), c++; v[i] = s/p[i])); k++); v;} \\ Amiram Eldar, Feb 15 2025

A227470 Least k such that n divides sigma(n*k).

Original entry on oeis.org

1, 3, 2, 3, 8, 1, 4, 7, 10, 4, 43, 2, 9, 2, 8, 21, 67, 5, 37, 6, 20, 43, 137, 5, 149, 9, 34, 1, 173, 4, 16, 21, 27, 64, 76, 22, 73, 37, 6, 3, 163, 10, 257, 43, 6, 137, 281, 11, 52, 76, 67, 45, 211, 17, 109, 4, 49, 173, 353, 2, 169, 8, 32, 93, 72, 27, 401, 67
Offset: 1

Views

Author

Alex Ratushnyak, Jul 12 2013

Keywords

Comments

Theorem: a(n) always exists.
Proof: If n is a power of a prime, say n = p^a, then, by Euler's generalization of Fermat's little theorem and the multiplicative property of sigma, one can take k = x^(p^a-p^(a-1)-1) where x is a different prime from p. Similarly. if n = p^a*q^b, then take k = x^(p^a-p^(a-1)-1)*y^(q^b-q^(b-1)-1) where {x,y} are primes different from {p,q}. And so on. These k's have the desired property, and so there is always at least one candidate for the minimal k. - N. J. A. Sloane, May 01 2016

Examples

			Least k such that 9 divides sigma(9*k) is k = 10: sigma(90) = 234 = 9*26. So a(9) = 10.
Least k such that 89 divides sigma(89*k) is k = 1024: sigma(89*1024) = 184230 = 89*2070. So a(89) = 1024.

Links

Crossrefs

Indices of 1's: A007691.
Cf. A000203, A227302, A227303, A097018.
See A272349 for the sequence [n*a(n)]. - N. J. A. Sloane, May 01 2016

Programs

  • Maple
    A227470 := proc(n)
    local k;
    for k from 1 do
        if modp(numtheory[sigma](k*n),n) =0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, May 06 2016
  • Mathematica
    lknds[n_]:=Module[{k=1},While[!Divisible[DivisorSigma[1,k*n],n],k++];k]; Array[lknds,70] (* Harvey P. Dale, Jul 10 2014 *)
  • PARI
    a227470(n) = {k=1; while(sigma(n*k)%n != 0, k++); k} \\ Michael B. Porter, Jul 15 2013

Formula

a(n) = A272349(n)/n. - R. J. Mathar, May 06 2016

A272349 Least multiple of n whose sum of divisors is divisible by n.

Original entry on oeis.org

1, 6, 6, 12, 40, 6, 28, 56, 90, 40, 473, 24, 117, 28, 120, 336, 1139, 90, 703, 120, 420, 946, 3151, 120, 3725, 234, 918, 28, 5017, 120, 496, 672, 891, 2176, 2660, 792, 2701, 1406, 234, 120, 6683, 420, 11051, 1892, 270, 6302, 13207, 528, 2548, 3800, 3417, 2340
Offset: 1

Views

Author

Waldemar Puszkarz, Apr 26 2016

Keywords

Comments

See A227470(n) for the sequence a(n)/n. If n = prime(i) is a prime then A097018 gives the answer: a(n) = n*A097018(i). One can show that a(n) always exists - see A227470 for the proof. - N. J. A. Sloane, May 01 2016

Examples

			For n = 2, a(2) = 6 because it is the smallest number divisible by 2 whose sum of divisors (12) is also divisible by 2; 3 and 5 are not divisible by 2 and the sum of divisors of 2 and 4 is 3 and 7, hence also not divisible by 2.

Links

Crossrefs

Cf. A000203, A097018 (if n is a prime), A227470.

Programs

  • Maple
    A272349 := proc(n)
    local k;
    for k from 1 do
        if modp(numtheory[sigma](k*n),n) =0 then
            return k*n;
        end if;
    end do:
end proc: # R. J. Mathar, May 02 2016
  • Mathematica
    A272349 = {}; Do[k = n; While[!(Divisible[k, n] && Divisible[DivisorSigma[1, k], n]), k++]; AppendTo[A272349, k], {n, 65}]; A272349
  • PARI
    for(n=1, 65, k=n; while(!(k%n==0&&sigma(k)%n==0), k++); print1(k ", "))
  • PARI
    a(n)=my(k=n); while(sigma(k)%n,k+=n); k \\ Charles R Greathouse IV, Apr 28 2016

Formula

a(n) = n*A227470(n). - R. J. Mathar, May 02 2016
Showing 1-3 of 3 results.

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

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