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 1-6 of 6 results.

A342925 a(n) = A003415(sigma(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

0, 1, 4, 1, 5, 16, 12, 8, 1, 21, 16, 32, 9, 44, 44, 1, 21, 16, 24, 41, 80, 60, 44, 92, 1, 41, 68, 92, 31, 156, 80, 51, 112, 81, 112, 20, 21, 92, 92, 123, 41, 272, 48, 124, 71, 156, 112, 128, 22, 34, 156, 77, 81, 244, 156, 244, 176, 123, 92, 332, 33, 272, 164, 1, 124, 384, 72, 165, 272, 384, 156, 119, 39, 101, 128, 188
Offset: 1

Views

Author

Antti Karttunen, Apr 07 2021

Keywords

Links

Crossrefs

Cf. A023194 (positions of ones, which is a subsequence of prime powers, A000961).
Cf. A342922, A342923, A342924, A342926, A351568, A351569, A351570, A351571.
Cf. A342021 (fixed points), A343216 [positions k where a(k) < k], A343217 [a(k) >= k], A343218 [a(k) > k].
Cf. A347870 (parity of terms), A347872, A347873, A347877 (positions of odd terms), A347878 (of even terms), A343218, A343220, A344024.

Programs

  • Mathematica
    Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ DivisorSigma[1, #] &, 76] (* Michael De Vlieger, Apr 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));

Formula

a(A023194(n)) = 1.
If gcd(m,n) = 1, a(m*n) = sigma(m)*A003415(sigma(n)) + sigma(n)*A003415(sigma(m)) = sigma(m)*a(n) + sigma(n)*a(m).
a(n) = (A351568(n)*A351571(n)) + (A351569(n)*A351570(n)). - Antti Karttunen, Feb 23 2022

A342926 a(n) = A003415(sigma(n)) - n, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

-1, -1, 1, -3, 0, 10, 5, 0, -8, 11, 5, 20, -4, 30, 29, -15, 4, -2, 5, 21, 59, 38, 21, 68, -24, 15, 41, 64, 2, 126, 49, 19, 79, 47, 77, -16, -16, 54, 53, 83, 0, 230, 5, 80, 26, 110, 65, 80, -27, -16, 105, 25, 28, 190, 101, 188, 119, 65, 33, 272, -28, 210, 101, -63, 59, 318, 5, 97, 203, 314, 85, 47, -34, 27, 53, 112, 195
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Links

Crossrefs

Cf. A342925, A342924, A343223 [= gcd(A003415(n), a(n))].
Cf. A342021 (positions of 0's), A343216 (of negative terms), A343217 (of nonnegative terms), A343218 (of positive terms).

Programs

  • Mathematica
    Array[If[#2 < 2, 0, #2 Total[#2/#1 & @@@ FactorInteger[#2]]] - #1 & @@ {#, DivisorSigma[1, #]} &, 77] (* Michael De Vlieger, Apr 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);

Formula

a(n) = A342925(n) - n = A003415(A000203(n)) - n.

A343217 Numbers k such that A003415(sigma(k)) >= k, where A003415(x) gives the arithmetic derivative of x.

Original entry on oeis.org

3, 5, 6, 7, 8, 10, 11, 12, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Links

Crossrefs

Cf. A000203, A003415, A342925, A343216 (complement).
Disjoint union of A342021 and A343218.
Positions of nonnegative terms in A342926.

Programs

  • Mathematica
    Select[Range[100], If[#2 < 2, 0, #2 Total[#2/#1 & @@@ FactorInteger[#2]]] >= #1 & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Apr 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA343217(n) = (A003415(sigma(n))>=n);

A347890 Odd numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.

Original entry on oeis.org

245025, 540225, 893025, 2205225, 3080025, 4862025, 6125625, 6890625, 7868025, 10989225, 13505625, 14402025, 19847025, 22896225, 23474025, 26471025, 27720225, 29648025, 43758225, 45765225, 55130625, 57836025, 60140025, 65367225, 70812225, 72335025, 76475025, 77000625, 94770225, 121550625, 153140625, 156125025
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Odd numbers k such that A033880(k) is positive but A342926(k) is negative.
This is a subsequence of A156942, "odd abundant numbers whose abundance is odd". Proof: If sigma(k) > 2*k, and sigma(k) were even, then sigma(k)/2 would be an integer and a divisor of sigma(k), and we could compute A003415(sigma(k)) as A003415(2)*(sigma(k)/2) + 2*A003415(sigma(k)/2) by the definition of the arithmetic derivative. But that value is certainly larger than k, because sigma(k)/2 > k, therefore sigma(k) must be an odd number, with also its abundance sigma(k)-(2k) odd. This also entails that all terms are squares. See A347891 for the square roots.
The first term that is not a multiple of 25 is a(146) = 6800806089 = 82467^2.
This is not a subsequence of A325311. The first term that is not present there is a(5) = 3080025.

Links

Crossrefs

Intersection of A005231 and A343216.
Subsequence of A016754, of A156942 and of A347889 (its odd terms).
Cf. A000203, A003415, A033880, A325311, A342926, A347891 (the square roots).

Programs

  • PARI
    \\ Using the program given in A347891 would be much faster than this:
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347890(n) = ((n%2)&&(A003415(sigma(n))(2*n)));

A347889 Numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.

Original entry on oeis.org

18, 36, 100, 144, 324, 400, 576, 784, 900, 1296, 1458, 1600, 1936, 2304, 2500, 2704, 2916, 3136, 3600, 4624, 5184, 5202, 5776, 6400, 7744, 8464, 9216, 9604, 10000, 10404, 10816, 11664, 12100, 13122, 13456, 14400, 15376, 17424, 18496, 19044, 23104, 25600, 26244, 28900, 30258, 30276, 30976, 32400, 33856, 36864, 38416
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Numbers k such that A033880(k) is positive but A342926(k) is negative.

Links

Crossrefs

Intersection of A005101 and A343216. Subsequence A347890 gives the odd terms.
Cf. A000203, A003415, A033880, A342926.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 40000], DivisorSigma[1, #] > 2*# && ad[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347889(n) = ((A003415(sigma(n))(2*n)));

A347891 Odd numbers k such that sigma(k^2) > 2*k^2 and A003415(sigma(k^2)) < k^2.

Original entry on oeis.org

495, 735, 945, 1485, 1755, 2205, 2475, 2625, 2805, 3315, 3675, 3795, 4455, 4785, 4845, 5145, 5265, 5445, 6615, 6765, 7425, 7605, 7755, 8085, 8415, 8505, 8745, 8775, 9735, 11025, 12375, 12495, 13365, 13965, 14025, 15435, 15795, 16065, 16335, 16905, 17595, 18375, 19845, 20295, 21315, 22185, 22275, 22785, 22815, 23265
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Odd numbers whose square is abundant and present in A343216.
The first term that is not a multiple of 5 is a(146) = 82467.

Links

Crossrefs

Square roots of A347890. Subsequence of A174830.
Cf. A000196, A000203, A003415, A005231, A343216.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 24000, 2], DivisorSigma[1, #^2] > 2*#^2 && ad[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347891(n) = if(!(n%2),0,my(u=n*n); (A003415(sigma(u))(2*u)));

Formula

a(n) = A000196(A347890(n)).
Showing 1-6 of 6 results.

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