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 1-5 of 5 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.

A065997 Numbers n such that sigma(n) / n is prime.

Original entry on oeis.org

6, 28, 120, 496, 672, 8128, 523776, 33550336, 459818240, 1476304896, 8589869056, 14182439040, 31998395520, 51001180160, 137438691328, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 2305843008139952128
Offset: 1

Views

Author

Joseph L. Pe, Dec 10 2001

Keywords

Comments

This is a subsequence of the sequence of multiply perfect numbers A007691.
The prime values of sigma(n) / n are A219545.
Numbers whose abundancy index is a prime. There are two visible bends (sudden changes in the growth rate) in the scatter plot. Compare also to the scatter plot of A336702. - Antti Karttunen, Feb 25 2022

Links

Crossrefs

Subsequence of A007691 and of A342924.
Cf. A000396, A005820, A046060 (subsequences).
Cf. A219545, A336702.

Programs

  • PARI
    isA065997(n) = { my(p=sigma(n)/n); (1==denominator(p) && isprime(p)); }; \\ Antti Karttunen, Feb 25 2022

Extensions

Terms a(10) to a(14) from Jonathan Sondow, Nov 22 2012
Extended by T. D. Noe, Nov 26 2012

A342922 Numbers k such that A342925(k) = k + 2*A003415(k).

Original entry on oeis.org

6, 28, 29, 496, 857, 1721, 8128, 164284, 6511664, 33550336, 400902412, 8589869056
Offset: 1

Views

Author

Antti Karttunen, Apr 07 2021

Keywords

Comments

Question: Are all odd terms in A001359?
Certainly any prime p such that A003415(p+1) = p + 2 satisfies the equation. See comments in A007850.

Links

Crossrefs

Cf. A000396 (subsequence), A001359, A003415, A007850.
Cf. A203617, A342923, A342924, A342925.

Programs

  • Mathematica
    Select[Range[2*10^5], #3 == #1 + 2 #2 & @@ Prepend[Map[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &, {#, DivisorSigma[1, #]}], #] &] (* Michael De Vlieger, Feb 25 2022 *)
  • 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));
isA342922(n) = (A342925(n)==(n+(2*A003415(n))));

Extensions

Terms a(11) and a(12) from Antti Karttunen, Feb 25 2022

A347884 Odd composites k for which A003415(sigma(k))-k is strictly positive and a multiple of A003415(k). Here A003415 is the arithmetic derivative.

Original entry on oeis.org

963, 969, 5871, 10479, 2308203, 41240261, 52024391, 69989429, 75384301, 319255721, 634457761, 781718149, 1184197307, 1190942957, 1195786661, 2114464203
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Odd nonprimes k for which A343223(k) = A003415(k).
Any odd terms of A065997, including odd perfect numbers, odd triperfect numbers and odd 5-multiperfect numbers, should occur in this sequence, if such numbers exist at all.
The odd terms present in A342924 form a subsequence of this sequence.

Links

Crossrefs

Cf. A000203, A003415, A007691, A065997, A342925, A342926, A343223.
Subsequence of A343218.
Cf. also A000396, A005820, A046060, A342922, A342923, A342924, A347872, A347880.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 2.5*10^6, 2], CompositeQ[#] && (d = ad[DivisorSigma[1, #]] - #) > 0 && Divisible[d, ad[#]] &] (* 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]));
isA347884(n) = if(!(n%2)||isprime(n),0,my(u=(A003415(sigma(n))-(n))); ((u>0)&&!(u%A003415(n))));
Showing 1-5 of 5 results.

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