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 1-4 of 4 results.

A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

Original entry on oeis.org

1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

The corresponding quotients are 1, 3, 3, 5, 7, 7, 7, 7, 11, 13, 13, 13, 13, ...
a(14) > 7.5*10^10, if it exists. - Amiram Eldar, Jun 04 2021

Examples

			For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.

Crossrefs

Cf. A064608.
Analogous "integer-mean" sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A048290, A063986, A063971, A064911, A062982, A045345.

Programs

  • Mathematica
    s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)

Formula

{n: A064608(n) == 0 (mod n)}.

Extensions

a(10)-a(13) from Donovan Johnson, Jul 20 2012

A145191 Numbers m such that Sum_{i=1..m} omega(i)^2 is divisible by m, where omega is A001221.

Original entry on oeis.org

1, 20, 68, 903, 3876, 3890, 19096, 19122, 19127, 110990, 111004, 111007, 111010, 111013, 774276, 774277, 774278, 774279, 774303, 774313, 774314, 774315, 6615593, 70607550, 70607559, 959878582, 959878737, 959878753, 959878836, 959878846, 959878888, 959878902, 959878914
Offset: 1

Views

Author

Ctibor O. Zizka, Oct 03 2008

Keywords

Comments

If for some m is c square, then we have RootMeanSquare(omega(1),...,omega(n)) = c.

Crossrefs

Cf. A001221, A063971, A013939, A140480.

Programs

  • Mathematica
    With[{max = 10^5}, Position[Accumulate[PrimeNu[Range[max]]^2]/Range[max], ?IntegerQ] // Flatten] (* _Amiram Eldar, Sep 22 2024 *)
  • PARI
    isok(m) = !frac(sum(i=1, m, omega(i)^2)/m); \\ Michel Marcus, Mar 15 2022
  • PARI
    lista(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", ")););} \\ Michel Marcus, Mar 16 2022
  • PARI
    listaa(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", "));); for (m=1, 100, last = v[nn]; v = vector(nn, k, omega(k+m*nn)^2); v[1] += last; for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/(m*nn+n)), print1(n+m*nn, ", "));););} \\ Michel Marcus, Mar 16 2022

Extensions

a(7)-a(9) from Michel Marcus, Mar 15 2022
a(10)-a(25) from Michel Marcus, Mar 16 2022
a(26)-a(33) from Amiram Eldar, Sep 22 2024

A309272 Numbers m such that m divides A173290(m) = Sum_{k=1..m} psi(k), where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 5, 15, 31, 40, 66, 81, 315, 966, 1398, 1768, 30166, 32335, 98734, 388033, 591597, 1375056, 14966304, 15160528, 50793208, 51302236, 99253376, 110994356, 230465053, 402340268, 497982399, 2027319577, 2879855394, 18450762682, 29922126368, 31711273834, 40583934786
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2019

Keywords

Comments

The corresponding quotients are 1, 2, 4, 12, 24, 31, 51, 62, 240, 735, 1063, 1344, 22924, 24572, 75029, 294870, 449560, 1044918, 11373028, 11520620, 38598210, 38985025, 75423522, 84345597, 175132440, 305741942, 378421246, 1540578144, 2188427680, 14020898356, 22738089456, 24097678498, 30840092321, ...

Examples

			2 is in the sequence since psi(1) + psi(2) = 1 + 3 = 4 is divisible by 2.
5 is in the sequence since psi(1) + psi(2) + ... + psi(5) = 1 + 3 + 4 + 6 + 6 = 20 is divisible by 5.

Crossrefs

Cf. A001615, A173290.
Cf. A045345, A048290, A050226, A056550, A063971, A064605, A064606, A064607, A064610, A064611, A064612, A306950, A307043, A307161, A326488.

Programs

  • Mathematica
    psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); seq = {}; s = 0; Do[s += psi[n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^4}]; seq

Extensions

a(31)-a(33) from Giovanni Resta, Oct 24 2019

A339009 Numbers k such that the average number of odd divisors of {1..k} is an integer.

Original entry on oeis.org

1, 2, 165, 170, 1274, 9437, 69720, 69732, 69734, 69736, 515230, 515236, 515246, 28132043, 28132063, 28132079
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 18 2020

Keywords

Comments

Numbers k that divide A060831(k) where A060831(k) = Sum_{j=1..k} A001227(j).
The sequence also includes: 83860580242, 4578632504347, 4578632504465, 4578632504515. - Daniel Suteu, Nov 24 2020

Examples

			165 is in the sequence because the average number of odd divisors of {1..165} is an integer: A060831(165) / 165 = 495 / 165 = 3.

Links

Crossrefs

Cf. A001227, A048290, A050226, A060831, A063971, A064610, A064612.

Programs

  • Mathematica
    s[n_] := Module[{c = 0, k = 1, sum = 0, seq = {}}, While[c < n, sum += DivisorSigma[0, k/2^IntegerExponent[k, 2]]; If[Divisible[sum, k], c++; AppendTo[seq, k]]; k++]; seq]; s[13] (* Amiram Eldar, Nov 18 2020 *)
  • PARI
    f(n) = my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2; \\ A060831
isok(k) = (f(k) % k) == 0; \\ Michel Marcus, Nov 25 2020
Showing 1-4 of 4 results.

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