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.

A064510 Numbers m such that the sum of the first k divisors of m is equal to m for some k.

Original entry on oeis.org

1, 6, 24, 28, 496, 2016, 8128, 8190, 42336, 45864, 392448, 714240, 1571328, 33550336, 61900800, 91963648, 211891200, 1931236608, 2013143040, 4428914688, 8589869056, 10200236032, 137438691328, 214204956672
Offset: 1

Views

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 06 2001

Keywords

Comments

Obviously all perfect numbers are included in this sequence.
a(25) > 5*10^11. Other than perfect numbers, 104828758917120, 916858574438400, 967609154764800, 93076753068441600, 215131015678525440 and 1371332329173024768 are also terms. - Donovan Johnson, Dec 26 2012
a(25) > 10^12. - Giovanni Resta, Apr 15 2017

Examples

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. 1+2+3+4+6+8 = 24.

Crossrefs

Union of A000396 and A194472.
Cf. A109883, A109884, A109886, A185584, A185960, A190940, A307137, A318528, A378313, A378314.

Programs

  • Mathematica
    subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n]; lst = {}; Do[ If[ f[n] == 0, AppendTo[lst, n]], {n, 10^8}]; lst (* Bobby R. Treat and Robert G. Wilson v, Jul 14 2005 *)
Select[Range[2000000],MemberQ[Accumulate[Divisors[#]],#]&] (* Harvey P. Dale, Mar 22 2012 *)
  • PARI
    isok(n) = {my(d = divisors(n)); my(k = 1); while ((k <= #d) && ((sd = sum(j=1, k, d[j])) != n), k++;); (sd == n);} \\ Michel Marcus, Jan 16 2014

Extensions

More terms from Don Reble, Dec 17 2001
a(19)-a(23) from Donovan Johnson, Aug 31 2008
a(24) from Donovan Johnson, Aug 11 2011

A190940 Number of divisors of LCM(1,2,...,n)/n.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 12, 8, 16, 18, 48, 32, 96, 72, 64, 48, 240, 128, 480, 288, 320, 384, 960, 512, 960, 1152, 960, 1152, 3840, 3072, 7680, 3072, 6912, 7680, 6144, 6144, 18432, 15360, 13824, 12288, 36864, 23040, 73728, 49152, 49152, 61440, 147456, 73728, 147456, 122880
Offset: 1

Views

Author

Naohiro Nomoto, May 24 2011

Keywords

Comments

Also, number of sequences of d1 = 1 < d2 < ... < dk = n for some k >= 1 that are the first k divisors of some integer (cf. A378314). - Max Alekseyev, Nov 22 2024
Also, the number of distinct values taken by lcm(a,a+b,a+b+c,...,n), where positive integers a,b,c,... run over the compositions a+b+c+...=n. - Conjectured by Ridouane Oudra, Aug 24 2019; proved by Max Alekseyev, Nov 22 2024
Proof. It is clear that n | lcm(a,a+b,...,n) | lcm(1,2,...,n). Hence, lcm(a,a+b,...,n) = d*n for some d | lcm(1,2,...,n)/n. We'll show that each such d is achievable. Suppose d*n has prime factorization p1^e1 * ... * pk^ek with p1^e1 < ... < pk^ek. It is clear that pk^ek <= n, and we can take a composition (a,b,c,...) = (p1^e1, p2^e2 - p1^e1, p3^e3 - p2^e2, ..., pk^ek - p(k-1)^e(k-1), n - pk^ek), which delivers lcm(a,a+b,a+b+c,...,n) = p1^e1 * ... * pk^ek = d*n. QED - Max Alekseyev, Nov 22 2024

Examples

			Examples: for n=3 the a(3) = 2 distinct values are 3, 6. The compositions are 3, 1+2, 2+1, and 1+1+1. The values of the lcm are lcm(3)=3, lcm(1,1+2)=3, lcm(2,2+1)=6, and lcm(1,1+1,1+1+1)=6.

Crossrefs

First difference of A378314.
Cf. A000005, A002944, A003418, A056793, A101207.

Programs

  • Maple
    Lpsum := proc(L) local ps,k ; ps := [op(1,L)] ; for i from 2 to nops(L) do ps := [op(ps), op(-1,ps)+op(i,L)] ; end do: ps ; end proc:
A190940 := proc(n) local lc,k,c ; lc := {} ; for k from 1 to n do for c in combinat[composition](n,k) do lc := lc union { ilcm( op(Lpsum(c))) }; end do: end do: nops(lc) ; end proc: # R. J. Mathar, Jun 02 2011
  • Mathematica
    a[n_] := LCM @@@ (Accumulate /@ (Permutations /@ Rest[IntegerPartitions[n]] // Flatten[#, 1]&)) // Union // Length; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Feb 27 2014 *)

Formula

a(n) = A000005(A002944(n)).

Extensions

a(12)-a(20) from R. J. Mathar, Jun 02 2011
a(21)-a(24) from Alois P. Heinz, Nov 03 2011
Edited and terms a(25) onward added by Max Alekseyev, Nov 22 2024

A378313 Numbers m such that such that Sum_{i=1..k} A027750(m,i)^i = m for some k <= A000005(m).

Original entry on oeis.org

1, 130, 135, 288, 5083, 8064, 10130, 374057639685, 3138436947541900183, 5386775810449231243, 74220449392444960903, 153525475816743446263, 1388286039882808958923, 8020029492466254993943, 8756593744534084572523, 16468366959402667137403
Offset: 1

Views

Author

Max Alekseyev, Nov 22 2024

Keywords

Comments

There are many terms of the form 1 + p^2 + q^3 with primes p < q. Next known term not of this form is 1254382690393861635950014154836028.

Crossrefs

Cf. A001248, A027750, A064510, A180851, A190940, A194269, A307137, A378314.

A194269 Numbers j such that Sum_{i=1..k} d(i)^i = j+1 for some k where d(i) is the sorted list of divisors of j.

Original entry on oeis.org

4, 9, 25, 49, 68, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 17500, 18769, 19321, 22201, 22801, 24649, 26569, 27889
Offset: 1

Views

Author

Michel Lagneau, Aug 27 2011

Keywords

Comments

Equivalently, numbers j such that Sum_{i=2..k} A027750(j,i)^i = j for some k.
The majority of these numbers are squares.
The sequence of numbers j such that Sum_{i=1..k} A027750(j,i)^i = j for some k is given by A378313.
All prime squares p^2 (A001248) are terms because the partial sum 1^1 + p^2 satisfy the condition. The terms that are not squares are given by A307137. - Michel Marcus, Mar 25 2019

Examples

			The divisors of 68 are 1, 2, 4, 17, 34, 68; 1^1 + 2^2 + 4^3 = 69, so 68 is a term.

Crossrefs

Cf. A001248, A027750, A064510, A180851, A190940, A307137, A378313, A378314.

Programs

  • Maple
    isA194269 := proc(n) local dgs ,i,k; dgs := sort(convert(numtheory[divisors](n),list)) ; for k from 1 to nops(dgs) do if add(op(i,dgs)^i,i=1..k) = n+1 then return true; end if; end do; false ; end proc:
for n from 1 to 30000 do if isA194269(n) then print(n); end if; end do: # R. J. Mathar, Aug 27 2011
  • PARI
    isok(n) = {my(d=divisors(n), s=0); for(k=1, #d, s += d[k]^k; if(s == n+1, return(1)); if(s > n+1, break););0;} \\ Michel Marcus, Mar 25 2019

Extensions

Edited by Max Alekseyev, Nov 22 2024

A318528 a(n) = least number > 1 that equals the sum of the n-th powers of its first k divisors for some k.

Original entry on oeis.org

6, 130, 36, 41860, 276, 1015690, 2316, 921951940, 20196, 10009766650, 179196, 2387003305930334914, 1602516, 100006103532010, 14381676, 1880100018939820249188604888836, 129271236, 1000003814697527770, 1162785756, 19105043663614041367780, 10462450356, 10000002384185795209930, 94151567436, 226500219158007133816826003223992308820431641700
Offset: 1

Views

Author

Amiram Eldar, Aug 28 2018

Keywords

Comments

a(48) > 10^90. - Max Alekseyev, Jan 17 2025

Examples

			a(2) = 130 since 130 has the divisors 1, 2, 5, 10, ... and 1^2 + 2^2 + 5^2 + 10^2 = 130.

Links

Crossrefs

Cf. A027750, A185584, A185960, A190940, A307137, A378313, A378314.

Programs

  • Mathematica
    a[k_] := Module[{n = 2}, While[! MemberQ[Accumulate[Divisors[n]^k], n], n++]; n]; Do[Print[a[n]], {n, 1, 10}]
  • PARI
    a(n) = for(x=2, oo, my(div=divisors(x), s=0); for(k=1, #div, s=sum(i=1, k, div[i]^n); if(s==x, return(x)))) \\ Felix Fröhlich, Aug 28 2018

Formula

a(n) = 1 + 2^n + 3^n for n = p^k with prime p > 2. - Giovanni Resta, Aug 28 2018
From Charlie Neder, Jan 24 2019: (Start)
a(n) = 1 + 2^n + 3^n for n odd,
a(n) = 1 + 2^n + 5^n + 10^n for n congruent to 2 modulo 4,
a(n) = 1 + 2^n + 4^n + 5^n + 7^n + 10^n + 13^n for n congruent to 4 or 8 modulo 12 and not 16 modulo 20.
All other a(n) contain a term at least 24^n. (End)

Extensions

a(12)-a(24) from Giovanni Resta confirmed by Max Alekseyev, Jan 04 2025
Showing 1-5 of 5 results.

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