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.

A378314 a(n) = number of subsets of {1, 2, ..., n} that represent the first k divisors of m for some positive integers m and 1 <= k <= A000005(m).

Original entry on oeis.org

0, 1, 2, 4, 6, 12, 16, 28, 36, 52, 70, 118, 150, 246, 318, 382, 430, 670, 798, 1278, 1566, 1886, 2270, 3230, 3742, 4702, 5854, 6814, 7966, 11806, 14878, 22558, 25630, 32542, 40222, 46366, 52510, 70942, 86302, 100126, 112414, 149278, 172318, 246046, 295198, 344350, 405790, 553246, 626974, 774430, 897310
Offset: 0

Views

Author

Max Alekseyev, Nov 22 2024

Keywords

Examples

			a(4) = 6 enumerates subsets {1}, {1,2}, {1,2,3}, {1,2,4}, {1,2,3,4}, {1,3}.

Crossrefs

Partial sums of A190940.
Cf. A001248, A027750, A064510, A180851, A194269, A307137, A378313.

Programs

  • PARI
    a378314(n) = my(L=1); sum(i=1,n, L=lcm(L,i); sigma(L/i));

Formula

a(n) = Sum_{i=1..n} sigma(LCM(1,2,...,i)/i).

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

A264786 Let { d_1, d_2, ..., d_k } be the divisors of n. Then a(n) = d_k^1 + d_(k-1)^2 + ... + d_1^k.

Original entry on oeis.org

1, 3, 4, 9, 6, 24, 8, 33, 19, 44, 12, 226, 14, 72, 68, 161, 18, 429, 20, 534, 98, 152, 24, 3858, 51, 204, 136, 856, 30, 6534, 32, 1089, 182, 332, 210, 22965, 38, 408, 236, 12886, 42, 14262, 44, 2148, 1868, 584, 48, 128338, 99, 2333, 368, 3214, 54, 21810, 302
Offset: 1

Views

Author

Carlos Eduardo Olivieri, Nov 29 2015

Keywords

Examples

			For n = 4: a(4) = 4^1 + 2^2 + 1^3 = 9.
For n = 5: a(5) = 5^1 + 1^2 = 6.
For n = 6: a(6) = 6^1 + 3^2 + 2^3 + 1^4 = 24.

Links

Crossrefs

Cf. A027750, A180851.

Programs

  • Mathematica
    a[n_] := Sum[Sort[Divisors[n], #1 > #2 &][[i]]^i, {i, DivisorSigma[0, n]}]; Table[a[n], {n, 60}]
  • PARI
    a(n) = my(d = divisors(n)); sum(k=1, #d, d[k]^(#d-k+1)); \\ Michel Marcus, Jan 01 2016
Showing 1-4 of 4 results.

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