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.

A378210 Dirichlet inverse of A191150, hypersigma variant 1.

Original entry on oeis.org

1, -3, -4, -1, -6, 5, -8, 5, -1, 9, -12, 22, -14, 13, 14, 7, -18, 28, -20, 32, 20, 21, -24, -1, -1, 25, 11, 42, -30, 51, -32, -3, 32, 33, 34, -28, -38, 37, 38, -9, -42, 63, -44, 62, 49, 45, -48, -76, -1, 56, 50, 72, -54, -6, 54, -17, 56, 57, -60, -182, -62, 61, 63, -17, 64, 87, -68, 92, 68, 75, -72, -183, -74, 73
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2024

Keywords

Links

Crossrefs

Cf. A191150.
Cf. also A378211.

Programs

  • PARI
    A191150(n) = (sigma(n)+sumdiv(n,d,if((d>1)&&(dA191150(d), 0)));
memoA378210 = Map();
A378210(n) = if(1==n,1,my(v); if(mapisdefined(memoA378210,n,&v), v, v = -sumdiv(n,d,if(dA191150(n/d)*A378210(d),0)); mapput(memoA378210,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA191150(n/d) * a(d).

A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

Original entry on oeis.org

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55
Offset: 1

Views

Author

Alonso del Arte, May 26 2011

Keywords

Comments

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Links

Crossrefs

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).
Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

Programs

  • Mathematica
    hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)
  • PARI
    a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

Formula

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011
From Antti Karttunen, Nov 22 2024: (Start)
Following formulas were conjectured by Sequence Machine:
For n > 1, a(n) = A191150(n) + A074206(n).
a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.
a(n) = Sum_{d|n} A330575(d).
a(n) = Sum_{d|n} d*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A074206(n/d).
a(n) = Sum_{d|n} A007429(d)*A174725(n/d).
a(n) = Sum_{d|n} A000010(d)*A253249(n/d).
a(n) = Sum_{d|n} A038040(d)*A323912(n/d).
a(n) = Sum_{d|n} A060640(d)*A323910(n/d).
(End)

A192279 Anti-hypersigma(n): sum of the anti-divisors of n plus the recursive sum of the anti-divisors of the anti-divisors until 2 is reached.

Original entry on oeis.org

2, 5, 7, 9, 19, 17, 17, 40, 33, 37, 45, 40, 67, 49, 89, 96, 65, 88, 71, 134, 127, 91, 189, 120, 187, 170, 91, 166, 151, 219, 243, 164, 261, 140, 315, 392, 233, 310, 247, 374, 245, 150, 461, 280, 285, 347, 407, 468, 215, 538, 515, 234, 565, 422, 609, 532, 495
Offset: 3

Views

Author

Paolo P. Lava, Jul 13 2011

Keywords

Comments

Similar to A191150 but using anti-divisors. The recursion is stopped when 2 is reached because 2 has no anti-divisors.

Examples

			n=14 -> anti-divisors are 3,4,9. We start with 3+4+9=16.
Now for 3, 4 and 9 we repeat the procedure:
3-> 2 -> no anti-divisors. To add: 2.
4-> 3 -> 2 -> no anti-divisors. To add: 3+2=5.
9-> 2,6. To add: 2+6=8.
--- 2 -> no anti-divisors.
--- 6 -> 4 -> 3 -> 2 -> no anti-divisors. To add: 4+3+2=9.
Total is 16+2+5+8+9=40.

Links

Crossrefs

Cf. A066272, A191150.

Programs

  • Maple
    with(numtheory);
P:=proc(n)
local a,b,c,k,s;
a:={};
for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a:=a union {k}; fi;
od;
b:=nops(a); c:=op(a); s:=0;
if b>1 then
  for k from 1 to b do s:=s+c[k]; od;
else s:=c;
fi;
b:=nops(a); c:=(sort([op(a)]));
for k from 1 to b do if c[k]>2 then s:=s+P(c[k]); fi; od;
s;
end:
Antihps:=proc(i)
local n;
for n from 1 to i do print(P(n)); od;
end:

A237428 Numbers k with following property: List all proper divisors of k. Replace any composite number in the list with its proper divisors. Repeat. Sum of remaining numbers (1's and primes) is equal to k.

Original entry on oeis.org

6, 126, 3808, 19360, 104320, 4317184, 126764640, 1920554064, 3710312448
Offset: 1

Views

Author

Lechoslaw Ratajczak, Feb 07 2014

Keywords

Comments

Is there a largest term? Is there any odd term?
a(8) if it exists is greater than 10^9. - Giovanni Resta, Feb 07 2014
Composite numbers k such that k = A074206(k) + Sum_{p|k} (p-1)*A074206(k/p). - Charlie Neder, Jun 02 2019

Examples

			6 is a term because: 1 + 2 + 3 = 6.
126 is a term because: [1 + 2 + 3 + (6 - 6) + 7 + (9 - 9) + (14 - 14) + (18 - 18) + (21 - 21) + (42 - 42) + (63 - 63)] + [1 + 2 + 3] + [1 + 3] + [1 + 2 + 7] + [1 + 2 + 3 + (6 - 6) + (9 - 9)] + [1 + 3 + 7] + [1 + 2 + 3 + (6 - 6) + 7 + (14 - 14) + (21 - 21)] + [1 + 3 + 7 + (9 - 9) + (21 - 21)] + [1 + 2 + 3] + [1 + 3] + [1 + 2 + 3] + [1 + 2 + 7] + [1 + 3 + 7] + [1 + 3] + [1 + 3 + 7] = 126.

Crossrefs

Cf. A000203, A000396, A001065, A002033, A032741, A074206, A191150.

Programs

  • Mathematica
    v[n_] := If[PrimeQ@n, 1, Block[{s = Sum[If[e == 1 || PrimeQ@e, e, v@e], {e, Most@ Divisors@n}]}, If[n < 1000, v[n] = s, s]]]; Select[Range@ 20000, # == v@# &] (* Giovanni Resta, Feb 07 2014 *)

Extensions

a(6)-a(7) from Giovanni Resta, Feb 07 2014
a(8)-a(9) from Amiram Eldar, Jun 28 2025
Showing 1-4 of 4 results.

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