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.
8 is a term since 8*tau(8) - sigma(8) = 8*4 - 15 = 32 - 15 = 17 is a prime.
A094471:= n -> n*numtheory:-tau(n) - numtheory:-sigma(n): select(t -> isprime(A094471(t)), 2*[3/2,$1..10^6]); # Robert Israel, Nov 11 2015
Do[s=n*DivisorSigma[0, n]-DivisorSigma[1, n]; If[PrimeQ[s], Print[{n, s}]; ta[[u]]=n; tb[[u]]=s; u=u+1], {n, 1, 1000000}]; ta
Select[Range[215000],PrimeQ[# DivisorSigma[0,#]-DivisorSigma[1,#]]&] (* Harvey P. Dale, Dec 07 2021 *)
q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; PrimeQ[n * Times @@ (e + 1) - Times @@ ((p^(e + 1) - 1)/(p - 1))]]; seq[lim_] := Module[{m1 = Floor[Sqrt[lim/2]], m2 = Floor[Sqrt[lim]/2]}, Join[{3}, Union[Select[2*Range[m1]^2, q], Select[4*Range[m2]^2, q]]]]; seq[220000] (* Amiram Eldar, Feb 14 2025 *)
isok(n) = isprime(n*numdiv(n)-sigma(n)); \\ Michel Marcus, Nov 12 2015
isok(k) = if(k % 2, k == 3, if(!issquare(k) && !issquare(2*k), 0, my(f = factor(k)); isprime(k * numdiv(f) - sigma(f)))); \\ Amiram Eldar, Feb 14 2025
Table[(n + 1) DivisorSigma[0, n] - DivisorSigma[1, n], {n, 100}] (* Wesley Ivan Hurt, Nov 15 2021 *)
a(n)=if(n<1,0,polcoeff(sum(k=1,n,sum(l=1,k,1/(1-x^l)),x*O(x^n)),n))
a(n)=sum(j=1, n, sum(k=1, j, n%k==0)) \\ Hugo Pfoertner, Jul 09 2025
For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 + 7 = 19. - _Omar E. Pol_, Nov 30 2019
Array[Total[{#, 1} DivisorSigma[{0, 1}, #]] &, 48] (* Michael De Vlieger, Dec 01 2019 *)
a(n) = n*numdiv(n) + sigma(n) \\ Michel Marcus, Jun 02 2013
First few rows of the array: 1, 1, 1, 1, 1, 1, 1, ... 1, 3, 1, 3, 1, 3, 1, ... 1, 1, 4, 1, 1, 4, 1, ... 1, 3, 1, 7, 1, 3, 1, ... 1, 1, 1, 1, 6, 1, 1, ... ... First antidiagonals: 1; 1, 1; 1, 3, 1; 1, 1, 1, 1; 1, 3, 4, 3, 1; 1, 1, 1, 1, 1, 1; 1, 3, 1, 7, 1, 3, 1; 1, 1, 4, 1, 1, 4, 1, 1; 1, 3, 1, 3, 6, 3, 1, 3, 1; ...
Table[DivisorSigma[1, GCD[#, k]] &[n - k + 1], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 19 2022 *)
T(n, k) = sigma(gcd(n, k)); \\ Michel Marcus, Dec 19 2022
using AbstractAlgebra function A367326(n) sum(k^2 for k in 0:n if is_divisible_by(n, n - k)) end [A367326(n) for n in 0:48] |> println
divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0): a := n -> local d; add(`if`(divides(n - d, n), d^2, 0), d = 0..n-1): seq(a(n), n = 0..61);
a[0] = 0; a[n_] := DivisorSum[n, (n - #)^2 &]; Array[a, 50, 0] (* Amiram Eldar, Nov 14 2023 *) a[n_] := Sum[If[Divisible[n, n - k], k^2, 0], {k, 0, n - 1}] Table[a[n], {n, 0, 48}] (* Peter Luschny, Nov 14 2023 *)
a(n) = if(n == 0, 0, sumdiv(n, d, (n-d)^2)); \\ Michel Marcus, Nov 14 2023
from math import prod from sympy import factorint def A367326(n): f = factorint(n).items() return n*(n*prod(e+1 for p,e in f) - (prod((p**(e+1)-1)//(p-1) for p,e in f)<<1))+prod((p**(e+1<<1)-1)//(p**2-1) for p,e in f) if n else 0 # Chai Wah Wu, Nov 14 2023
def A367326(n): return sum(k^2 for k in (0..n) if (n-k).divides(n)) print([A367326(n) for n in range(50)])
Written as a triangle starting from n = 2: 1; 2; 3, 2; 4; 5, 4, 3; 6; 7, 6, 4; 8, 6; 9, 8, 5; 10; 11, 10, 9, 8, 6; 12;
with (numtheory):
T:= n-> map(x-> n-x, sort([(divisors(n) minus {n})[]]))[]:
seq (T(n), n=2..50); # Alois P. Heinz, Apr 11 2012
T[n_] := Most[n-Divisors[n]]; Table[T[n], {n, 2, 50}] // Flatten (* Jean-François Alcover, Feb 21 2017 *)
Do[s=n*DivisorSigma[0, n]-DivisorSigma[1, n];If[PrimeQ[s], Print[{n, s}];ta[[u]]=n;tb[[u]]=s;u=u+1], {n, 1, 1000000}];ta
s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; n * Times @@ (e + 1) - Times @@ ((p^(e + 1) - 1)/(p - 1))]; q[n_] := PrimeQ[s[n]]; seq[lim_] := Module[{m1 = Floor[Sqrt[lim/2]], m2 = Floor[Sqrt[lim]/2]}, s /@ Join[{3}, Union[Select[2*Range[m1]^2, q], Select[4*Range[m2]^2, q]]]]; seq[200000] (* Amiram Eldar, Feb 14 2025 *)
using AbstractAlgebra function A367326(n) sum(k for k in 0:n if ! is_divisible_by(n, n - k)) end [A367326(n) for n in 0:54] |> println
# Warning: Be careful when using the deprecated 'numtheory' package. # It might not handle the case n = 0 correctly. A better solution is: divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0): A367368 := n -> local k; add(`if`(divides(n - k, n), 0, k), k = 0..n): seq(A367368(n), n = 0..61);
a[n_]:=n+Sum[k*Boole[!Divisible[n,n-k]],{k,0,n-1}]; Array[a,55,0] (* Stefano Spezia, Nov 15 2023 *)
def divides(k, n): return k == n or ((k > 0) and (n % k == 0)) def A367368(n): return sum(k for k in range(n + 1) if not divides(n - k, n)) print([A367368(n) for n in range(55)])
from math import prod from sympy import factorint def A367368(n): f = factorint(n).items() return (n*(n+1)>>1)-n*prod(e+1 for p,e in f)+prod((p**(e+1)-1)//(p-1) for p,e in f) if n else 0 # Chai Wah Wu, Nov 17 2023
def A367368(n): return sum(k for k in (0..n) if not (n - k).divides(n)) print([A367368(n) for n in range(55)])
a[n_]:=Sum[(n-d)^n,{d,Divisors[n]}]; Array[a,20] (* Stefano Spezia, Nov 20 2023 *)
from sympy import divisors def A367493(n): return sum((n-d)**n for d in divisors(n, generator=True))
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