A290478 Triangle read by rows in which row n lists the sum of the divisors of each divisor of n.

1, 1, 3, 1, 4, 1, 3, 7, 1, 6, 1, 3, 4, 12, 1, 8, 1, 3, 7, 15, 1, 4, 13, 1, 3, 6, 18, 1, 12, 1, 3, 4, 7, 12, 28, 1, 14, 1, 3, 8, 24, 1, 4, 6, 24, 1, 3, 7, 15, 31, 1, 18, 1, 3, 4, 12, 13, 39, 1, 20, 1, 3, 7, 6, 18, 42, 1, 4, 8, 32, 1, 3, 12, 36, 1, 24, 1, 3, 4, 7

Offset: 1

Author

Michel Lagneau, Aug 03 2017

Keywords

• nonn
• tabf

Comments

Or, in the triangle A027750(n), replace each element with the sum of its divisors.

The row whose index x is a prime power p^m (p prime and m >= 0) is equal to (1, sigma(p), sigma(p^2), ..., sigma(p^(m-1))).

We observe the following properties of row n when n is the product of k distinct primes, k = 1,2,...:

when n = prime(m), row n = (1, prime(m)+1);

when n is the product of two distinct primes p < q, row n = (1, p+1, q+1,(p+1)(q+1));

when n is the product of three distinct primes p < q < r, row n = (1, p+1, q+1, r+1, (p+1)(q+1), (p+1)(r+1), (q+1)(r+1), sigma(p*q*r));

Examples

			Row 6 is (a(11), a(12), a(13), a(14)) = (1, 3, 4, 12) because sigma(A027750(11))= sigma(1) = 1, sigma(A027750(12))= sigma(2) = 3, sigma(A027750(13))= sigma(3) = 4 and sigma(A027750(14)) = sigma(6) = 12.
Triangle begins:
  1;
  1,  3;
  1,  4;
  1,  3,  7;
  1,  6;
  1,  3,  4, 12;
  1,  8;
  1,  3,  7, 15;
  1,  4, 13;
  1,  3,  6, 18;
  ...
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened).

Crossrefs

Cf. A000203, A027750, A290532.

Cf. A007429 (row sums), A206032 (row products).

Programs

Magma

[[SumOfDivisors(d): d in Divisors(n)]: n in [1..20]]; // Vincenzo Librandi, Sep 08 2017

Maple

with(numtheory):nn:=100:
for n from 1 to nn do:
  d1:=divisors(n):n1:=nops(d1):
   for i from 1 to n1 do:
     s:=sigma(d1[i]):
     printf(`%d, `,s):
   od:
od:

Mathematica

Array[DivisorSigma[1, Divisors@ #] &, 24] // Flatten (* Michael De Vlieger, Aug 07 2017 *)

PARI

row(n) = apply(sigma, divisors(n)); \\ Michel Marcus, Dec 27 2021

Formula

a(n) = sigma(A027750(n)).