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

1,3

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));

Links

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

Formula

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

Example

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;
  ...

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 *)

Prog

(PARI) row(n) = apply(sigma, divisors(n)); \\ Michel Marcus, Dec 27 2021
(Magma) [[SumOfDivisors(d): d in Divisors(n)]: n in [1..20]]; // Vincenzo Librandi, Sep 08 2017

Crossrefs

Cf. A000203, A027750, A290532.

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

Sequence in context: A325610 A278536 A143825 * A240698 A010602 A120731

Adjacent sequences: A290475 A290476 A290477 * A290479 A290480 A290481

Keyword

nonn,tabf

Author

Michel Lagneau, Aug 03 2017

Status

approved