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
KEYWORD
nonn,tabf
AUTHOR
Michel Lagneau, Aug 03 2017
STATUS
approved