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A290478 Triangle read by rows in which row n lists the sum of the divisors of each divisor of n. 1
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 (list; graph; refs; listen; history; text; internal format)
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));

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

Table of n, a(n) for n=1..80.

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.

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) tabf(nn) = for (n=1, nn, fordiv(n, d, print1(sigma(d), ", ")); print(); ); \\ Michel Marcus, Aug 05 2017

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

CROSSREFS

Cf. A000203, A027750, A290532.

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

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Last modified June 25 04:52 EDT 2021. Contains 345452 sequences. (Running on oeis4.)