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 A290532 Irregular triangle read by rows in which row n lists the number of divisors of each divisor of n. 2
 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 2, 4, 1, 2, 1, 2, 2, 3, 4, 6, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 2, 4, 3, 6, 1, 2, 1, 2, 3, 2, 4, 6, 1, 2, 2, 4, 1, 2, 2, 4, 1, 2, 1, 2, 2, 3, 4, 4, 6, 8, 1, 2, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Or, in the triangle A027750, replace each element with the number of its divisors. The row of index n = p^m (p prime and m >= 0) is equal to (1, 2, ..., m + 1); We observe an interesting property when the index n of the row n is the product of k distinct primes, k = 1,2,... For example: The index n is prime => row n = (1, 2); The index n equals the product of two distinct primes => row n = (1, 2, 2, 4); The index n equals the product of three distinct primes => row n = (1, 2, 2, 2, 4, 4, 4, 8) or a permutation of the elements; ... Let us now consider Pascal's triangle (A007318(n) for n > 0): 1, 1; 1, 2, 1; 1, 3, 3, 1; 1, 4, 6, 4, 1; ... Row 1 of Pascal's triangle gives the number of "1" and the number of "2" respectively belonging to the row of index n = prime(m) of the sequence; Row 2 of Pascal's triangle gives the number of "1", the number of "2" and the number of "4" respectively belonging to the row of index n = p*q of the sequence, where p and q are distinct primes; Row 3 of Pascal's triangle gives the number of "1", the number of "2", the number of "4" and the number of "8" respectively belonging to the row of index n = p*q*r of the sequence, where p, q and r are distinct primes; ... It is now easy to generalize this process by the following proposition. Proposition: binomial(m,k) is the number of terms of the form 2^k belonging to the row of index n in the sequence when n is the product of m distinct primes. LINKS Robert Israel, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened) FORMULA T(n, k) = tau(A027750(n, k)). EXAMPLE Row 6 is (1, 2, 2, 4) because the 6th row of A027750 is [1, 2, 3, 6] and tau(1) = 1, tau(2) = 2, tau(3) = 2 and tau(6) = 4. Triangle begins: 1; 1, 2; 1, 2; 1, 2, 3; 1, 2; 1, 2, 2, 4; 1, 2; 1, 2, 3, 4; 1, 2, 3; 1, 2, 2, 4; ... 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: n2:=tau(d1[i]): printf(`%d, `, n2): od: od: MATHEMATICA Table[DivisorSigma[0, Divisors@ n], {n, 25}] // Flatten (* Michael De Vlieger, Aug 07 2017 *) PROG (PARI) row(n) = apply(numdiv, divisors(n)); \\ Michel Marcus, Dec 27 2021 CROSSREFS Cf. A000005, A007318, A027750, A084997, A290478. Sequence in context: A067815 A133780 A270808 * A080237 A136109 A105265 Adjacent sequences: A290529 A290530 A290531 * A290533 A290534 A290535 KEYWORD nonn,tabf AUTHOR Michel Lagneau, Aug 05 2017 STATUS approved

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Last modified December 2 07:55 EST 2022. Contains 358493 sequences. (Running on oeis4.)