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A064364 Positive integers sorted by A001414(n), the sum of their prime divisors, as the major key and n as the minor key. 8
1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 14, 20, 24, 27, 21, 25, 30, 32, 36, 11, 28, 40, 45, 48, 54, 35, 42, 50, 60, 64, 72, 81, 13, 22, 56, 63, 75, 80, 90, 96, 108, 33, 49, 70, 84, 100, 120, 128, 135, 144, 162, 26, 44, 105, 112, 125, 126, 150, 160, 180, 192, 216, 243 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is a permutation of the positive integers.

a(1) could be taken as 0 because 1 is not a member of A001414 and one could start with a(0)=1 (see the W. Lang link).

The row length sequence of this array is A000607(n), n>=2.

If the array is [1,0,2,3,4,5,6,6,...] with offset 0 then the row length sequence is A000607(n), n>=0.

From David James Sycamore, May 11 2018: (Start)

For n>1 a(n) is the smallest number not yet seen such that Sopfr(a(n)) is the least possible integer >= Sopfr(a(n-1)). The sequence lists consecutively, elements of the finite ordered sets S(k)={x: Sopfr(x) = k; k>=2}. When a(n) = A056240(k) for some k>=2 then Sopfr(n) = k and a(n) is the first of A000607(k) successive terms, all of which have Sopfr = k. Consequently the sequence follows a sawtooth profile, rising from a(n) = A056240(k) to A000792(k), the greatest number whose Sopfr is k, then falling abruptly as k indexes to k+1. (End)

LINKS

Reinhard Zumkeller and Alois P. Heinz, Rows n = 1..60, flattened (first 32 rows from Reinhard Zumkeller)

H. Havermann: The first 100 sums (complete, a 6 MB file)

H. Havermann: Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums)

W. Lang, First 16 rows.

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

The triangle reads:

1,

(0,) (see comment in link to "first 16 rows" by W. Lang)

2,

3,

4,

5,  6,

8,  9,

7,  10, 12,

15, 16, 18,

14, 20, 24, 27,

21, 25, 30, 32, 36,

11, 28, 40, 45, 48, 54,

35, 42, 50, 60, 64, 72, 81,

13, 22, 56, 63, 75, 80, 90, 96, 108,

...

MATHEMATICA

terms = 1000; nmax0 = 100000 (* a rough estimate of max sopfr *);

sopfr[n_] := sopfr[n] = Total[Times @@@ FactorInteger[n]];

f[n1_, n2_] := Which[t1 = sopfr[n1]; t2 = sopfr[n2]; t1 < t2, True, t1 == t2, n1 <= n2, True, False];

Clear[g];

g[nmax_] := g[nmax] = Sort[Range[nmax], f][[1 ;; terms]];

g[nmax = nmax0];

g[nmax += nmax0];

While[g[nmax] != g[nmax - nmax0], Print[nmax]; nmax += nmax0];

A064364 = g[nmax] (* Jean-Fran├žois Alcover, Mar 13 2019 *)

PROG

(Haskell)

import Data.List (partition, union)

a064364 n k = a064364_tabf !! (n-1) !! (k-1)

a064364_row n = a064364_tabf !! (n-1)

a064364_tabf = [1] : tail (f 1 [] 1 (map a000792 [2..])) where

   f k pqs v (w:ws) = (map snd pqs') :

     f (k + 1) (union pqs'' (zip (map a001414 us) us )) w ws where

       us = [v + 1 .. w]

       (pqs', pqs'') = partition ((== k) . fst) pqs

a064364_list = concat a064364_tabf

-- Reinhard Zumkeller, Jun 11 2015

CROSSREFS

Cf. A001414.

Cf. A000607 (row lengths), A002098 (row sums), A056240 (least = first term in the n-th row), A000792 (greatest term in the n-th row).

Cf. A257815 (inverse).

Sequence in context: A265568 A265552 A303936 * A303645 A269855 A209274

Adjacent sequences:  A064361 A064362 A064363 * A064365 A064366 A064367

KEYWORD

easy,nonn,look,tabf,changed

AUTHOR

Howard A. Landman, Sep 25 2001

EXTENSIONS

More terms from Vladeta Jovovic, Sep 27 2005

STATUS

approved

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Last modified March 25 10:31 EDT 2019. Contains 321470 sequences. (Running on oeis4.)