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A226853 Numbers n such that Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of n for some q and n is primitive (the set {d(1), d(2), ..., d(q)} appears only once). 1
1, 6, 28, 120, 180, 496, 672, 1890, 8128, 8415, 20482, 20496, 25410, 30240, 32760, 33345, 34155, 38430, 40128, 47804, 72800, 90720, 103530, 407715, 523776 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Subsequence of A225110 where we find classes of numbers having the same first q divisors; for example, each of the numbers 6, 18, 42, 54, 66, ... has {1, 2, 3, 6} as its first four divisors, and 1/1 + 1/2 + 1/3 + 1/6 = 2; similarly, each of the numbers 28, 196, 812, 868, ... has {1, 2, 4, 7, 14, 28} as its first six divisors, and 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2.

This sequence includes only the smallest number having any given set of first divisors {d(1), d(2), ..., d(q)}, i.e., the set of first divisors corresponding to each term occurs only once.

The sets of first divisors (such that Sum_{i = 1..q} 1/d(i) is integer) corresponding to the first few terms are as follows:

a(1) =   1: [1];

a(2) =   6: [1, 2, 3, 6];

a(3) =  28: [1, 2, 4, 7, 14, 28];

a(4) = 120: [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120];

a(5) = 180: [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45];

a(6) = 496: [1, 2, 4, 8, 16, 31, 62, 124, 248, 496];

a(7) = 672: [1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672].

LINKS

Table of n, a(n) for n=1..25.

EXAMPLE

180 is in the sequence because the divisors are {1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180} and the sum of the reciprocals of the first q = 15 divisors is 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 + 1/30 + 1/36 + 1/45 = 3, which is integer.

Although the first 4 divisors of 18 are {1, 2, 3, 6} and the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/6 = 2, 18 is not in the sequence because 6 has those same first four divisors and is the smallest (i.e., primitive) number having that set of first 4 divisors. Thus, the primitive number 6 is in the sequence, so the non-primitive number 18 is not.

MAPLE

with(numtheory): printf ( "%d %d \n", 1, 6):lst:={6}:for n from 1 to 10000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 and {x[q]} intersect lst <>{x[q]} then lst:=lst union {x[q]}:ii:=1: printf(`%d, `, n):else fi:od:od:

CROSSREFS

Cf. A225110.

Sequence in context: A230492 A026014 A332208 * A259307 A183019 A183016

Adjacent sequences:  A226850 A226851 A226852 * A226854 A226855 A226856

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jun 19 2013

EXTENSIONS

Edited by Jon E. Schoenfield, Oct 02 2017

STATUS

approved

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Last modified September 22 21:07 EDT 2020. Contains 337291 sequences. (Running on oeis4.)