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, 6, 28, 120, 180, 496, 672, 1890, 8128, 8415, 20482, 20496, 25410, 30240, 32760, 33345, 34155, 38430, 40128, 47804, 72800, 90720, 103530, 407715, 523776, 806190, 979992, 1084160, 1273725, 1274100, 2178540, 3571050, 7441280, 10782216, 12499150, 23569920, 28360464, 33550336, 45532800

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 an 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..39.

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 an 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:

Prog

(PARI) isok(k) = if (k==1, return([1])); my(d=divisors(k), s=1); for (i=2, #d, s += 1/d[i]; if (denominator(s)==1, return(Vec(d, i)); ));
already(list, v) = for (i=1, #list, if (list[i] == v, return(1)); );
lista(nn) = my(listv=List(), listi=List()); for (n=1, nn, my(v=isok(n)); if (v && !already(listv, v), listput(listi, n); listput(listv, v); ); ); Vec(listi); \\ Michel Marcus, Feb 22 2025

Crossrefs

Subsequence of 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

More terms from Michel Marcus, Feb 22 2025

Status

approved