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

Author

Michel Lagneau, Jun 19 2013

Keywords

• nonn

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].

Examples

			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.
		

Crossrefs

Subsequence of A225110.

Programs

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:

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

Extensions

Edited by Jon E. Schoenfield, Oct 02 2017

More terms from Michel Marcus, Feb 22 2025