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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).

%I A226853 #24 Feb 22 2025 13:03:37
%S A226853 1,6,28,120,180,496,672,1890,8128,8415,20482,20496,25410,30240,32760,
%T A226853 33345,34155,38430,40128,47804,72800,90720,103530,407715,523776,
%U A226853 806190,979992,1084160,1273725,1274100,2178540,3571050,7441280,10782216,12499150,23569920,28360464,33550336,45532800
%N 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).
%C A226853 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.
%C A226853 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.
%C A226853 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:
%C A226853 a(1) =   1: [1];
%C A226853 a(2) =   6: [1, 2, 3, 6];
%C A226853 a(3) =  28: [1, 2, 4, 7, 14, 28];
%C A226853 a(4) = 120: [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120];
%C A226853 a(5) = 180: [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45];
%C A226853 a(6) = 496: [1, 2, 4, 8, 16, 31, 62, 124, 248, 496];
%C A226853 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].
%e A226853 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.
%e A226853 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.
%p A226853 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:
%o A226853 (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));));
%o A226853 already(list, v) = for (i=1, #list, if (list[i] == v, return(1)););
%o A226853 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
%Y A226853 Subsequence of A225110.
%K A226853 nonn
%O A226853 1,2
%A A226853 _Michel Lagneau_, Jun 19 2013
%E A226853 Edited by _Jon E. Schoenfield_, Oct 02 2017
%E A226853 More terms from _Michel Marcus_, Feb 22 2025