A226364 Composite squarefree numbers n such that the ratios (n - 1/3)/(p - 1/3) are integers for each prime p dividing n.
308267, 1420467, 1445995, 46874667, 153810067, 324218667, 355724747, 393253747, 471957547, 618729307, 886489707, 901990059, 1062803467, 1525582667, 1735517355, 4306362667, 4815895467, 6528285867, 6634856107, 11460166667, 12364885867, 13330858667, 20628538667
Offset: 1
Keywords
Examples
The prime factors of 1445995 are 5, 19, 31 and 491. We see that (1445995 - 1/3)/(5 - 1/3) = 309856, (1445995 - 1/3)/(19 - 1/3) = 77464, (1445995 - 1/3)/(31 - 1/3) = 47152 and (1445995 - 1/3)/(491 - 1/3) = 2947. Hence 1445995 is in the sequence. The prime factors of 1112307 are 3, 7 and 52967. We see that (1112307 - 1/3)/(3 - 1/3) = 417115, (1112307 - 1/3)/(7 - 1/3) = 166846 but (1112307 - 1/3)/(52967 - 1/3) = 166846/7945. Hence 1112307 is not in the sequence.
Links
- Giovanni Resta, Table of n, a(n) for n = 1..68 (terms < 2*10^12)
Programs
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Maple
with(numtheory); ListA226364:=proc(i, j) local c, d, n, ok, p; for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1; for d from 1 to nops(p) do if p[d][2]>1 or not type((n-j)/(p[d][1]-j),integer) then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; od; end: ListA226364(10^9,1/3);
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PARI
is(n,P)=n=3*n-1; for(i=1,#P,if(n%(3*P[i]-1),return(0))); 1 list(lim,P=[],n=1,mx=lim\2)=my(v=[],t);if(#P>1&&is(n,P), v=[n]); P=concat(P,0); forprime(p=2,min(lim,mx),P[#P]=p;t=list(lim\p,P,n*p,p-1);if(#t,v=concat(v,t))); v \\ Charles R Greathouse IV, Jun 07 2013
Extensions
a(5)-a(23) from Giovanni Resta, Jun 07 2013
Comments