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A226364 Composite squarefree numbers n such that the ratios (n - 1/3)/(p - 1/3) are integers for each prime p dividing n. 6
308267, 1420467, 1445995, 46874667, 153810067, 324218667, 355724747, 393253747, 471957547, 618729307, 886489707, 901990059, 1062803467, 1525582667, 1735517355, 4306362667, 4815895467, 6528285867, 6634856107, 11460166667, 12364885867, 13330858667, 20628538667 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also composite squarefree numbers n such that (3p - 1) | (3n - 1).

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..68 (terms < 2*10^12)

EXAMPLE

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.

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);

PROG

(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

CROSSREFS

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226448.

Sequence in context: A270764 A236818 A234188 * A206276 A251304 A127491

Adjacent sequences:  A226361 A226362 A226363 * A226365 A226366 A226367

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Jun 05 2013

EXTENSIONS

a(5)-a(23) from Giovanni Resta, Jun 07 2013

STATUS

approved

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Last modified June 12 10:58 EDT 2021. Contains 344947 sequences. (Running on oeis4.)