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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A226112 Composite squarefree numbers n such that the ratio (n + 1/3)/(p(i) + 1/3) is an integer, where p(i) are the prime factors of n.

Original entry on oeis.org

133653, 1280533, 193638133, 514276565, 1421486733, 1567953933, 3857178453, 3973200933, 5411272533, 7694639213, 8021152533, 8469827669, 9820706133, 15832804533, 18238619373, 22356801133, 23037766613, 25136796813, 27315827733, 32434329685, 39817016633
Offset: 1

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Author

Paolo P. Lava, May 29 2013

Keywords

Comments

Also composite squarefree numbers n such that (3*p(i)+1) | (3*n+1).

Examples

			The prime factors of 133653 are 3, 13, 23 and 149. We see that (133653 + 1/3)/(3 + 1/3) = 40096, (133653 + 1/3)/(13 + 1/3) = 10024, (133653 + 1/3)/(23 + 1/3) = 5728 and (133653 + 1/3)/(149 + 1/3) = 895. Hence 133653 is in the sequence.
The prime factors of 1127749 are 7, 31 and 5197. We see that
(1127749 + 1/3)/(7 + 1/3) = 153784, (1127749 + 1/3)/(31 + 1/3) = 35992 but (1127749 + 1/3)/(5197 + 1/3) = 422906/1949. Hence 1127749 is not in the sequence.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111, A226113, A226114.

Programs

  • Maple
    with(numtheory); A226112:=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: A226112(10^9,1/3);

Extensions

a(4)-a(21) from Giovanni Resta, Jun 02 2013
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