A261656 Numbers with 3 prime factors a < b < c such that 2c = a^4 + b^2 and a = round(sqrt(b)).

3333, 66355, 106285, 22913341, 72145333, 3154579927, 3303006103, 16443947749, 16762851701, 18000215291, 93827227037, 280747032637, 512149867007, 520524616433, 1159728301763, 2460177504259, 2502980807149, 2531825000909, 3109094928581, 3122521499989

Offset

1,1

Comments

This sequence is interesting because the sequence of any member's divisors increases approximately exponentially, without members having any repeated prime factors.

Removing the limitation a=round(sqrt(b)) results in A261657.

This sequence would be particularly interesting if the b and c of one member were the a and b of another, as it could give us a number with 4 prime factors and similar properties as here. The resulting compound would be a member of A261658. No such pairs exist in this sequence with any lowest factor < 10000.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

The prime factors of 3333 are 3, 11, and 101. 3^4=81, 11^2=121, and the average of 81 and 121 gives 101. Thus, 3333 is a member.
The divisors of 3333 are 1, 3, 11, 33, 101, 303, 1111, 3333. This sequence is approximately exponential.

Maple

n := 100; for a from 3 to n do if isprime(a) then for b from ceil((a-.5)^2) to floor((a+.5)^2) do if isprime(b) then c := (a^4+b^2)*(1/2); if isprime(c) then print(a*b*c) end if end if end do end if end do

Prog

(PARI) list(lim)=my(v=List(), t, a, c); forprime(b=2, , a=round(sqrt(b)); c=(a^4+b^2)/2; if(isprime(a) && denominator(c)==1 && isprime(c), t=a*b*c; if(t>lim, break); listput(v, a*b*c))); Set(v) \\ Charles R Greathouse IV, Aug 29 2015

Crossrefs

Cf. A261657, A261658.

Sequence in context: A230307 A043504 A251331 * A231955 A255657 A252188

Adjacent sequences: A261653 A261654 A261655 * A261657 A261658 A261659

Keyword

nonn

Author

David Ferris, Aug 28 2015

Status

approved