

A215197


Numbers n such that n and n + 1 are both of the form p*q^4 where p and q are distinct primes.


4



2511, 7856, 10287, 15471, 15632, 18063, 20816, 28592, 36368, 40816, 54512, 75248, 88047, 93231, 101168, 126927, 134703, 160624, 163376, 170991, 178767, 210032, 215216, 217808, 220624, 254096, 256527, 274671, 280624, 292976, 334448, 347408, 443151, 482192
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OFFSET

1,1


COMMENTS

The smaller of adjacent terms in A178739.  R. J. Mathar, Aug 08 2012
These are numbers n such that n and n+1 both have 10 divisors. Proof: clearly n and n+1 cannot both be of the form p^9, so for contradiction assume either n and n+1 is of the form p*q^4 and the other is of the form r^9 where p, q, and r are prime. So p*q^4 is either r^9  1 = (r1)(r^2+r+1)(r^6+r^3+1) or r^9 + 1 = (r+1)(r^2r+1)(r^6r^3+1). But these factors are relatively prime and so cannot represent p*q^4 unless one or more factors are units. But this does not happen for r > 2, and the case r = 2 does not work since neither 511 not 513 is of the form p*q^4.  Charles R Greathouse IV, Jun 19 2016


LINKS

Table of n, a(n) for n=1..34.


EXAMPLE

2511 is a member as 2511 = 31*3^4 and 2512 = 157*2^4.


MAPLE

with(numtheory):for n from 3 to 500000 do:x:=factorset(n):y:=factorset(n+1):n1:=nops(x):n2:=nops(y):if n1=2 and n2=2 then xx1:=x[1]*x[2]^4 : xx2:=x[2]*x[1]^4:yy1:=y[1]*y[2]^4: yy2:=y[2]*y[1]^4:if (xx1=n or xx2=n) and (yy1=n+1 or yy2=n+1) then printf("%a, ", n):else fi:fi:od:


MATHEMATICA

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 4}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 4}, AppendTo[lst, n]]], {n, 3, 55000}]; lst (* Lagneau *)
(* First run program for A178739 *) Select[A178739, MemberQ[A178739, # + 1] &] (* Alonso del Arte, Aug 05 2012 *)


PROG

(PARI) is(n)=numdiv(n)==10 && numdiv(n)==10 \\ Charles R Greathouse IV, Jun 19 2016
(PARI) is(n)=vecsort(factor(n)[, 2])==[1, 4]~ && vecsort(factor(n+1)[, 2])==[1, 4]~ \\ Charles R Greathouse IV, Jun 19 2016


CROSSREFS

Intersection of A005237 and A030628.
Cf. A074172, A215173.
Sequence in context: A203055 A031621 A212400 * A250694 A250679 A145534
Adjacent sequences: A215194 A215195 A215196 * A215198 A215199 A215200


KEYWORD

nonn


AUTHOR

Michel Lagneau, Aug 05 2012


STATUS

approved



