A229108 Increasing a(n)is the smallest number of the form p^a*q^b, where a,b are positive integers and p < q are odd primes such that max( p^a, q^b)/min( p^a, q^b) <= 1 + 2/prime(n).

15, 35, 63, 143, 323, 575, 675, 783, 899, 1763, 2303, 3599, 5183, 6399, 6723, 10403, 11663, 15875, 19043, 22499, 27221, 28223, 32399, 36863, 39203, 50621, 51983, 53357, 57599, 58563, 72899, 77837, 79523, 95477, 97343, 119021, 121103, 123197, 129599

Offset: 2

Author

Vladimir Shevelev and Peter J. C. Moses, Sep 13 2013

Keywords

• nonn

Examples

			15 is the least number of considered form, and 5/3 = 1 + 2/prime(2). So a(2)=15; in case of n=23, not only 28223 but also 29237 satisfies required inequality and we choose the smallest from them.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 2..1001

Crossrefs

Cf. A037074.

Programs

Mathematica

tmp={1}; Do[test=1+2/Prime[n]; AppendTo[tmp, NestWhile[#+2&, Last[tmp]+2, !((Max[#]/Min[#]&[Map[#[[1]]^#[[2]]&, FactorInteger[#]]] <= test) && (Length[FactorInteger[#]]==2))&]], {n,2,30}]; Rest[tmp]

PARI

factorPP(n)=my(f=factor(n)); vecsort(vector(#f~,i,f[i,1]^f[i,2]))
list(n)=my(v=primes(n),t=1,f);for(i=1,n,while(1, f=factorPP(t += 2); if(#f==2 && f[2]/f[1] <= 1+2/v[i], v[i]=t; break))); v \\ Charles R Greathouse IV, Sep 13 2013

Formula

If [prime(n), prime(n+1)] is a twin pair, then a(n) <= prime(n)*prime(n+1).