A275474 Numbers n for which e_n(2*i)=e_n(2*i+1), for all i>=1, where e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n.

1, 2, 4, 8, 15, 16, 30, 32, 60, 64, 77, 120, 128, 154, 221, 225, 240, 256, 308, 437, 442, 450, 480, 512, 616, 874, 884, 899, 900, 960, 1024, 1155, 1232, 1517, 1748, 1768, 1798, 1800, 1920, 2021, 2048, 2310, 2464, 3034, 3127, 3315, 3375, 3496, 3536, 3596, 3600

Offset: 1

Author

Vladimir Shevelev, Jul 29 2016

Keywords

• nonn

Comments

There exists a permutation alpha of the sequence such that {alpha(a(n))} is a completely multiplicative function.

Numbers which are the product of zero or more of {2, 3*5, 7*11, 13*17, 19*23, ...} with multiplicity. - Charles R Greathouse IV, Jul 30 2016

Examples

			1 is a member, since all e_1(k)=0;
Powers 2^m, m>=1, are members, since e_2^m(k)=0, for all k>=2;
15 is a member, since e_15(2)*e_15(3)=1;
n = 2983500 is a member, since e_n(1)=2, e_n(2)=e_n(3)=3 and e_n(6)=e_n(7)=1, all other e_n(k)=0.
		

Links

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

Crossrefs

Cf. A000040, A001248, A089581, A275246, A275248, A275249, A275251, A275252, A275253, A275407.

Programs

PARI

is(n)=my(f=factor(n>>valuation(n,2))); if (#f~%2, return(0)); for(i=1,#f~/2, if(f[2*i-1,2]!=f[2*i,2] || nextprime(f[2*i-1,1]+1)!=f[2*i,1], return(0))); for(i=1,#f~/2, if(primepi(f[2*i,1])%2==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 30 2016

PARI

list(lim)=my(v=List([1,2]),p=3,pStart=2,pEnd,start=2,end,nStart,t); lim\=1; forprime(q=5,sqrtint(lim+1)+1, p=if(p, listput(v,p*q); 0, q)); end=pEnd=#v; for(n=2,logint(lim,2), nStart=end+1; for(i=start,end, for(j=pStart,pEnd, t=v[i]*v[j]; if(t>lim, break); listput(v, t))); start=nStart; end=#v); Set(v) \\ Charles R Greathouse IV, Jul 30 2016

Extensions

More terms from Peter J. C. Moses, Jul 29 2016