cp's OEIS Frontend

A275407 Let e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n. The sequence lists 1 followed by numbers n for which e_n(2*i-1)=e_n(2*i), for all i>=1.

%I A275407 #36 Feb 15 2023 12:44:56
%S A275407 1,6,35,36,143,210,216,323,667,858,1147,1225,1260,1296,1763,1938,2491,
%T A275407 3599,4002,4757,5005,5148,5767,6882,7350,7387,7560,7776,9797,10578,
%U A275407 11021,11305,11628,12317,14946,16637,19043,20449,21594,22499,23345,24012,25591,28542
%N A275407 Let e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n. The sequence lists 1 followed by numbers n for which e_n(2*i-1)=e_n(2*i), for all i>=1.
%C A275407 There exists a permutation alpha of the sequence such that {alpha(a(n))} is a completely multiplicative function.
%C A275407 This sequence corresponds to the fixed points of A061898. - _Rémy Sigrist_, Feb 15 2023
%H A275407 Peter J. C. Moses, <a href="/A275407/b275407.txt">Table of n, a(n) for n = 1..1000</a>
%e A275407 15 is not in the sequence, since 15 = 3*5 and the prime index of 5 is odd.
%e A275407 5148 is in the sequence, since 5148 = 2^2*3^2*11*13 and
%e A275407 (1) 3 is the next prime after 2,
%e A275407 (2) the exponents of 2 and 3 are equal,
%e A275407 (3) the prime index of 3 is even,
%e A275407 (4) 13 is the next prime after 11,
%e A275407 (5) the exponents of 11 and 13 are equal,
%e A275407 (6) the prime index of 13 is even.
%t A275407 inA275407Q:=If[EvenQ[Length[#]],Apply[And,Join[Map[#[[1]]+1==#[[2]]&&EvenQ[#[[2]]]&,PrimePi[#[[1]]]],Map[#[[1]]==#[[2]]&,#[[2]]]]]&[Map[Partition[#,2]&,Transpose[#]]],False]&[FactorInteger[#]]&;
%t A275407 Join[{1},Select[Range[10000],inA275407Q]] (* _Peter J. C. Moses_, Jul 29 2016 *)
%o A275407 (PARI) isok(n) = {f = factor(n); nbpok = 0; for (k=1, #f~, ip = primepi(f[k, 1]); if ((ip % 2) && (kk = vecsearch(f[,1]~, prime(ip+1))) && (f[kk, 2] == f[k,2]), nbpok++;)); nbpok == #f~/2;} \\ _Michel Marcus_, Jul 27 2016
%o A275407 (Sage)
%o A275407 def is_A275407(n):
%o A275407     L = list(factor(n))
%o A275407     if is_odd(len(L)): return False
%o A275407     for i in range(0,len(L)//2+1,2):
%o A275407         if L[i][1] != L[i+1][1]: return False
%o A275407         if L[i][0] != previous_prime(L[i+1][0]): return False
%o A275407         if is_even(len(prime_range(1, L[i+1][0]))): return False
%o A275407     return True
%o A275407 [n for n in (2..5000) if is_A275407(n)] # _Peter Luschny_, Jul 27 2016
%Y A275407 Cf. A000040, A001248, A061898, A089581, A275246, A275248, A275249, A275251, A275252, A275253.
%K A275407 nonn
%O A275407 1,2
%A A275407 _Vladimir Shevelev_, Jul 26 2016
%E A275407 More terms from _Peter J. C. Moses_, Jul 26 2016