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.

1, 6, 35, 36, 143, 210, 216, 323, 667, 858, 1147, 1225, 1260, 1296, 1763, 1938, 2491, 3599, 4002, 4757, 5005, 5148, 5767, 6882, 7350, 7387, 7560, 7776, 9797, 10578, 11021, 11305, 11628, 12317, 14946, 16637, 19043, 20449, 21594, 22499, 23345, 24012, 25591, 28542

Offset

1,2

Comments

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

This sequence corresponds to the fixed points of A061898. - Rémy Sigrist, Feb 15 2023

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Example

15 is not in the sequence, since 15 = 3*5 and the prime index of 5 is odd.
5148 is in the sequence, since 5148 = 2^2*3^2*11*13 and
(1) 3 is the next prime after 2,
(2) the exponents of 2 and 3 are equal,
(3) the prime index of 3 is even,
(4) 13 is the next prime after 11,
(5) the exponents of 11 and 13 are equal,
(6) the prime index of 13 is even.

Mathematica

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[#]]&;
Join[{1}, Select[Range[10000], inA275407Q]] (* Peter J. C. Moses, Jul 29 2016 *)

Prog

(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
(Sage)
def is_A275407(n):
    L = list(factor(n))
    if is_odd(len(L)): return False
    for i in range(0, len(L)//2+1, 2):
        if L[i][1] != L[i+1][1]: return False
        if L[i][0] != previous_prime(L[i+1][0]): return False
        if is_even(len(prime_range(1, L[i+1][0]))): return False
    return True
[n for n in (2..5000) if is_A275407(n)] # Peter Luschny, Jul 27 2016

Crossrefs

Cf. A000040, A001248, A061898, A089581, A275246, A275248, A275249, A275251, A275252, A275253.

Sequence in context: A145000 A354792 A209182 * A362306 A046025 A198327

Adjacent sequences: A275404 A275405 A275406 * A275408 A275409 A275410

Keyword

nonn

Author

Vladimir Shevelev, Jul 26 2016

Extensions

More terms from Peter J. C. Moses, Jul 26 2016

Status

approved