A317087 Numbers whose prime factors span an initial interval of prime numbers and whose sequence of prime multiplicities is a palindrome.

1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 90, 128, 210, 216, 256, 270, 300, 512, 810, 900, 1024, 1296, 2048, 2310, 2430, 2700, 2940, 3000, 3150, 4096, 7290, 7776, 8100, 8192, 9000, 11550, 16384, 21870, 24300, 27000, 30000, 30030, 32768, 41160, 44100, 46656, 47250, 48510

Offset

1,2

Comments

3^m*10^k for k, m > 0 are terms of this sequence. - Chai Wah Wu, Jun 23 2020

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Wikipedia, Palindrome

Example

The sequence of rows of A296150 indexed by the terms of this sequence begins: (1), (11), (21), (111), (1111), (321), (11111), (2211), (111111), (3221), (1111111), (4321), (222111), (11111111), (32221), (33211), (111111111), (322221), (332211).

Mathematica

nrmpalQ[n_]:=With[{f=If[n==1, {}, FactorInteger[n]]}, And[PrimePi/@ Sort[First/@f] == Range[ Length[f]], Reverse[Last/@f] == Last/@f]]; Select[Range[100], nrmpalQ]
upto = 10^20; pL[n_] := Block[{p = Prime@Range@n, h = Ceiling[n/2]}, Take[p, h] Reverse@ If[n == 2 h, Take[p, -h], Prepend[ Take[p, 1-h], 1]]]; ric[v_, p_] := If[p == {}, AppendTo[L, v], Block[{w = v}, While[w <= upto, ric[w, Rest@ p]; w *= First@ p]]]; np = 1; L = {1}; While[(b = Times @@ Prime[Range@ np]) <= upto, ric[b, pL[np++]]]; Sort[L] (* Giovanni Resta, Jun 23 2020 *)

Prog

(Python)
from sympy import factorint, primepi
A317087_list = [1]
for n in range(1, 10**5):
    d = factorint(n)
    k, l = sorted(d.keys()), len(d)
    if l > 0 and l == primepi(max(d)):
        for i in range(l//2):
            if d[k[i]] != d[k[l-i-1]]:
                break
        else:
            A317087_list.append(n) # Chai Wah Wu, Jun 23 2020

Crossrefs

Cf. A025065, A055932, A124010, A133808, A242414, A296150, A317085, A317086.

Sequence in context: A296993 A068799 A317306 * A100778 A368325 A324579

Adjacent sequences: A317084 A317085 A317086 * A317088 A317089 A317090

Keyword

nonn

Author

Gus Wiseman, Jul 21 2018

Status

approved