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

%I A317087 #16 Jun 23 2020 05:52:23
%S A317087 1,2,4,6,8,16,30,32,36,64,90,128,210,216,256,270,300,512,810,900,1024,
%T A317087 1296,2048,2310,2430,2700,2940,3000,3150,4096,7290,7776,8100,8192,
%U A317087 9000,11550,16384,21870,24300,27000,30000,30030,32768,41160,44100,46656,47250,48510
%N A317087 Numbers whose prime factors span an initial interval of prime numbers and whose sequence of prime multiplicities is a palindrome.
%C A317087 3^m*10^k for k, m > 0 are terms of this sequence. - _Chai Wah Wu_, Jun 23 2020
%H A317087 Giovanni Resta, <a href="/A317087/b317087.txt">Table of n, a(n) for n = 1..10000</a>
%H A317087 Wikipedia, <a href="https://en.wikipedia.org/wiki/Palindrome">Palindrome</a>
%e A317087 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).
%t A317087 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]
%t A317087 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 *)
%o A317087 (Python)
%o A317087 from sympy import factorint, primepi
%o A317087 A317087_list = [1]
%o A317087 for n in range(1,10**5):
%o A317087     d = factorint(n)
%o A317087     k, l = sorted(d.keys()), len(d)
%o A317087     if l > 0 and l == primepi(max(d)):
%o A317087         for i in range(l//2):
%o A317087             if d[k[i]] != d[k[l-i-1]]:
%o A317087                 break
%o A317087         else:
%o A317087             A317087_list.append(n) # _Chai Wah Wu_, Jun 23 2020
%Y A317087 Cf. A025065, A055932, A124010, A133808, A242414, A296150, A317085, A317086.
%K A317087 nonn
%O A317087 1,2
%A A317087 _Gus Wiseman_, Jul 21 2018