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Numbers k such that no x with an odd arithmetic derivative is encountered when repeatedly prime shifting from k down to 1 with the map x -> A064989(x).
2

%I #16 Apr 23 2022 18:12:25

%S 1,4,8,9,16,25,32,36,49,64,72,81,100,108,121,128,144,169,196,200,216,

%T 225,256,288,289,324,361,392,400,432,441,484,512,529,576,625,648,676,

%U 729,784,800,841,864,900,961,968,972,1024,1089,1152,1156,1225,1296,1352,1369,1444,1521,1568,1600,1681,1728,1764,1800,1849

%N Numbers k such that no x with an odd arithmetic derivative is encountered when repeatedly prime shifting from k down to 1 with the map x -> A064989(x).

%C Note that k itself must also be in A235992 to be included here. All terms must be powerful (in A001694) because otherwise at some point in the shifting process a number of the form 4u+2 would be encountered, and they are all in A235991.

%H Antti Karttunen, <a href="/A353485/b353485.txt">Table of n, a(n) for n = 1..10000</a>

%o (PARI)

%o A000265(n) = (n>>valuation(n,2));

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };

%o isA353485(n) = { while(n>1, if(A003415(n)%2, return(0)); n = A064989(n)); (1); };

%Y Positions of zeros in A353484.

%Y Cf. A000290 (subsequence), A003415, A064989, A165560, A235991, A235992.

%Y Subsequence of A001694.

%K nonn

%O 1,2

%A _Antti Karttunen_, Apr 23 2022