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Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.
2

%I #5 Jan 14 2024 15:06:02

%S 14,186,258,322,338,3318,3962,5334,6106,7674,8970,9186,9978,10930,

%T 11994,12154,12614,12970,13218,13234,14626,15226,15914,16378,17122,

%U 18226,18658,19058,19874,20194,20962,21082,21106,21218,44718,49358,57346,58354,75442,76162,81802,87814,95114,102794,113505,114918,130802

%N Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.

%C Numbers k for which A003415(k) is one of the terms of A327978.

%o (PARI)

%o A002620(n) = ((n^2)>>2);

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

%o A068346(n) = A003415(A003415(n));

%o A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };

%o isA368702flat(n) = { my(u=A068346(n)); ((u>1)&&(1==A276150(u))); };

%o k=0; for(n=1, A002620(A002110(7)), if(isA368702flat(n), k++; write("b368702.txt", k, " ", n)));

%Y Cf. A002110, A003415, A068346, A327978.

%Y Subsequence of A046100 and of A328244.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jan 09 2024