cp's OEIS Frontend

A369061 Numbers k such that k + k'*2 is equal to a partial sum of primorial numbers (a term of A143293), where k' stands for the arithmetic derivative of k, A003415.

%I A369061 #16 Jan 18 2024 11:01:22
%S A369061 1,7,37,99,2557,32587,543097,10242787,232889539,146710424885,
%T A369061 207263519017
%N A369061 Numbers k such that k + k'*2 is equal to a partial sum of primorial numbers (a term of A143293), where k' stands for the arithmetic derivative of k, A003415.
%C A369061 Numbers k such that A068719(k) = A143293(n), for some n >= 0.
%C A369061 Numbers k for which A276087(A068719(k)) is a prime.
%C A369061 All terms are odd.
%C A369061 Notably each of the terms a(2) .. a(9) map (in the same order) to A143293(2..9), but then k for A143293(10) = 6703028889 is missing, and a(10) and a(11) both map to A143293(11) = 207263519019.
%e A369061 For 99, A068719(99) = 99 + 99'*2 = 99 + 75*2 = 249 = 1 + 2 + 6 + 30 + 210 = A143293(4), therefore 99 is included in this sequence.
%e A369061 For 2557, which is a prime, 2557 + 2557' * 2 = 2557+2 = 2559 = 1 + 2 + 6 + 30 + 210 + 2310 = A143293(5), therefore 2557 is included in this sequence.
%o A369061 (PARI)
%o A369061 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369061 A068719(n) = (n+2*A003415(n));
%o A369061 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 A369061 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A369061 isA369061(n) = (1==A276150(A276086(A068719(n))));
%Y A369061 After the initial 1, the even terms of A328243 halved.
%Y A369061 Cf. A003415, A068719, A143293, A276086, A276087.
%K A369061 nonn
%O A369061 1,2
%A A369061 _Antti Karttunen_, Jan 17 2024