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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.
0
1, 7, 37, 99, 2557, 32587, 543097, 10242787, 232889539, 146710424885, 207263519017
OFFSET
1,2
COMMENTS
Numbers k such that A068719(k) = A143293(n), for some n >= 0.
Numbers k for which A276087(A068719(k)) is a prime.
All terms are odd.
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.
EXAMPLE
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.
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.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068719(n) = (n+2*A003415(n));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369061(n) = (1==A276150(A276086(A068719(n))));
CROSSREFS
After the initial 1, the even terms of A328243 halved.
Sequence in context: A154105 A159491 A106064 * A282001 A038862 A136204
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 17 2024
STATUS
approved