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.

1, 7, 37, 99, 2557, 32587, 543097, 10242787, 232889539, 146710424885, 207263519017

Offset: 1

Author

Antti Karttunen, Jan 17 2024

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

After the initial 1, the even terms of A328243 halved.

Cf. A003415, A068719, A143293, A276086, A276087.

Programs

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))));