A007445 Inverse Moebius transform of primes.

2, 5, 7, 12, 13, 23, 19, 31, 30, 45, 33, 67, 43, 65, 65, 84, 61, 107, 69, 123, 97, 115, 85, 175, 110, 147, 133, 179, 111, 223, 129, 215, 175, 203, 179, 302, 159, 235, 215, 315, 181, 337, 193, 315, 285, 287, 213, 451, 246, 371, 299, 393, 243, 461, 301, 461, 343

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From Davide Rotondo, Mar 09 2022: (Start)

Can be constructed by writing the sequence of prime numbers, then the sequence of prime numbers spaced by a zero, then the sequence of prime numbers spaced by two zeros, and so on. Finally add the values of the columns.

2 3 5 7 11 13 17 19 23 29 ...

0 2 0 3 0 5 0 7 0 11 ...

0 0 2 0 0 3 0 0 5 0 ...

0 0 0 2 0 0 0 3 0 0 ...

0 0 0 0 2 0 0 0 0 3 ...

0 0 0 0 0 2 0 0 0 0 ...

0 0 0 0 0 0 2 0 0 0 ...

0 0 0 0 0 0 0 2 0 0 ...

0 0 0 0 0 0 0 0 2 0 ...

0 0 0 0 0 0 0 0 0 2 ...

...

----------------------------------

Tot. 2 5 7 12 13 23 19 31 30 45 ... (End)

Examples

			a(6)=23 because the divisors of 6 are: 1, 2, 3 and 6; and prime(1) + prime(2) + prime(3) + prime(6) = 2 + 3 + 5 + 13 = 23.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• N. J. A. Sloane, Transforms

Programs

Mathematica

a[n_] := DivisorSum[n, Prime]; Array[a, 60] (* Jean-François Alcover, Dec 01 2015 *)

PARI

je=[]; for(n=1,150,je=concat(je,sumdiv(n,d, prime(d)))); j

Formula

a(n) = Sum_{d|n} prime(d).

G.f.: Sum_{k>=1} prime(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 02 2017

Extensions

More terms from Jason Earls, Jul 08 2001