A007445 Inverse Moebius transform of primes. (Formerly M1335)

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

1,1

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 ...

...

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

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A039679 A088823 A302294 * A350129 A159699 A063217

Adjacent sequences: A007442 A007443 A007444 * A007446 A007447 A007448

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Jason Earls, Jul 08 2001

Status

approved