A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.

0, 1, 3, 2, 7, 4, 15, 3, 6, 8, 31, 5, 63, 16, 10, 4, 127, 7, 255, 9, 18, 32, 511, 6, 14, 64, 9, 17, 1023, 11, 2047, 5, 34, 128, 22, 8, 4095, 256, 66, 10, 8191, 19, 16383, 33, 13, 512, 32767, 7, 30, 15, 130, 65, 65535, 10, 38, 18, 258, 1024, 131071, 12, 262143, 2048, 21, 6, 70, 35, 524287, 129, 514, 23, 1048575, 9, 2097151, 4096, 17, 257, 46, 67, 4194303, 11, 12

Offset: 1

Author

Antti Karttunen, Nov 09 2017

Keywords

• nonn

Comments

Original, equal definition: totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000225, A000720, A003557, A005187, A087207.

Cf. also A046645, A048675, A293442, A331591, A331740.

Programs

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; }; \\ This function from Charles R Greathouse IV
A293447(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2] * A005187(2^(primepi(f[k, 1])-1))); }

Scheme

(define (A293447 n) (cond ((= 1 n) 0) (else (+ (A005187 (A087207 n)) (A293447 (A003557 n))))))
;; Alternatively:
(define (A293447 n) (if (= 1 n) 0 (+ (A005187 (A000079 (+ -1 (A061395 n)))) (A293447 (/ n (A006530 n))))))

Formula

Totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).

a(1) = 0, and for n > 1, a(n) = A005187(A087207(n)) + a(A003557(n)).

Other identities:

For all n >= 1, a(A293442(n)) = A046645(n).

For all n >= 2 and all k >= 0, a(n^k) = k*a(n).

For all n >= 1, a(n) >= A048675(n) >= A331740(n) >= A331591(n).

Extensions

Definition simplified by Antti Karttunen, Feb 05 2020