A283985 - OEIS

%I #22 May 08 2021 23:05:46

%S 0,1,3,4,9,10,12,13,39,40,42,43,48,49,51,52,249,250,252,253,258,259,

%T 261,262,288,289,291,292,297,298,300,301,2559,2560,2562,2563,2568,

%U 2569,2571,2572,2598,2599,2601,2602,2607,2608,2610,2611,2808,2809,2811,2812,2817,2818,2820,2821,2847,2848,2850,2851,2856,2857,2859,2860,32589

%N Sums of distinct terms of A143293: a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

%C Indexing starts from zero, with a(0) = 0.

%H Antti Karttunen, <a href="/A283985/b283985.txt">Table of n, a(n) for n = 0..8191</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

%F a(n) = A276085(A283477(n)).

%F Other identities. For all n >= 0:

%F a(2^n) = A143293(n).

%o (PARI)

%o A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; }; \\ This function from _Charles R Greathouse IV_, Feb 05 2014

%o A030308(n,k) = bittest(n,k);

%o A283985(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A143293(i));

%o (Scheme) (define (A283985 n) (A276085 (A283477 n)))

%o (Python)

%o from sympy import primorial, primepi, prime, primerange, factorint

%o from operator import mul

%o from functools import reduce

%o def a002110(n): return 1 if n<1 else primorial(n)

%o def a276085(n):

%o     f=factorint(n)

%o     return sum([f[i]*a002110(primepi(i) - 1) for i in f])

%o def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])

%o def a108951(n):

%o     f = factorint(n)

%o     return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])

%o def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after _Chai Wah Wu_

%o def a(n): return a276085(a108951(a019565(n)))

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 22 2017

%Y Cf. A002110, A030308, A143293, A276085, A276156, A283477, A283984.

%K nonn

%O 0,3

%A _Antti Karttunen_, Mar 19 2017