A372428 - OEIS

%I #22 Oct 19 2024 08:33:00

%S 1,1,1,1,1,2,2,1,1,2,2,3,2,4,5,1,-1,2,0,3,3,4,2,4,4,4,6,6,3,8,4,1,0,0,

%T 2,3,-2,2,4,4,-2,5,-1,6,7,5,1,5,4,6,5,6,-1,9,9,8,6,6,1,11,1,8,13,1,-1,

%U 1,-9,1,0,4,-7,4,-9,0,6,4,6,7,-5,5,5,0,-8

%N Sum of binary indices of n minus sum of prime indices of n.

%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%H John Tyler Rascoe, <a href="/A372428/b372428.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A029931(n) - A056239(n).

%e The binary indices of 65 are {1,7}, and the prime indices are {3,6}, so a(65) = 8 - 9 = -1.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Table[Total[bix[n]]-Total[prix[n]],{n,100}]

%o (Python)

%o from itertools import count, islice

%o from sympy import sieve, factorint

%o def a_gen():

%o     for n in count(1):

%o         b = sum((i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1')

%o         p = sum(sieve.search(i)[0] for i in factorint(n, multiple=True))

%o         yield(b-p)

%o A372428_list = list(islice(a_gen(), 83)) # _John Tyler Rascoe_, May 04 2024

%o (Python)

%o from sympy import primepi, factorint

%o def A372428(n): return int(sum(i for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')-sum(primepi(p)*e for p, e in factorint(n).items())) # _Chai Wah Wu_, Oct 18 2024

%Y Positions of zeros are A372427.

%Y For minimum instead of sum we have A372437.

%Y For length instead of sum we have A372441, zeros A071814.

%Y For maximum instead of sum we have A372442, zeros A372436.

%Y Positions of odd terms are A372586, even A372587.

%Y A003963 gives product of prime indices.

%Y A019565 gives Heinz number of binary indices, adjoint A048675.

%Y A029837 gives greatest binary index, least A001511.

%Y A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

%Y A061395 gives greatest prime index, least A055396.

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y A112798 lists prime indices, length A001222, reverse A296150, sum A056239.

%Y A326031 gives weight of the set-system with BII-number n.

%Y Cf. A000720, A001221, A014499, A030101, A059893, A066099, A243055, A304818, A359495, A372429-A372432.

%K sign,base

%O 1,6

%A _Gus Wiseman_, May 02 2024