cp's OEIS Frontend

A372427 Numbers whose binary indices and prime indices have the same sum.

%I A372427 #18 May 23 2024 00:49:29
%S A372427 19,33,34,69,74,82,130,133,305,412,428,436,533,721,755,808,917,978,
%T A372427 1036,1058,1062,1121,1133,1143,1341,1356,1630,1639,1784,1807,1837,
%U A372427 1990,2057,2115,2130,2133,2163,2260,2324,2328,2354,2358,2512,2534,2627,2771,2825
%N A372427 Numbers whose binary indices and prime indices have the same sum.
%C A372427 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 A372427 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 A372427 John Tyler Rascoe, <a href="/A372427/b372427.txt">Table of n, a(n) for n = 1..10000</a>
%e A372427 The binary indices of 130 are {2,8}, and the prime indices are {1,3,6}. Both sum to 10, so 130 is in the sequence.
%e A372427 The terms together with their prime indices begin:
%e A372427    19: {8}
%e A372427    33: {2,5}
%e A372427    34: {1,7}
%e A372427    69: {2,9}
%e A372427    74: {1,12}
%e A372427    82: {1,13}
%e A372427   130: {1,3,6}
%e A372427   133: {4,8}
%e A372427   305: {3,18}
%e A372427   412: {1,1,27}
%e A372427   428: {1,1,28}
%e A372427 The terms together with their binary expansions and binary indices begin:
%e A372427    19:      10011 ~ {1,2,5}
%e A372427    33:     100001 ~ {1,6}
%e A372427    34:     100010 ~ {2,6}
%e A372427    69:    1000101 ~ {1,3,7}
%e A372427    74:    1001010 ~ {2,4,7}
%e A372427    82:    1010010 ~ {2,5,7}
%e A372427   130:   10000010 ~ {2,8}
%e A372427   133:   10000101 ~ {1,3,8}
%e A372427   305:  100110001 ~ {1,5,6,9}
%e A372427   412:  110011100 ~ {3,4,5,8,9}
%e A372427   428:  110101100 ~ {3,4,6,8,9}
%t A372427 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A372427 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A372427 Select[Range[100],Total[prix[#]]==Total[bix[#]]&]
%Y A372427 For length instead of sum we get A071814.
%Y A372427 Positions of zeros in A372428.
%Y A372427 For maximum instead of sum we have A372436.
%Y A372427 A003963 gives product of prime indices.
%Y A372427 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372427 A029837 gives greatest binary index, least A001511.
%Y A372427 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A372427 A061395 gives greatest prime index, least A055396.
%Y A372427 A070939 gives length of binary expansion.
%Y A372427 A096111 gives product of binary indices.
%Y A372427 A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
%Y A372427 A326031 gives weight of the set-system with BII-number n.
%Y A372427 Cf. A000720, A001221, A014499, A030101, A066099, A304818, A318283, A355536, A359401, A359402, A372429-A372433, A372441.
%K A372427 nonn,base
%O A372427 1,1
%A A372427 _Gus Wiseman_, May 01 2024