cp's OEIS Frontend

A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

%I A380220 #6 Jan 22 2025 22:26:30
%S A380220 2,1,21,25,39,35,57,55,49,65,75,77,129,95,91,105,183,119,125,143,133,
%T A380220 185,147,161,169,195,175,209,339,217,255,253,259,305,247,285,273,245,
%U A380220 301,299,345,323,325,357,371,435,669,391,361,403,399,473,343,469,481
%N A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.
%C A380220 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. The sum and product of prime indices are A056239 and A003963 respectively.
%F A380220 Satisfies A003963(a(n)) - A056239(a(n)) = n.
%e A380220 The least number whose prime indices satisfy (product) - (sum) = 3 is 25 (prime indices {3,3}), so a(3) = 25.
%t A380220 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380220 pp=Table[Total[prix[n]]-Times@@prix[n],{n,100}];
%t A380220 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A380220 Table[Position[pp,-i][[1,1]],{i,0,mnrm[-DeleteCases[pp,0|_?Positive]]}]
%Y A380220 Position of first appearance of n in A325036.
%Y A380220 For sum instead of difference we have A379682, firsts of A379681.
%Y A380220 A000040 lists the primes, differences A001223.
%Y A380220 A003963 multiplies together prime indices.
%Y A380220 A055396 gives least prime index, greatest A061395.
%Y A380220 A056239 adds up prime indices, row sums of A112798.
%Y A380220 The subtraction A325036 takes the following values:
%Y A380220 - zero: A301987, counted by A001055 (strict A045778).
%Y A380220 - negative: A325037, counted by A114324, see A318029
%Y A380220 - positive: A325038, counted by A096276 shifted right
%Y A380220 - negative one: A325041, counted by A028422
%Y A380220 - one: A325042, counted by A001055 shifted right
%Y A380220 - nonnegative: A325044, counted by A096276
%Y A380220 - nonpositive: A379721, counted by A319005
%Y A380220 Cf. A000720, A178503, A175508, A318950, A319000, A325034, A326151, A326153/A326154, A379319, A379722.
%K A380220 nonn,look
%O A380220 0,1
%A A380220 _Gus Wiseman_, Jan 21 2025