A367105 - OEIS

%I #7 Nov 09 2023 08:51:09

%S 1,12,24,48,60,192,144,120,180,336,240,630,420,360,900,1344,960,1008,

%T 720,840,2340,1980,1260,1440,3120,2640,1680,4032,2880,6840,3600,4620,

%U 3780,2520,6480,11700,8820,6300,7200,10560,6720,12240,9360,7920,5040,10920,9240

%N Least positive integer with n more divisors than distinct subset-sums of prime indices.

%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.An integer n is a subset-sum (A299701, A304792) of a multiset y if there exists a submultiset of y with sum n.

%F A000005(a(n)) - A299701(a(n)) = n.

%e The divisors of 60 are {1,2,3,4,5,6,10,12,15,20,30,60}, and the distinct subset-sums of its prime indices {1,1,2,3} are {0,1,2,3,4,5,6,7}, so the difference is 12 - 8 = 4. Since 60 is the first number with this difference, we have a(4) = 60.

%e The terms together with their prime indices begin:

%e      1: {}

%e     12: {1,1,2}

%e     24: {1,1,1,2}

%e     48: {1,1,1,1,2}

%e     60: {1,1,2,3}

%e    120: {1,1,1,2,3}

%e    144: {1,1,1,1,2,2}

%e    180: {1,1,2,2,3}

%e    192: {1,1,1,1,1,1,2}

%e    240: {1,1,1,1,2,3}

%e    336: {1,1,1,1,2,4}

%e    360: {1,1,1,2,2,3}

%e    420: {1,1,2,3,4}

%e    630: {1,2,2,3,4}

%e    720: {1,1,1,1,2,2,3}

%e    840: {1,1,1,2,3,4}

%e    900: {1,1,2,2,3,3}

%e    960: {1,1,1,1,1,1,2,3}

%t nn=1000;

%t w=Table[DivisorSigma[0,n]-Length[Union[Total/@Subsets[prix[n]]]],{n,nn}];

%t spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[0,i]]];

%t Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

%Y The first part (divisors) is A000005.

%Y The second part (subset-sums of prime indices) is A299701, positive A304793.

%Y These are the positions of first appearances in the difference A325801.

%Y The binary version is A367093, firsts of  A086971 - A366739.

%Y A001222 counts prime factors (or prime indices), distinct A001221.

%Y A056239 adds up prime indices, row sums of A112798.

%Y Cf. A000720, A220264, A304792, A365920, A366740, A367097.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 09 2023