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A367105
Least positive integer with n more divisors than distinct subset-sums of prime indices.
1
1, 12, 24, 48, 60, 192, 144, 120, 180, 336, 240, 630, 420, 360, 900, 1344, 960, 1008, 720, 840, 2340, 1980, 1260, 1440, 3120, 2640, 1680, 4032, 2880, 6840, 3600, 4620, 3780, 2520, 6480, 11700, 8820, 6300, 7200, 10560, 6720, 12240, 9360, 7920, 5040, 10920, 9240
OFFSET
1,2
COMMENTS
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.
FORMULA
A000005(a(n)) - A299701(a(n)) = n.
EXAMPLE
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.
The terms together with their prime indices begin:
1: {}
12: {1,1,2}
24: {1,1,1,2}
48: {1,1,1,1,2}
60: {1,1,2,3}
120: {1,1,1,2,3}
144: {1,1,1,1,2,2}
180: {1,1,2,2,3}
192: {1,1,1,1,1,1,2}
240: {1,1,1,1,2,3}
336: {1,1,1,1,2,4}
360: {1,1,1,2,2,3}
420: {1,1,2,3,4}
630: {1,2,2,3,4}
720: {1,1,1,1,2,2,3}
840: {1,1,1,2,3,4}
900: {1,1,2,2,3,3}
960: {1,1,1,1,1,1,2,3}
MATHEMATICA
nn=1000;
w=Table[DivisorSigma[0, n]-Length[Union[Total/@Subsets[prix[n]]]], {n, nn}];
spnm[y_]:=Max@@Select[Union[y], Function[i, Union[Select[y, #<=i&]]==Range[0, i]]];
Table[Position[w, k][[1, 1]], {k, 0, spnm[w]}]
CROSSREFS
The first part (divisors) is A000005.
The second part (subset-sums of prime indices) is A299701, positive A304793.
These are the positions of first appearances in the difference A325801.
The binary version is A367093, firsts of A086971 - A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Sequence in context: A377590 A102067 A181924 * A270257 A367361 A180617
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 09 2023
STATUS
approved