A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

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.

Links

Table of n, a(n) for n=1..47.

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.

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

Sequence in context: A213739 A102067 A181924 * A270257 A367361 A180617

Adjacent sequences: A367102 A367103 A367104 * A367106 A367107 A367108

Keyword

nonn

Author

Gus Wiseman, Nov 09 2023

Status

approved