A379721 - OEIS

%I #5 Jan 05 2025 22:35:02

%S 1,2,3,5,7,9,11,13,15,17,19,21,23,25,27,29,30,31,33,35,37,39,41,42,43,

%T 45,47,49,50,51,53,54,55,57,59,61,63,65,66,67,69,70,71,73,75,77,78,79,

%U 81,83,84,85,87,89,90,91,93,95,97,98,99,100,101,102,103

%N Numbers whose prime indices have sum <= product.

%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.

%C Partitions of this type are counted by A319005.

%C The complement is A325038.

%F Number k such that A056239(k) <= A003963(k).

%e The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     3: {2}

%e     5: {3}

%e     7: {4}

%e     9: {2,2}

%e    11: {5}

%e    13: {6}

%e    15: {2,3}

%e    17: {7}

%e    19: {8}

%e    21: {2,4}

%e    23: {9}

%e    25: {3,3}

%e    27: {2,2,2}

%e    29: {10}

%e    30: {1,2,3}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Total[prix[#]]<=Times@@prix[#]&]

%Y The case of equality is A301987, inequality A325037.

%Y Nonpositive positions in A325036.

%Y A000040 lists the primes, differences A001223.

%Y A055396 gives least prime index, greatest A061395.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A379681 gives sum plus product of prime indices, firsts A379682.

%Y Counting and ranking multisets by comparing sum and product:

%Y - same: A001055 (strict A045778), ranks A301987

%Y - divisible: A057567, ranks A326155

%Y - divisor: A057568, ranks A326149, see A326156, A326172, A379733

%Y - greater: A096276 shifted right, ranks A325038

%Y - greater or equal: A096276, ranks A325044

%Y - less: A114324, ranks A325037, see A318029

%Y - less or equal: A319005, ranks A379721 (this)

%Y - different: A379736, ranks A379722, see A111133

%Y Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.

%K nonn,new

%O 1,2

%A _Gus Wiseman_, Jan 05 2025