cp's OEIS Frontend

A379721 Numbers whose prime indices have sum <= product.

%I A379721 #5 Jan 05 2025 22:35:02
%S A379721 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 A379721 45,47,49,50,51,53,54,55,57,59,61,63,65,66,67,69,70,71,73,75,77,78,79,
%U A379721 81,83,84,85,87,89,90,91,93,95,97,98,99,100,101,102,103
%N A379721 Numbers whose prime indices have sum <= product.
%C A379721 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 A379721 Partitions of this type are counted by A319005.
%C A379721 The complement is A325038.
%F A379721 Number k such that A056239(k) <= A003963(k).
%e A379721 The terms together with their prime indices begin:
%e A379721     1: {}
%e A379721     2: {1}
%e A379721     3: {2}
%e A379721     5: {3}
%e A379721     7: {4}
%e A379721     9: {2,2}
%e A379721    11: {5}
%e A379721    13: {6}
%e A379721    15: {2,3}
%e A379721    17: {7}
%e A379721    19: {8}
%e A379721    21: {2,4}
%e A379721    23: {9}
%e A379721    25: {3,3}
%e A379721    27: {2,2,2}
%e A379721    29: {10}
%e A379721    30: {1,2,3}
%t A379721 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379721 Select[Range[100],Total[prix[#]]<=Times@@prix[#]&]
%Y A379721 The case of equality is A301987, inequality A325037.
%Y A379721 Nonpositive positions in A325036.
%Y A379721 A000040 lists the primes, differences A001223.
%Y A379721 A055396 gives least prime index, greatest A061395.
%Y A379721 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379721 A379681 gives sum plus product of prime indices, firsts A379682.
%Y A379721 Counting and ranking multisets by comparing sum and product:
%Y A379721 - same: A001055 (strict A045778), ranks A301987
%Y A379721 - divisible: A057567, ranks A326155
%Y A379721 - divisor: A057568, ranks A326149, see A326156, A326172, A379733
%Y A379721 - greater: A096276 shifted right, ranks A325038
%Y A379721 - greater or equal: A096276, ranks A325044
%Y A379721 - less: A114324, ranks A325037, see A318029
%Y A379721 - less or equal: A319005, ranks A379721 (this)
%Y A379721 - different: A379736, ranks A379722, see A111133
%Y A379721 Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.
%K A379721 nonn
%O A379721 1,2
%A A379721 _Gus Wiseman_, Jan 05 2025