cp's OEIS Frontend

A379722 Numbers whose prime indices do not have the same sum as product.

%I A379722 #6 Jan 08 2025 19:37:48
%S A379722 1,4,6,8,10,12,14,15,16,18,20,21,22,24,25,26,27,28,32,33,34,35,36,38,
%T A379722 39,40,42,44,45,46,48,49,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,
%U A379722 69,70,72,74,75,76,77,78,80,81,82,85,86,87,88,90,91,92,93
%N A379722 Numbers whose prime indices do not have the same sum as product.
%C A379722 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 A379722 Partitions of this type are counted by A379736.
%C A379722 The complement is A301987, counted by A001055.
%e A379722 The terms together with their prime indices begin:
%e A379722     1: {}
%e A379722     4: {1,1}
%e A379722     6: {1,2}
%e A379722     8: {1,1,1}
%e A379722    10: {1,3}
%e A379722    12: {1,1,2}
%e A379722    14: {1,4}
%e A379722    15: {2,3}
%e A379722    16: {1,1,1,1}
%e A379722    18: {1,2,2}
%e A379722    20: {1,1,3}
%e A379722    21: {2,4}
%e A379722    22: {1,5}
%e A379722    24: {1,1,1,2}
%e A379722    25: {3,3}
%e A379722    26: {1,6}
%e A379722    27: {2,2,2}
%e A379722    28: {1,1,4}
%e A379722    32: {1,1,1,1,1}
%t A379722 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379722 Select[Range[100],Times@@prix[#]!=Total[prix[#]]&]
%Y A379722 Nonzeros of A325036.
%Y A379722 A000040 lists the primes, differences A001223.
%Y A379722 A055396 gives least prime index, greatest A061395.
%Y A379722 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A379722 A324851 finds numbers > 1 divisible by the sum of their prime indices.
%Y A379722 A379666 counts partitions by sum and product, without 1's A379668.
%Y A379722 A379681 gives sum plus product of prime indices, firsts A379682.
%Y A379722 Counting and ranking multisets by comparing sum and product:
%Y A379722 - same: A001055 (strict A045778), ranks A301987
%Y A379722 - divisible: A057567, ranks A326155
%Y A379722 - divisor: A057568, ranks A326149, see A379733
%Y A379722 - greater: A096276 shifted right, ranks A325038
%Y A379722 - greater or equal: A096276, ranks A325044
%Y A379722 - less: A114324, ranks A325037, see A318029
%Y A379722 - less or equal: A319005, ranks A379721
%Y A379722 - different: A379736, ranks A379722 (this)
%Y A379722 Cf. A000720, A003963, A025147, A075254, A111133, A178503, A319000, A325041.
%K A379722 nonn
%O A379722 1,2
%A A379722 _Gus Wiseman_, Jan 08 2025