cp's OEIS Frontend

A380410 Numbers with greater product of prime indices than sum of prime factors.

%I A380410 #5 Jan 26 2025 09:12:28
%S A380410 1,45,49,63,75,77,81,91,99,105,117,119,121,125,126,133,135,143,147,
%T A380410 150,153,161,162,165,169,171,175,182,187,189,195,198,203,207,209,210,
%U A380410 217,221,225,231,234,238,242,243,245,247,250,253,255,259,261,266,270,273
%N A380410 Numbers with greater product of prime indices than sum of prime factors.
%C A380410 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, with product A003963.
%F A380410 A003963(a(n)) > A001414(a(n)).
%e A380410 126 has prime indices {1,2,2,4} and prime factors {2,3,3,7}, and 16 > 15, so 126 is in the sequence.
%e A380410 The terms together with their prime indices begin:
%e A380410      1: {}
%e A380410     45: {2,2,3}
%e A380410     49: {4,4}
%e A380410     63: {2,2,4}
%e A380410     75: {2,3,3}
%e A380410     77: {4,5}
%e A380410     81: {2,2,2,2}
%e A380410     91: {4,6}
%e A380410     99: {2,2,5}
%e A380410    105: {2,3,4}
%e A380410    117: {2,2,6}
%e A380410    119: {4,7}
%e A380410    121: {5,5}
%e A380410    125: {3,3,3}
%e A380410    126: {1,2,2,4}
%e A380410    133: {4,8}
%e A380410    135: {2,2,2,3}
%t A380410 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380410 Select[Range[100],Times@@prix[#]>Plus@@Prime/@prix[#]&]
%Y A380410 For factors instead of indices we have A002808.
%Y A380410 The case of prime powers is A244623.
%Y A380410 For indices instead of factors we have A325037, see also A325038.
%Y A380410 The version for equality is A331384, counted by A331383.
%Y A380410 Positions of positive terms in A380344.
%Y A380410 Partitions of this type are counted by A380411.
%Y A380410 A000040 lists the primes, differences A001223.
%Y A380410 A001222 counts prime factors with multiplicity.
%Y A380410 A055396 gives least prime index, greatest A061395.
%Y A380410 Triangles:
%Y A380410 - A027746 = prime factors
%Y A380410 - A112798 = prime indices
%Y A380410 Statistics:
%Y A380410 - A000027 = product of prime factors = row products of A027746
%Y A380410 - A001414 = sum of prime factors = row sums of A027746
%Y A380410 - A003963 = product of prime indices = row products of A112798
%Y A380410 - A056239 = sum of prime indices = row sums of A112798
%Y A380410 Combinations:
%Y A380410 - A075254 = product of factors + sum of factors = A000027 + A001414
%Y A380410 - A075255 = product of factors - sum of factors = A000027 - A001414
%Y A380410 - A178503 = product of factors - sum of indices = A000027 - A056239
%Y A380410 - A325036 = product of indices - sum of indices = A003963 - A056239
%Y A380410 - A379681 = product of indices + sum of indices = A003963 + A056239
%Y A380410 - A380344 = product of indices - sum of factors = A003963 - A001414
%Y A380410 - A380345 = product of factors + sum of indices = A000027 + A056239
%Y A380410 - A380409 = product of indices + sum of factors = A003963 + A001414
%Y A380410 Cf. A000720, A175508, A301987, A319000, A325040, A325041, A325044, A379682, A380220.
%K A380410 nonn
%O A380410 1,2
%A A380410 _Gus Wiseman_, Jan 25 2025