cp's OEIS Frontend

A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

%I A380989 #33 Feb 21 2025 09:32:17
%S A380989 1,6,30,210,900,7776,27000,279936,810000,9261000,24300000,362797056,
%T A380989 729000000,13060694016,21870000000,408410100000,656100000000,
%U A380989 16926659444736,19683000000000,609359740010496,590490000000000,18010885410000000,17714700000000000
%N A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).
%C A380989 Is this sequence strictly increasing?
%C A380989 From _David Consiglio, Jr._, Feb 20 2025: (Start)
%C A380989 The answer to the question above is: no, a(21) < a(20). And all subsequent odd indexed terms are lower than their even predecessors.
%C A380989 All terms must be a product of x primes (with multiplicity) to the y power where x-y = n and x mod y = 0. There are very few combinations of numbers that meet these criteria, so checking all of them to find the minimum outcome is quite fast.
%C A380989 Example --> n=5
%C A380989 6 primes to the 1 power --> 6 distinct primes
%C A380989   2*3*5*7*11*13 = 30030
%C A380989 7 primes to the 2 power -- disallowed (5 mod 2 = 1)
%C A380989 8 primes to the 3 power -- disallowed (4 mod 3 = 1)
%C A380989 9 primes to the 4 power -- disallowed (9 mod 4 = 1)
%C A380989 10 primes to the 5 power --> 2 distinct primes
%C A380989   2*2*2*2*2*3*3*3*3*3 = 7776
%C A380989 The minimum value is 7776 and thus a(5) = 7776. (End)
%H A380989 David Consiglio, Jr., <a href="/A380989/b380989.txt">Table of n, a(n) for n = 0..100</a>
%H A380989 David Consiglio, Jr., <a href="/A380989/a380989_2.py.txt">Python program</a>
%e A380989 The terms together with their prime indices begin:
%e A380989         1: {}
%e A380989         6: {1,2}
%e A380989        30: {1,2,3}
%e A380989       210: {1,2,3,4}
%e A380989       900: {1,1,2,2,3,3}
%e A380989      7776: {1,1,1,1,1,2,2,2,2,2}
%e A380989     27000: {1,1,1,2,2,2,3,3,3}
%e A380989    279936: {1,1,1,1,1,1,1,2,2,2,2,2,2,2}
%e A380989    810000: {1,1,1,1,2,2,2,2,3,3,3,3}
%e A380989   9261000: {1,1,1,2,2,2,3,3,3,4,4,4}
%t A380989 prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
%t A380989 q=Table[Total[prisig[n]]-Total[Union[prisig[n]]],{n,10000}];
%t A380989 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A380989 Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]
%Y A380989 Position of first appearance of n in A001222 - A136565.
%Y A380989 For factors instead of exponents we have A280286 (sorted A381075), firsts of A280292.
%Y A380989 For indices instead of exponents we have A380956 (sorted A380957), firsts of A380955.
%Y A380989 A000040 lists the primes, differences A001223.
%Y A380989 A005361 gives product of prime exponents.
%Y A380989 A055396 gives least prime index, greatest A061395.
%Y A380989 A056239 (reverse A296150) adds up prime indices, row sums of A112798.
%Y A380989 A124010 lists prime exponents (signature); A001221, A051903, A051904.
%Y A380989 Cf. A046660, A071625, A075254, A075255, A076694, A130091, A130092, A290106, A380986.
%K A380989 nonn
%O A380989 0,2
%A A380989 _Gus Wiseman_, Feb 18 2025
%E A380989 a(10)-a(11) from _Michel Marcus_, Feb 20 2025
%E A380989 a(12) and beyond from _David Consiglio, Jr._, Feb 20 2025