cp's OEIS Frontend

A287142 Least k such that the number of pairs of consecutive divisors of k equals n.

%I A287142 #23 Feb 18 2025 11:23:50
%S A287142 1,2,6,12,72,60,180,360,420,840,1260,3780,2520,5040,13860,36960,41580,
%T A287142 27720,55440,83160,166320,277200,491400,471240,360360,1113840,720720,
%U A287142 1081080,3341520,2162160,2827440,5405400,4324320,12972960,6126120,16576560,28274400
%N A287142 Least k such that the number of pairs of consecutive divisors of k equals n.
%C A287142 a(n) is even for n > 0.
%C A287142 We observe numbers of the decimal form (abcabc) = 360360, 720720 and numbers of the decimal form (abcabc0) = 1081080, 2162160, 5405400, 4324320, 6126120.
%C A287142 Observation and questions: many terms are products of powers of a contiguous set of the smallest primes. Many early terms of a(n) are in A002182; e.g., a(35) - A002182(44). The smallest exception outside of the empty product a(0) = 1 is a(22) = 491400 = 2^3 * 3^3 * 5^2 * 7 * 13. In other words, many terms have A006530(a(n)) < A053669(a(n)); a(22) is the smallest exception. Other exceptions include {471240, 1113840, 3341520, 2827440, 16576560, 28274400, ...}. A000720(A053669(a(22))) - A000720(A006530(a(22))) = 1, but the first instance of 2 for this function is a(35) = 16576560. This is evident by mapping A054841 across a(n). Are there a finite number of terms of a(n) that are also in A002182? Are there a finite number of terms of a(n) that have A006530(a(n)) < A053669(a(n)); are they becoming less frequent as n increases? - _Michael De Vlieger_, May 20 2017
%C A287142 In other words, a(n) is the least integer with exactly n divisors that are oblong (A002378). - _Bernard Schott_, Jul 30 2022
%F A287142 a(n) = 2*A130317(n) for n >= 1. - _Bernard Schott_, Jul 30 2022
%e A287142 a(3) = 12 because the divisors of 12 are {1, 2, 3, 4, 6, 12} with three pairs of consecutive divisors (1, 2), (2, 3) and (3, 4).
%p A287142 with(numtheory):
%p A287142 for n from 0 to 60 do:
%p A287142 ii:=0:
%p A287142   for k from 1 to 10^8 while(ii=0) do:
%p A287142     d0:=divisors(k):n0:=nops(d0):c0:=0:
%p A287142       for i from 1 to n0-1 do:
%p A287142         if d0[i+1]=d0[i]+1
%p A287142          then
%p A287142           c0:=c0+1:
%p A287142           else
%p A287142          fi:
%p A287142        od:
%p A287142        if c0=n
%p A287142        then
%p A287142      ii:=1:printf(“%d %d \n”,n,k):
%p A287142      else
%p A287142      fi:
%p A287142    od:
%p A287142   od:
%t A287142 Function[s, Function[t, ReplacePart[t, Map[#1 -> #2 & @@ # &, Transpose@{1 + Keys@ s, Values[s][[All, 1]]}]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Table[DivisorSum[n, 1 &, Divisible[n, # + 1] &], {n, 2 * 10^6}] (* _Michael De Vlieger_, May 20 2017, Version 10 *)
%o A287142 (PARI) isok(n,k) = {dk = divisors(k); ddk = vector(#dk-1, j, dk[j+1] - dk[j]); #select(x->x==1, ddk) == n;}
%o A287142 a(n) = {my(k=1); while (!isok(n, k), k++); k;} \\ _Michel Marcus_, May 20 2017
%Y A287142 Cf. A000005, A002378, A027750, A129308, A130317.
%Y A287142 Essentially the same as A088726.
%K A287142 nonn
%O A287142 0,2
%A A287142 _Michel Lagneau_, May 20 2017