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A287142
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Least k such that the number of pairs of consecutive divisors of k equals n.
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3
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1, 2, 6, 12, 72, 60, 180, 360, 420, 840, 1260, 3780, 2520, 5040, 13860, 36960, 41580, 27720, 55440, 83160, 166320, 277200, 491400, 471240, 360360, 1113840, 720720, 1081080, 3341520, 2162160, 2827440, 5405400, 4324320, 12972960, 6126120, 16576560, 28274400
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OFFSET
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0,2
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COMMENTS
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a(n) is even for n > 0.
We observe numbers of the decimal form (abcabc) = 360360, 720720 and numbers of the decimal form (abcabc0) = 1081080, 2162160, 5405400, 4324320, 6126120.
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
In other words, a(n) is the least integer with exactly n divisors that are oblong (A002378). - Bernard Schott, Jul 30 2022
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LINKS
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FORMULA
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EXAMPLE
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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).
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MAPLE
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with(numtheory):
for n from 0 to 60 do:
ii:=0:
for k from 1 to 10^8 while(ii=0) do:
d0:=divisors(k):n0:=nops(d0):c0:=0:
for i from 1 to n0-1 do:
if d0[i+1]=d0[i]+1
then
c0:=c0+1:
else
fi:
od:
if c0=n
then
ii:=1:printf(“%d %d \n”, n, k):
else
fi:
od:
od:
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MATHEMATICA
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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 *)
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PROG
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(PARI) isok(n, k) = {dk = divisors(k); ddk = vector(#dk-1, j, dk[j+1] - dk[j]); #select(x->x==1, ddk) == n; }
a(n) = {my(k=1); while (!isok(n, k), k++); k; } \\ Michel Marcus, May 20 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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