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A367449
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Numbers k for which there are exactly k pairs (i, j), 1 <= i < j < k, such that i + j is a divisor of k.
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0
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30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 208, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1312, 1338, 1362, 1374
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OFFSET
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1,1
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COMMENTS
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Numbers k >= 1 for which A367588(k) = Sum_{d|k} floor((d-1)/2) = k;
The sequence is infinite because all numbers of the form m = 6*p, p >= 5 prime (A138636), are terms.
Indeed: sigma(6*p) - tau(6(p) - A183063(6*p) = 3*4*(p + 1) - 8 - 4 = 12*p = 2*m.
If m = 2^k*p, p = 2^(k + 1) - 4*k - 3 prime number, then m is a term. Indeed: sigma(m) - tau(m) - A183063(m) = (2^(k + 1) - 1)*(p + 1) - 2*(k + 1) - 2*k = 2*m.
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LINKS
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EXAMPLE
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30 is a term since it has exactly 30 pairs (i,j): (1, 2), (2, 3), (1, 4), (2, 4), (1, 5), (4, 6), (3, 7), (2, 8), (7, 8), (1,9), (6, 9), (5, 10), (4, 11), (3, 12), (2, 13), (1, 14), (14, 16), (13, 17),(12, 18), (11, 19), (10, 20), (9, 21), (8, 22), (7, 23), (6, 24), (5, 25), (4,26), (3, 27), (2, 28), (1, 29).
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MAPLE
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filter:= proc(n) uses numtheory;
sigma(n) - tau(n) - `if`(n::even, tau(n/2), 0) = 2*n
end proc:
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MATHEMATICA
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f1[p_, e_] := e+1; f1[2, e_] := 2*e+1; f2[p_, e_] := (p^(e+1)-1)/(p-1); s[1] = 0; s[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct - Times @@ f1 @@@ fct]; Select[Range[1400], s[#] == 2*# &] (* Amiram Eldar, Dec 16 2023 *)
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PROG
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(Magma) [k:k in [1..1000]|(DivisorSigma(1, k)-#Divisors(k)-#[d:d in Divisors(k)| IsEven(d)]) eq 2*k ];
(PARI) isok(k) = sumdiv(k, d, (d-1)\2) == k; \\ Michel Marcus, Dec 19 2023
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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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