310 Jean-Marie De Koninck
36 721 681
the smallest number n such that σ(n) σ(n+1) σ(n+2) σ(n+3) σ(n+
4); it is also the smallest number n such that σ(n) σ(n + 1) . . . σ(n + 5);
here 37085364 55082526 56162808 64262954 64350720 76876128;
the numbers in question can be factored as follows:
36 721 681 = 101 · 363581,
36 721 682 = 2 · 18360841,
36 721 683 =
32
· 17 · 240011,
36 721 684 =
22
· 9180421,
36 721 685 = 5 · 7 ·
112
· 13 · 23 · 29,
36 721 686 = 2 · 3 · 37 · 53 · 3121;
denoting by nk the smallest number n such that σ(n) σ(n + 1) . . .
σ(n + k 1), we have the following table:
k 2 3 4 5 6 7
nk 9 13 13 36 721 681 36 721 681 ? ?
36 865 412
the smallest number n which allows the sum
i≤n
1
i
to exceed 18 (see the number
83).
37 033 919
the smallest number n such that σ(n + 1) = 4σ(n): observe that 37 033 919 =
13 · 863 · 3301 and 37 033 920 =
26
·
32
· 5 · 7 · 11 · 167; the sequence of numbers
satisfying this property begins as follows: 37033919, 141162839, 264995639,
596672999, 606523679, 630777839, . . . (see the number 1253 for the smallest
numbers n = nk satisfying σ(n + 1) = kσ(n)).
37 156 667
the exponent of the
45th
known Mersenne prime
237 156 667
1 (a 11 185 272
digit number) discovered by Hans-Michael Elvenich in September 2008, using
the programme developed by G. Woltman (see the number 1 398 269).
37 387 980
the ninth even number n such that σI (n) = σI (n + 2) (see the number 54 178).
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