Those Fascinating Numbers 103
719
the smallest prime factor of the Mersenne number
2359
1, whose complete
factorization is given by
2359
1 = 719 · 855857 · 778165529 · 65877330027880703
·370906580744492785430299503112990447
·100361196281293745682520861860411315001;
the smallest prime number of height 6 (see the number 283).
720 (=
24
·
32
· 5)
the
14th
highly composite number (see the number 180);
the sixth number n 1 such that φ(σ(n)) = n (see the number 128).
726
the first of the six smallest consecutive numbers at which the Ω(n) function
takes distinct values, namely here the values 4, 1, 5, 6, 3 and 2: if nk, k 2,
stands
for107
the first of the k smallest consecutive numbers at which the Ω(n)
function takes k distinct values, then n2 = 3, n3 = 6, n4 = 15, n5 = 60,
n6 = 726, n7 = 6 318, n8 = 189 375, n9 = 755 968 and n10 = 683 441 871 (see
the number 417 for the same matter, but this time with the ω(n) function).
727
the smallest prime number p such that p + 2 is108 a cube: here 727 + 2 = 93;
the sequence of numbers satisfying this property begins as follows: 727, 3373,
6857, 19681, 29789, 50651, 300761, 753569, 970297, 1295027, . . . ;
the third prime number of the form n!+ n +1, here with n = 6 (see the number
52).
729
the fifth solution of φ(σ(n)) = σ(φ(n)) (see the number 516).
732
the tenth number n for which the distance from en to the nearest integer is the
smallest (see the numbers 178 and 58).
107See the footnote tied to the number 417.
108It
would be interesting if one could prove that this sequence is infinite. This could be a difficult
task since on the one hand it is not known if the sequence (n3 2)n≥3 contains infinitely many
prime numbers, while on the other hand the challenge could be similar to that of the twin prime
conjecture.
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