242 Jean-Marie De Koninck

150 287

• the smallest prime factor of the Mersenne number

2163

− 1, whose complete

factorization is given by

2163

− 1 = 150287 · 704161 · 110211473 · 27669118297 · 36230454570129675721.

151 023

• the

26th

number n such that n ·

2n

− 1 is prime (see the number 115).

153 720

• the third number n such that φ(n) + σ(n) = 4n (see the number 23 760 as well

as the number 312).

153 846

• the third number which quadruples when its last digit is moved to the first

position (see the number 102 564).

154 876

• the smallest Niven number n such that n +50 is also a Niven number, but with

no others in between (see the number 28 680).

155 863

• the smallest prime number q for which the value of the corresponding sum

∑

p≤q

p uses each of the ten digits once and only once: here the sum is equal

to 1 063 254 978; exactly 13 prime numbers q satisfy this

property174:

these

are the primes 155 863, 207 301, 260 539, 289 847, 309 977, 322 429, 334 427,

356 831, 376 291, 381 631, 382 873, 416 821 and 441 461, while the 13 correspond-

ing sums are 1063254978, 1829360475, 2835410967, 3478029561, 3954061782,

4271593608, 4583260179, 5190863247, 5741092638, 5897230146, 5932401786,

6980571324 and 7803615924.

157 951

• the smallest prime number p such that Ω(p−1) = Ω(p+1) = 9 : here 157 950 =

2 ·

35

·

52

· 13 and 157 952 =

28

· 617 (see the number 271).

174Interestingly,

there are no prime numbers q for which the value of the corresponding sum

∑

p≤q

p2 uses each of the digits 0,1,2,. . . ,9 once and only once. In fact, one can prove that there

exist 411 prime numbers q for which

109

∑

p≤q

p

1010;

but each of these 411 sums has a

decimal expansion where at least one digit is repeated.