A340157 - OEIS

%I #12 Sep 08 2022 08:46:26

%S 421,3013,5029,5223,5245,5893,6487,10533,11911,14677,17173,23077,

%T 23573,24613,25141,25213,27637,27973,28357,30661,32407,34117,37477,

%U 38282,39751,43495,45973,47365,48423,50821,50965,53413,53989,54421,55141,56103,57877,58165

%N Numbers m such that numbers m, m + 1, m + 2 and m + 3 have k, 2k, 3k and 4k divisors respectively.

%C Numbers m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/3 = tau(m + 3)/4, where tau(k) = the number of divisors of k (A000005).

%C Quadruplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2), tau(a(n) + 3)] = [tau(a(n)), 2*tau(a(n)), 3*tau(a(n)), 4*tau(a(n))]: [2, 4, 6, 8], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], [4, 8, 12, 16], ...

%C Corresponding values of tau(a(n)): 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...

%C Subsequence of A063446 and A339778. Supersequence of A340158.

%C Prime terms (primes p such that p, p + 1, p + 2 and p + 3 have 2, 4, 6 and 8 divisors respectively): 421, 30661, 50821, 54421, 130021, 195541, 423781, 635461, 1003381, 1577941, 1597381, 1883941, ...

%e tau(421) = 2, tau(422) = 4, tau(423) = 6, tau(424) = 8.

%t Select[Range[60000], Equal @@ (DivisorSigma[0, # + {0, 1, 2, 3}]/{1, 2, 3, 4}) &] (* _Amiram Eldar_, Dec 30 2020 *)

%o (Magma) [m: m in [1..10^5] | #Divisors(m) eq #Divisors(m + 1)/2 and #Divisors(m) eq #Divisors(m + 2)/3 and #Divisors(m) eq #Divisors(m + 3)/4]

%o (PARI) isok(m, n=4) = {my(k=numdiv(m)); for (i=1, n-1, if (numdiv(m+i) != (i+1)*k, return (0));); return(1);} \\ _Michel Marcus_, Dec 30 2020

%Y Cf. A000005, A063446, A294528, A303578, A339778, A340158, A340159.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Dec 29 2020