A340158 Numbers m such that m, m + 1, m + 2, m + 3 and m + 4 have k, 2k, 3k, 4k and 5k divisors respectively.

211082, 2364062, 2774165, 3379802, 3743573, 4390682, 5651042, 5845442, 6708578, 7326122, 7371482, 8566394, 8839202, 9056282, 10154642, 10301333, 10325621, 10446242, 10540202, 11238341, 11719562, 11978762, 12377282, 12871058, 13456202, 16840058, 16954562, 17155141

Offset

1,1

Comments

Numbers m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/3 = tau(m + 3)/4 = tau(m + 4)/5, where tau(k) = the number of divisors of k (A000005).

Quintuples of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2), tau(a(n) + 3), tau(a(n) + 4)] = [tau(a(n)), 2*tau(a(n)), 3*tau(a(n)), 4*tau(a(n)), 5*tau(a(n))]: [4, 8, 12, 16, 20], [4, 8, 12, 16, 20], [4, 8, 12, 16, 20], [8, 16, 24, 32, 40], [4, 8, 12, 16, 20], [4, 8, 12, 16, 20], ...

Corresponding values of numbers k: 4, 4, 4, 8, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, ...

1524085621 is the smallest prime term (see A294528).

Subsequence of A063446, A339778 and A340157.

Links

Table of n, a(n) for n=1..28.

Example

tau(211082) = 4, tau(211083) = 8, tau(211084) = 12, tau(211085) = 16, tau(211086) = 20.

Mathematica

Select[Range[5*10^6], Equal @@ (DivisorSigma[0, # + {0, 1, 2, 3, 4}]/{1, 2, 3, 4, 5}) &] (* Amiram Eldar, Dec 30 2020 *)

Prog

(Magma) [m: m in [1..10^6] | #Divisors(m) eq #Divisors(m + 1)/2 and #Divisors(m) eq #Divisors(m + 2)/3 and #Divisors(m) eq #Divisors(m + 3)/4 and #Divisors(m) eq #Divisors(m + 4)/5]
(PARI) isok(m) = my(k = numdiv(m)); (numdiv(m+1) == 2*k) && (numdiv(m+2) == 3*k) && (numdiv(m+3) == 4*k) && (numdiv(m+4) == 5*k); \\ Michel Marcus, Jan 16 2021

Crossrefs

Cf. A000005, A063446, A294528, A339778, A340157, A340159.

Sequence in context: A072760 A178421 A069175 * A097021 A236086 A123103

Adjacent sequences: A340155 A340156 A340157 * A340159 A340160 A340161

Keyword

nonn

Author

Jaroslav Krizek, Dec 29 2020

Status

approved