A340157 -id:A340157 - cp's OEIS frontend

A340159 a(n) is the smallest number m such that numbers m, m + 1, m + 2, ..., m + n - 1 have k, 2k, 3k, ..., n*k divisors respectively.

1, 1, 61, 421, 211082, 11238341, 16788951482, 41126483642

Offset: 1

Jaroslav Krizek, Dec 29 2020

a(n) is the smallest number m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/3 = tau(m + 3)/4 = ... = tau(m + n - 1)/n, where tau(k) = the number of divisors of k (A000005).
Corresponding values of tau(a(n)): 1, 1, 2, 2, 4, 4, 4, ...
a(8) <= 41126483642. - David A. Corneth, Dec 31 2020
Any subsequent terms are > 10^11. - Lucas A. Brown, Mar 18 2024

			a(3) = 61 because 61, 62 and 63 have 2, 4, and 6 divisors respectively and there is no smaller number having this property.

Lucas A. Brown, Python program.

Cf. A294528 for similar sequence with primes.

Cf. A000005, A063446, A339778, A340157, A340158.

isok(m, n) = {my(k=numdiv(m)); for (i=1, n-1, if (numdiv(m+i) != (i+1)*k, return (0));); return(1);}
a(n) = my(m=1); while(!isok(m, n), m++); m; \\ Michel Marcus, Dec 30 2020

Python
```
# see LINKS
```

a(7) from Jinyuan Wang, Dec 31 2020

a(8) from Lucas A. Brown, Mar 18 2024

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

Original entry on oeis.org

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

Jaroslav Krizek, Dec 29 2020

nonn

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.

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

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

[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]

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

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

A340871 Primes p such that p, p + 1, p + 2 and p + 3 have 2, 4, 6 and 8 divisors respectively.

Original entry on oeis.org

421, 30661, 50821, 54421, 130021, 195541, 423781, 635461, 1003381, 1577941, 1597381, 1883941, 2070421, 2100661, 2162581, 2534821, 2585941, 2666581, 2851621, 3296581, 3658021, 3800581, 4657381, 4969141, 5739541, 5962741, 6188821, 6537301, 6556741, 7090261

Offset: 1

Jaroslav Krizek, Jan 24 2021

nonn

Primes from A340157.

Term 1524085621 is the smallest prime p such that p, p + 1, p + 2, p + 3 and p + 4 have 2, 4, 6, 8 and 10 divisors respectively. No such run exists for any 6 consecutive integers with prime p with this property (see A294528).

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

Cf. A000005, A294528, A340157.

[m: m in [1..10^7] | IsPrime(m) and #Divisors(m + 1) eq 4 and #Divisors(m + 2) eq 6 and #Divisors(m + 3) eq 8];

Select[Range[10^6], DivisorSigma[0, # + {0, 1, 2, 3}] == {2, 4, 6, 8} &] (* Amiram Eldar, Jan 25 2021 *)
Select[Prime[Range[490000]],DivisorSigma[0,#+{1,2,3}]=={4,6,8}&] (* Harvey P. Dale, Oct 02 2021 *)

cp's OEIS Frontend

A340159 a(n) is the smallest number m such that numbers m, m + 1, m + 2, ..., m + n - 1 have k, 2k, 3k, ..., n*k divisors respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A340871 Primes p such that p, p + 1, p + 2 and p + 3 have 2, 4, 6 and 8 divisors respectively.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

cp's OEIS Frontend

A340159 a(n) is the smallest number m such that numbers m, m + 1, m + 2, ..., m + n - 1 have k, 2*k, 3*k, ..., n*k divisors respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A340871 Primes p such that p, p + 1, p + 2 and p + 3 have 2, 4, 6 and 8 divisors respectively.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A340159 a(n) is the smallest number m such that numbers m, m + 1, m + 2, ..., m + n - 1 have k, 2k, 3k, ..., n*k divisors respectively.