A294528 -id:A294528 - 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

A340157 Numbers m such that numbers m, m + 1, m + 2 and m + 3 have k, 2k, 3k and 4k divisors respectively.

Original entry on oeis.org

421, 3013, 5029, 5223, 5245, 5893, 6487, 10533, 11911, 14677, 17173, 23077, 23573, 24613, 25141, 25213, 27637, 27973, 28357, 30661, 32407, 34117, 37477, 38282, 39751, 43495, 45973, 47365, 48423, 50821, 50965, 53413, 53989, 54421, 55141, 56103, 57877, 58165

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, where tau(k) = the number of divisors of k (A000005).
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], ...
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, ...
Subsequence of A063446 and A339778. Supersequence of A340158.
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, ...

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

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

[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]

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

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

A341214 a(n) is the smallest prime p such that p, p - 1, p - 2, ..., p - n + 1 have 2, 4, 6, ..., 2*n divisors respectively.

Original entry on oeis.org

2, 7, 47, 1019, 55414379

Offset: 1

Jaroslav Krizek, Feb 07 2021

a(n) is the smallest prime p such that tau(p) = tau(p - 1)/2 = tau(p - 2)/3 = ... = tau(p - n + 1)/n = 2, where tau(k) = the number of divisors of k (A000005).

No such prime p exists for n > 5, so a(5) is the final term. - Jon E. Schoenfield, Feb 07 2021

			a(4) = 1019 because 1016, 1017, 1018 and 1019 have 8, 6, 4, and 2 divisors respectively and there is no smaller prime having this property (see A340872).

Jon E. Schoenfield, A proof that a(5) is the final term

Cf. A341213 (similar sequence for natural numbers).

Cf. A000005, A294528, A340871, A340872.

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 *)

A363335 Irregular table read by rows: T(n,k) is the smallest m that has 2*n divisors and is at the beginning of a run of exactly k consecutive integers whose number of divisors increases by 2, or -1 if no such m exists.

Original entry on oeis.org

2, 5, 61, 421, 1524085621, 10, 27, 187, 2545622, 12, 153, 35557, 363121, 223577456873, 44753756873

Offset: 1

Jon E. Schoenfield, May 29 2023

Equivalently, T(n,k) is the smallest m such that tau(m+j) = 2*(n+j) for each j in 0..k-1 but not for j = -1 or j = k, where tau is the number-of-divisors function, A000005.
Row 1 is A294528.
No row can have more than 15 terms: if some row n were to have 16 (or more) terms, then T(n,16) would begin a run of 16 consecutive integers, two of which (let's call them x and y, with x < y) would be 4 times an odd number, and thus x and y would each have a number of divisors that is a multiple of 3, but this is impossible since y - x = 8 so tau(y) - tau(x) = 16.
Row 3 has no more than 11 terms.
T(3,7) <= 9392318479793089373.
T(3,8) <= 42121593968048969225.
T(3,9) and T(3,10) seem likely to exist.
T(3,11) cannot exist unless there is a solution to 3*p^4 + 5 = 8*q^4 with p and q prime; if T(3,11) exists, then it exceeds 10^300000.

			T(2,3) is the smallest m such that tau(m+j) = 2*(2+j) for each j in 0..2 but not for j = -1 or j = 3; i.e., tau(m) = 4, tau(m+1) = 6, and tau(m+2) = 8, but tau(m-1) != 2 and tau(m+3) != 10. The smallest such m is 187:
    m = 187 = 11*17 (which has 4 divisors),
    m+1 = 188 = 2^2*47 (which has 6 divisors), and
    m+2 = 189 = 3^3*7 (which has 8 divisors), but
    m-1 = 186 = 2*3*31 (which has 8 divisors, not 2), and
    m+3 = 190 = 2*5*19 (which has 8 divisors, not 10).
The first several rows of the table are as follows:
  Row n=1:  2, 5, 61, 421, 1524085621; (A294528)
  Row n=2:  10, 27, 187, 2545622;
  Row n=3:  12, 153, 35557, 363121, 223577456873, 44753756873, ...
  Row n=4:  24, 890, 1615, 795056874, 718511874, ...
  Row n=5:  48, 1424, 84281875, 1578123, ...
  Row n=6:  60, 1215, 53216, ...
  Row n=7:  192, 2624, ...
  Row n=8:  120, 6699, 31310, ...
  Row n=9:  180, 16928, ...
  ...

Cf. A000005, A294528.

T(n,k) = min_{j : A000005(m+j) = 2*(n+j) for j = 0..k-1 but not for j = -1 or j = k}.

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.

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

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A340157 Numbers m such that numbers m, m + 1, m + 2 and m + 3 have k, 2k, 3k and 4k divisors respectively.

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A341214 a(n) is the smallest prime p such that p, p - 1, p - 2, ..., p - n + 1 have 2, 4, 6, ..., 2*n divisors respectively.

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

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A363335 Irregular table read by rows: T(n,k) is the smallest m that has 2*n divisors and is at the beginning of a run of exactly k consecutive integers whose number of divisors increases by 2, or -1 if no such m exists.

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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.