A063446 -id:A063446 - cp's OEIS frontend

A100363 Numbers n such that the numbers of divisors of n,n+1 and n+2 are k,2k,4k respectively for some k.

193, 397, 454, 457, 613, 614, 661, 757, 758, 997, 998, 1093, 1237, 1238, 1453, 1478, 1657, 1681, 1766, 2137, 2341, 2413, 2455, 2593, 2917, 2918, 2942, 2966, 3217, 3334, 3494, 3589, 4021, 4177, 4183, 4406, 4621, 5143, 5233, 5965, 6121, 6133, 6134, 6193

Offset: 1

Labos Elemer, Nov 19 2004

nonn

			n=193,195,195 have 2,4,8 divisors.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A063446.

d0[x_] :=DivisorSigma[0, x];ta={{0}}; Do[g=n;s=d0[n];s1=d0[n+1];s2=d0[n+2]; If[Equal[s1, 2*s]&&Equal[s2, 4*s], ta=Append[ta, n]; Print[{n, s, s1, s2}]], {n, 1, 10000}];{ta=Delete[ta, 1], g}
Flatten[Position[Partition[DivisorSigma[0,Range[6200]],3,1],?(Length[ Union[ #/{1,2,4}]] == 1&),{1},Heads->False]] (* _Harvey P. Dale, Aug 03 2014 *)

A100364 Numbers n such that the numbers of divisors of n,n+1,n+2 and n+3 are k,2k,4k,8k respectively for some k.

Original entry on oeis.org

613, 757, 997, 1237, 2917, 6133, 9853, 15733, 19477, 22501, 24421, 25237, 26293, 26437, 28117, 31477, 33301, 35317, 44101, 45061, 46357, 51637, 53077, 56701, 59221, 61653, 61717, 64381, 65917, 66853, 68583, 70501, 70887, 72901, 75133

Offset: 1

Labos Elemer, Nov 19 2004

nonn

			n=613,614,615,616 have 2,4,8,16 divisors respectively.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A063446, A100363.

d0[x_] :=DivisorSigma[0, x];ta={{0}}; Do[g=n;s=d0[n];s1=d0[n+1];s2=d0[n+2];s3=d0[n+3] If[Equal[s1, 2*s]&&Equal[s2, 4*s]&&Equal[s3, 8*s], ta=Append[ta, n]; Print[{n, s, s1, s2, s3}]], {n, 1, 1000000}];{ta=Delete[ta, 1], g}
nd2Q[{a_,b_,c_,d_}]:={b,c,d}/a=={2,4,8}; Position[Partition[ DivisorSigma[ 0,Range[80000]],4,1],?(nd2Q[#]&)]//Flatten (* _Harvey P. Dale, Feb 25 2020 *)

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.

Original entry on oeis.org

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

A063450 Numbers k such that d(k+1) < 2*d(k), where d() is the number of divisors function A000005.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Labos Elemer, Jul 24 2001

nonn

			d(k+1) < 2*d(k) holds mainly for composites and for the primes 2 and 3. E.g.:
For k = 10: 2*d(10) = 2*4 = 8 > 2 = d(11).
For k = 3: 2*d(3) = 2*2 = 4 > d(4) = 3.
For k = 2: 2*d(2) = 2*2 = 4 > d(3) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A002808, A063446, A063449.

SequencePosition[DivisorSigma[0,Range[110]],?(#[[2]]<2#[[1]]&)][[All,1]]// Quiet (* _Harvey P. Dale, Aug 19 2020 *)

is(m) = numdiv(m + 1) < 2*numdiv(m); \\ Harry J. Smith, Aug 21 2009

Formatting by Charles R Greathouse IV, Mar 24 2010

A100365 Prime numbers q such that q, q+1, q+2, q+3, q+4 have 2, 4, 8, 16, 32 divisors respectively.

Original entry on oeis.org

1124581, 2101621, 2135701, 3829381, 5801701, 6097381, 6453541, 6535861, 6609781, 6799621, 6972661, 7055317, 7527061, 8281381, 8524981, 9412981, 9895141, 11455141, 11901781, 12043621, 12929941, 13749061, 14747701, 15504661

Offset: 1

Labos Elemer, Nov 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A063446, A100363, A100364.

Select[Prime[Range[1002000]],DivisorSigma[0,Range[0,4]+#]=={2,4,8,16,32}&] (* Harvey P. Dale, Oct 04 2019 *)

A339778 Numbers m such that numbers m, m + 1 and m + 2 have k, 2k and 3k divisors respectively.

Original entry on oeis.org

61, 73, 277, 421, 458, 493, 583, 1234, 1393, 1418, 1658, 1909, 1954, 2066, 2138, 2234, 2329, 2386, 2533, 2594, 2773, 2797, 2846, 3013, 3073, 3265, 3394, 3841, 4322, 4333, 4538, 4586, 4633, 4717, 4754, 4766, 5029, 5223, 5245, 5342, 5378, 5554, 5893, 5906, 6169

Offset: 1

Jaroslav Krizek, Dec 16 2020

nonn

Numbers m such that tau(m) = tau(m + 1) / 2 = tau(m + 2) / 3, where tau(k) = the number of divisors of k (A000005).
Corresponding values of tau(a(n)): 2, 2, 2, 2, 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, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 4, ...
Triplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2)] = [tau(a(n)), 2*tau(a(n)), 3*tau(a(n))]: [2, 4, 6], [2, 4, 6], [2, 4, 6], [2, 4, 6], [4, 8, 12], [4, 8, 12], [4, 8, 12], [4, 8, 12], [4, 8, 12], ...

			tau(61) = 2, tau(62) = 4, tau(63) = 6.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A100363.

Subsequence of A063446.

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

Select[Range[6000], Equal @@ (DivisorSigma[0, # + {0, 1, 2}]/{1, 2, 3}) &] (* Amiram Eldar, Dec 16 2020 *)

isok(m) = my(nb = numdiv(m)); (numdiv(m+1) == 2*nb) && (numdiv(m+2) == 3*nb); \\ Michel Marcus, Dec 18 2020

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

A063449 Numbers k for which d(k+1) > 2*d(k), where d(j) = A000005(j).

Original entry on oeis.org

11, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 95, 97, 101, 103, 107, 109, 111, 113, 119, 125, 127, 131, 137, 139, 143, 149, 151, 155, 159, 161, 163, 167, 169, 173, 179, 181, 191, 197, 199, 203, 209, 211, 215, 219, 223, 227, 229, 233, 239

Offset: 1

Labos Elemer, Jul 24 2001

nonn

d(p+1) >= 2d(p) holds for all primes p and for some composite integers, as well.

			For k = 29: 2*d(29) = 2*2 = 4 < 8 = d(30).
For k = 95: 2*d(95) = 2*4 = 8 < 12 = d(96).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A063446, A063450.

Position[Partition[DivisorSigma[0,Range[250]],2,1],?(2#[[1]]<#[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Nov 16 2020 *)

is(m) = numdiv(m + 1) > 2*numdiv(m); \\ Harry J. Smith, Aug 21 2009

Edited by Jon E. Schoenfield, Sep 05 2017

A188540 Numbers k such that d(k+2) = 2*d(k) where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 13, 19, 22, 31, 37, 38, 53, 67, 83, 86, 89, 109, 113, 124, 127, 131, 133, 134, 139, 148, 157, 169, 181, 187, 199, 211, 233, 251, 253, 257, 263, 292, 293, 295, 307, 310, 317, 326, 328, 337, 338, 343, 353, 355, 361, 376, 379, 389, 401, 406, 409, 412, 422, 427, 438, 443, 449, 453

Offset: 1

Juri-Stepan Gerasimov, Apr 03 2011

			1 is a term since d(1+2) = d(3) = 2 = 2*d(1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A063446, A160921.

Select[Range[500], DivisorSigma[0, # + 2] == 2 * DivisorSigma[0, #] &] (* Amiram Eldar, Jan 17 2021 *)
Position[Partition[DivisorSigma[0,Range[500]],3,1],?(2#[[1]]== #[[3]]&),1,Heads-> False]// Flatten (* _Harvey P. Dale, Oct 19 2021 *)

isok(k) = numdiv(k+2) == 2*numdiv(k); \\ Michel Marcus, Jan 17 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

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A100363 Numbers n such that the numbers of divisors of n,n+1 and n+2 are k,2k,4k respectively for some k.

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A100364 Numbers n such that the numbers of divisors of n,n+1,n+2 and n+3 are k,2k,4k,8k respectively for some k.

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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, 2*k, 3*k, ..., n*k divisors respectively.

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A063450 Numbers k such that d(k+1) < 2*d(k), where d() is the number of divisors function A000005.

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A100365 Prime numbers q such that q, q+1, q+2, q+3, q+4 have 2, 4, 8, 16, 32 divisors respectively.

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A339778 Numbers m such that numbers m, m + 1 and m + 2 have k, 2k and 3k 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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A063449 Numbers k for which d(k+1) > 2*d(k), where d(j) = A000005(j).

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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.