A005237 -id:A005237 - cp's OEIS frontend

A274357 Numbers n such that n and n+1 both have 8 divisors.

104, 135, 189, 230, 231, 285, 296, 344, 374, 375, 429, 434, 609, 645, 663, 664, 741, 776, 782, 805, 874, 902, 903, 969, 986, 1001, 1015, 1022, 1029, 1065, 1085, 1095, 1105, 1106, 1112, 1130, 1161, 1208, 1221, 1245, 1265, 1269, 1309, 1310, 1334, 1335, 1374, 1406, 1431

Offset: 1

Charles R Greathouse IV, Jun 19 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005237 and A030626.
Numbers n such that n and n+1 both have k divisors: A039832 (k=4), A049103 (k=6), A274357 (k=8), A215197 (k=10), A174456 (k=12), A274358 (k=14), A274359 (k=16), A274360 (k=18), A274361 (k=20), A274366 (k=22), A274362 (k=24), A274363 (k=26), A274364 (k=28), A274365 (k=30).
Cf. A000005.

SequencePosition[DivisorSigma[0,Range[2000]],{8,8}][[All,1]] (* Harvey P. Dale, Sep 07 2021 *)

PARI
```
is(n)=numdiv(n)==8 && numdiv(n+1)==8
```

A049103 Numbers k such that k and k+1 both have 6 divisors.

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 242, 243, 244, 332, 387, 507, 548, 603, 604, 724, 844, 908, 931, 963, 1075, 1083, 1251, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 2523, 2524, 2636, 2644, 2763, 3283, 3356, 3411, 3508, 3788, 3987, 4075, 4203, 4204, 4418, 4491, 4804, 4868, 4923, 4924

Offset: 1

N. J. A. Sloane, Felice Russo, Olivier Gérard

nonn

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1986, entry 44, p. 103.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Intersection of A005237 and A030515.

Cf. A038400, A049104.

Flatten[Position[Partition[Table[DivisorSigma[0,n],{n,5000}],2,1],?(# == {6,6}&)]] (* _Harvey P. Dale, Apr 09 2012 *)

isok(n) = (numdiv(n) == 6) && (numdiv(n+1) == 6); \\ Michel Marcus, Feb 11 2016

a(n) = A049104(n) - 1. - Zak Seidov, Feb 11 2016

A141621 Numbers that begin a run of 5 consecutive integers of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

10093613546512321, 14414905793929921, 266667848769941521, 562672865058083521, 1579571757660876721, 1841337567664174321, 2737837351207392721, 4456162869973433521, 4683238426747860721, 4993613853242910721, 5037980611623036721, 5174116847290255921

Offset: 1

Matthijs Coster, Aug 23 2008

Old name was "The first number of a series of 5 consecutive numbers with the same signature, i.e., all numbers have the format p^2*q, where p and q are primes. Therefore the number of divisors is the same (6)." [That name could have been confusing in that not every sequence of 5 consecutive integers having the same prime signature has the prime signature p^2*q; e.g., 204323 is the first of 5 consecutive numbers of the form p^2*q*r. - Jon E. Schoenfield, Jun 05 2018]
Each of the five numbers in each such sequence has 6 divisors.
It is easy to prove that any number in this sequence must be congruent to 1 modulo 240. The program below calculates only an element of the sequence. Since the reference A119479 it is the smallest one. If we assume that the first element has the format 7^2*n49, the second number has the format 2*p^2, the third element has the format 3^2*n9 and the fifth element has the format 5^2*n25, then p must be modulo 22050 one out of 1181, 3719, 4219, 9119, 12931, 17831, 18331 or 20869.
It is unclear if these numbers are the smallest ones. - Matthijs Coster, Aug 28 2008 [The terms listed in the Data section are, in fact, the smallest numbers matching the definition. - Jon E. Schoenfield, Jun 05 2018]
The first quintuple not of the aforementioned form starts with 5344962129269790721 = 23^2*prime. - Ivan Neretin, Feb 08 2016
Among the first 200 terms, the frequency with which the squared prime factor p is {7, 17, 23, 31, 41, 47, 73, 127, 193, 1039, 1399} is {171, 10, 6, 4, 3, 1, 1, 1, 1, 1, 1}, respectively. - Jon E. Schoenfield, Jun 09 2018

			a(1) = 10093613546512321, because
10093613546512321 = 7^2 * 205992113194129,
10093613546512322 =   2 * 71040881^2,
10093613546512323 = 3^2 * 1121512616279147,
10093613546512324 = 2^2 * 2523403386628081, and
10093613546512325 = 5^2 * 403744541860493,
so each of the five consecutive integers is of the form p^2*q, and no smaller run of five consecutive integers has this property. [corrected by _Jon E. Schoenfield_, Jun 05 2018]

Ray Chandler, Table of n, a(n) for n = 1..2000 (first 25 terms from Ivan Neretin, through 200 terms from Jon E. Schoenfield)
Richard I. Hess, Problem 1231, Crux Mathematicorum, Vol. 13, No. 4, p. 118, 1987. (takes a long time to download)
Richard I. Hess, Puzzles from Around the World, p. 63, H17.
Carlos Rivera, Problem 20.- Divisors (II) K consecutive numbers with the same number of divisors, The Prime Puzzles & Problems Connection.
StackExchange, Sequence of numbers with prime factorization pq^2
wu :: forums, Same Number of Divisors, Oct 05 2007.
Index to sequences related to prime signature

Cf. A119479, A006558, A005237, A005238, A006601.

Cf. A054753, A074172, A178032, A308683.

## Warning: this program appears to be incorrect [Joerg Arndt, Feb 29 2016]
for m in range(5000):
    p = 22050*m+17831
    if is_prime(p):
        n = 2*p^2-2
        n4 = n/4+1
        if is_prime(n4):
            n49 = floor((n+1)/49)
            if (49*n49 == n+1) and is_prime(n49):
                n9 = floor((n+3)/9)
                if (9*n9 == n+3) and is_prime(n9):
                    n25 = floor((n+5)/25)
                    if (25*n25 == n+5) and is_prime(n25):
                        print(n+1, n49, p, n9, n4, n25)

Two more terms Matthijs Coster, Aug 28 2008
Missing terms added and extended by Ivan Neretin, Feb 08 2016
New name from Jon E. Schoenfield, Jun 05 2018

A039832 Numbers k such that k and k+1 both have 4 divisors.

Original entry on oeis.org

14, 21, 26, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697, 698, 706, 717, 745, 766, 778, 793, 802, 817

Offset: 1

N. J. A. Sloane, Felice Russo, Olivier Gérard, Dec 11 1999

nonn

			14 and 15 both have 4 as number of divisors and are consecutive.

David Wells, Curious and interesting numbers, Penguin Books, 1986, p. 91.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Intersection of A005237 and A030513.

Cf. A038456, A039833.

Flatten[Position[Partition[Table[DivisorSigma[0, n], {n, 1000}], 2, 1], ?(#=={4, 4}&)]] (* _Vincenzo Librandi, Oct 21 2012 *)

isA039832(n) = numdiv(n)==4 && numdiv(n+1)==4 \\ Michael B. Porter, Feb 03 2010

A075036 Smaller of two smallest consecutive numbers with 2n divisors.

Original entry on oeis.org

2, 14, 44, 104, 2511, 735, 29888, 2295, 6075, 5264, 2200933376, 5984, 689278976, 156735, 180224, 21735, 2035980763136, 223244, 9399153082499072, 458864, 41680575, 701443071, 2503092614937444351, 201824, 2707370000, 29785673727, 46977524, 5475519, 1737797404898095794225152

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

There cannot be two consecutive numbers with the same odd number of divisors as both cannot be squares.
These numbers have the property that a(n) * (a(n) + 1) has 4*n^2 divisors. - David A. Corneth, Jun 24 2016
Conjecture: if a term k is even, the highest p-adic order of k (the maximum may be attained by several p's) occurs at p=2 and the highest p-adic order of k+1 occurs at p=3. If a term k is odd, the highest p-adic order of k occurs at p=3 and the highest p-adic order of k+1 occurs at p=2. - Chai Wah Wu, Mar 12 2019
a(49) = 378401464109375, a(58) = 79921490583489592950783. - Jon E. Schoenfield, May 07 2022
a(51) = 34210814718574592, a(55) = 2481402804069375, a(57) = 394311388855795712. - Jon E. Schoenfield, Nov 06 2023 - Nov 08 2023

			a(4) = 104 as tau(104) = tau(105) = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..48
Ray Chandler, Known terms and upper limits.

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.

a[n_] := (For[k=1, ! (DivisorSigma[0, k] == 2*n && DivisorSigma[0, k+1] == 2*n), k++]; k); Array[a, 10] (* Giovanni Resta, Jun 24 2016 *)

a(n) = my(k=1); while(numdiv(k)!=2*n || numdiv(k+1)!=2*n, k++); k \\ Felix Fröhlich, Jun 24 2016

a(n) <= A215199(n-1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p-1). This conjecture is true if the conjecture in A215199 is true. The b-file of A215199 thus shows that a(p) = A215199(p-1) for prime p < 1279. - Chai Wah Wu, Mar 12 2019

a(5)-a(24) from Max Alekseyev, Mar 12 2009
a(25)-a(28) from Giovanni Resta, Jun 24 2016
a(29) from Chai Wah Wu, Mar 12 2019

A039665 Sets of 4 consecutive numbers with equal number of divisors.

Original entry on oeis.org

242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, 5943, 5944, 5945, 5946, 6853, 6854, 6855, 6856, 7256, 7257, 7258, 7259, 8392, 8393, 8394, 8395, 9367, 9368, 9369, 9370, 10983, 10984, 10985, 10986, 11605, 11606, 11607, 11608

Offset: 1

Felice Russo

nonn

Taking the first entry in each set gives A006601.

D. Wells, Curious and interesting numbers, Penguin Books, p. 134

Cf. A000005, A005237, A005238, A006601, A049051, A006558, A019273.

More terms from Patrick De Geest, Nov 15 1999

A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 4, 7, 20, 31, 57, 94, 98, 118, 122, 127, 201, 213, 218, 230, 242, 243, 244, 334, 384, 393, 423, 429, 481, 565, 603, 633, 694, 704, 729, 766, 844, 921, 1138, 1141, 1221, 1262, 1401, 1533, 1654, 1726, 1761, 1837, 1838, 1862, 1882, 1942, 2162, 2245, 2361, 2362

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1).

The Mersenne primes (A000668) are terms since if 2^p - 1 is a prime then A093653(2^p-1) = A093653(2^p) = p+1.

			3 is a term since A093653(3) = A093653(4) = 3.

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

Cf. A000120, A093653.
A000668 is a subsequence.
Similar sequences: A002961, A005237, A054004, A064125, A238380, A293183, A306985.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[s, n - 1]]; f1 = f2, {n, 2, 240}]; s

A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 124, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, May 14 2021

nonn

			1 is a term since A048105(1) = A048105(2) = 0.
27 is a term since A048105(27) = A048105(28) = 2.

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

Cf. A048105, A064115, A335328, A344315.

Similar sequences: A005237, A006049, A343819, A344312, A344313.

nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[200], nd[#] == nd[# + 1] &]

A049053 Numbers k such that k through k+6 all have the same number of divisors.

Original entry on oeis.org

171893, 180965, 647381, 1039493, 1071829, 1450261, 1563653, 1713413, 2129029, 2384101, 4704581, 4773301, 5440853, 5775365, 6627061, 6644405, 6697253, 8556661, 8833429, 10531253, 12101509, 12238453, 12307141, 13416661, 13970405

Offset: 1

David W. Wilson

nonn

Allan Swett found that the first term not congruent to 5 mod 16 is 67073285. - Ralf Stephan, Nov 15 2004

Since A119479(n) < 7 for n < 8, no term has fewer than 8 divisors; the first that has more is a(30)=17476613. - Ivan Neretin, Feb 05 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 244.

Cf. A000005, A006558, A019273, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A006601 (runs of 4), A049051 (runs of 5), A049052 (runs of 6).

isok(n) = {my(nb = numdiv(n)); for (k=1, 6, if (numdiv(n+k) != nb, return (0));); 1;} \\ Michel Marcus, Feb 06 2016

A057922 d(n) divides d(n+1), where d(n) is number of positive divisors of n.

Original entry on oeis.org

1, 2, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 47, 49, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 86, 87, 89, 93, 94, 95, 97, 98, 101, 103, 104, 107, 109, 113, 116, 118, 119, 122, 125, 127, 129, 131, 133, 134, 135

Offset: 0

Leroy Quet, Nov 11 2000

nonn

			11 is included because d(11) = 2 divides d(12) = 6.

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A057921, A005237.

Select[Range@150, Divisible @@ DivisorSigma[0, {# + 1, #}] &] (* Ivan Neretin, May 29 2015 *)
Position[Partition[DivisorSigma[0,Range[150]],2,1],?(Divisible[#[[2]], #[[1]]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, May 08 2019 *)

cp's OEIS Frontend

A274357 Numbers n such that n and n+1 both have 8 divisors.

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A049103 Numbers k such that k and k+1 both have 6 divisors.

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A141621 Numbers that begin a run of 5 consecutive integers of the form p^2*q where p and q are distinct primes.

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A039832 Numbers k such that k and k+1 both have 4 divisors.

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A075036 Smaller of two smallest consecutive numbers with 2n divisors.

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A039665 Sets of 4 consecutive numbers with equal number of divisors.

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A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

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A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

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