A039832 -id:A039832 - cp's OEIS frontend

A070552 Semiprimes k such that k+1 is also a semiprime.

9, 14, 21, 25, 33, 34, 38, 57, 85, 86, 93, 94, 118, 121, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 361, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 03 2002

nonn

A064911(a(n))*A064911(a(n)+1) = 1. - Reinhard Zumkeller, Dec 03 2009

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A001358, A007674, A039832, A100484.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..700] | IsSemiprime(n) and IsSemiprime(n+1) ]; // Vincenzo Librandi, Jan 22 2016

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2};lst={};Do[If[f[n]&&f[n+1],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)
Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],2,1],{1,1}]] (* Harvey P. Dale, Feb 04 2015 *)
Select[Range[700], PrimeOmega[#] == PrimeOmega[# + 1] == 2 &] (* Vincenzo Librandi, Jan 22 2016 *)

forprime(p=3,1e3,if(bigomega(2*p-1)==2,print1(2*p-1", "));if(bigomega(2*p+1)==2,print1(2*p", "))) \\ Charles R Greathouse IV, Nov 09 2011

is(n)=if(n%2, isprime((n+1)/2) && bigomega(n)==2, isprime(n/2) && bigomega(n+1)==2) \\ Charles R Greathouse IV, Sep 08 2015

from sympy import factorint
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(n): return is_semiprime(n) and is_semiprime(n+1)
print(list(filter(ok, range(698)))) # Michael S. Branicky, Sep 14 2021

a(n) >> n log n since either n or n+1 is in A100484. - Charles R Greathouse IV, Jul 21 2015

a(n) = A109373(n) - 1. - Zak Seidov Dec 19 2018

More terms from Vladeta Jovovic, May 03 2002

A039833 Smallest of three consecutive squarefree numbers k, k+1, k+2 of the form p*q where p and q are distinct primes.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285, 4413, 4533, 4593, 4881, 5601

Offset: 1

Olivier Gérard

Equivalently: k, k+1 and k+2 all have 4 divisors.
There cannot be four consecutive squarefree numbers as one of them is divisible by 2^2 = 4.
These 3 consecutive squarefree numbers of the form p*q have altogether 6 prime factors always including 2 and 3. E.g., if k = 99985, the six prime factors are {2,3,5,19997,33329,49993}. The middle term is even and not divisible by 3.
Nonsquare terms of A056809. First terms of A056809 absent here are A056809(4)=121=11^2, A056809(14)=841=29^2, A056809(55)=6241=79^2.
Cf. A179502 (Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes). - Zak Seidov, Oct 27 2015
The numbers k, k+1, k+2 have the form 2p-1, 2p, 2p+1 where p is an odd prime.  A195685 gives the sequence of odd primes that generates these maximal runs of three consecutive integers with four positive divisors. - Timothy L. Tiffin, Jul 05 2016
a(n) is always 1 or 9 mod 12. - Charles R Greathouse IV, Mar 19 2022

			33, 34 and 35 all have 4 divisors.
85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B18.
David Wells, Curious and interesting numbers, Penguin Books, 1986, p. 114.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
Roberto Conti, Pierluigi Contucci, and Vitalii Iudelevich, Bounds on tree distribution in number theory, arXiv:2401.03278 [math.NT], 2024. See page 13.

Subsequence of A006881 and A056809 and A263990.

Cf. A038456, A039832, A008683, A007675, A063736, A063838, A070552, A045939, A195685, A179502.

a039833 n = a039833_list !! (n-1)
a039833_list = f a006881_list where
   f (u : vs@(v : w : xs))
     | v == u+1 && w == v+1 = u : f vs
     | otherwise            = f vs
-- Reinhard Zumkeller, Aug 07 2011

lst = {}; Do[z = n^3 + 3*n^2 + 2*n; If[PrimeOmega[z/n] == PrimeOmega[z/(n + 2)] == 4 && PrimeNu[z] == 6, AppendTo[lst, n]], {n, 1, 5601, 2}]; lst (* Arkadiusz Wesolowski, Dec 11 2011 *)
okQ[n_]:=Module[{cl={n,n+1,n+2}},And@@SquareFreeQ/@cl && Union[ DivisorSigma[ 0,cl]]=={4}]; Select[Range[1,6001,2],okQ] (* Harvey P. Dale, Dec 17 2011 *)
SequencePosition[DivisorSigma[0,Range[6000]],{4,4,4}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 17 2017 *)

is(n)=n%4==1 && factor(n)[,2]==[1,1]~ && factor(n+1)[,2]==[1,1]~ && factor(n+2)[,2]==[1,1]~ \\ Charles R Greathouse IV, Aug 29 2016

is(n)=my(t=n%12); if(t==1, isprime((n+2)/3) && isprime((n+1)/2) && factor(n)[,2]==[1,1]~, t==9 && isprime(n/3) && isprime((n+1)/2) && factor(n+2)[,2]==[1,1]~) \\ Charles R Greathouse IV, Mar 19 2022

A008966(a(n)) * A064911(a(n)) * A008966(a(n)+1) * A064911(a(n)+1) * A008966(a(n)+2) * A064911(a(n)+2) = 1. - Reinhard Zumkeller, Feb 26 2011

Additional comments from Amarnath Murthy, Vladeta Jovovic, Labos Elemer and Benoit Cloitre, May 08 2002

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

Original entry on oeis.org

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
```

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

A038456 List of pairs of consecutive numbers each with 4 divisors (duplicates removed).

Original entry on oeis.org

14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 57, 58, 85, 86, 87, 93, 94, 95, 118, 119, 122, 123, 133, 134, 141, 142, 143, 145, 146, 158, 159, 177, 178, 201, 202, 203, 205, 206, 213, 214, 215, 217, 218, 219, 253, 254, 298, 299, 301, 302, 303, 326, 327, 334, 335, 381, 382, 393, 394, 395, 445, 446, 447, 453, 454

Offset: 1

Felice Russo

nonn

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

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

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

Cf. A039832, A039833.

Union[Flatten[Select[Partition[Range[500],2,1],DivisorSigma[0,First[#]] == DivisorSigma[0,Last[#]]==4&]]] (* Harvey P. Dale, Jul 22 2012 *)
SequencePosition[DivisorSigma[0,Range[500]],{4,4}]//Flatten//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

isA038456(n) = (numdiv(n)==4) && ((numdiv(n+1)==4) || (numdiv(n-1)==4)) \\ Michael B. Porter, Feb 03 2010

Corrected and extended by Olivier Gérard

Corrected by Rick L. Shepherd, Jun 07 2002

A356742 Numbers k such that k and k+2 both have exactly 4 divisors.

Original entry on oeis.org

6, 8, 33, 55, 85, 91, 93, 123, 141, 143, 159, 183, 185, 201, 203, 213, 215, 217, 219, 235, 247, 265, 299, 301, 303, 319, 321, 327, 339, 341, 391, 393, 411, 413, 415, 445, 451, 469, 471, 515, 517, 533, 535, 543, 551, 579, 581, 589, 633, 667, 669, 679, 685, 687, 695, 697

Offset: 1

Jianing Song, Aug 25 2022

nonn

6 and 8 are the only even terms: one of the two consecutive even numbers is divisible by 4, and the only multiple of 4 with exactly 4 divisors is 8.

			341 is a term since 341 and 343 both have 4 divisors.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), this sequence (m=4), A356743 (m=6), A356744 (m=8).

Cf. also A039832 (numbers k such that k and k+1 both have exactly 4 divisors).

SequencePosition[DivisorSigma[0,Range[700]],{4,,4}][[All,1]] (* _Harvey P. Dale, Oct 07 2022 *)

isA356742(n) = numdiv(n)==4 && numdiv(n+2)==4

A164119 Numbers k that are the smallest number that produces the ordered pair (d(k), d(k+1)), where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 20, 23, 24, 27, 30, 35, 36, 39, 44, 47, 48, 49, 54, 59, 60, 63, 64, 80, 81, 84, 95, 99, 104, 111, 112, 119, 120, 143, 144, 152, 153, 167, 169, 175, 176, 179, 180, 191, 192, 195, 210, 216, 224, 225, 239, 240, 252, 260, 272, 275

Offset: 1

T. D. Noe, Aug 10 2009

nonn

The set of numbers that produces a given ordered pair (i,j) is either empty, finite, or infinite. The pair (3,3) is produced by no number because d(k)=3 only if k is the square of a prime and no two consecutive numbers are squares of primes. Sequence A161460 lists numbers k that produce a unique ordered pair. It appears that the ordered pair (4,2) is produced by an infinite number of k, which is another way of conjecturing that there are an infinite number of safe primes, A005385. The pair (2,4) is produced by primes in A005383. The numbers in A039832 produce the pair (4,4).

			7 is not here because (d(7), d(8)) = (2,4), which is the same ordered pair produced by k=5.

T. D. Noe, Terms less than 10^8

Cf. A005385, A039832, A161460.

s={}; Reap[Do[pr=DivisorSigma[0,{n,n+1}]; If[ !MemberQ[s,pr], AppendTo[s,pr]; Sow[n]], {n,1000}]][[2,1]]

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A070552 Semiprimes k such that k+1 is also a semiprime.

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A039833 Smallest of three consecutive squarefree numbers k, k+1, k+2 of the form p*q where p and q are distinct primes.

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

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

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A038456 List of pairs of consecutive numbers each with 4 divisors (duplicates removed).

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A356742 Numbers k such that k and k+2 both have exactly 4 divisors.

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A164119 Numbers k that are the smallest number that produces the ordered pair (d(k), d(k+1)), where d(k) is the number of divisors of k.

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