A045939 -id:A045939 - cp's OEIS frontend

A045920 Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).

2, 9, 14, 21, 25, 27, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 121, 122, 124, 133, 135, 141, 142, 145, 147, 153, 158, 164, 170, 171, 174, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 245, 253, 284, 285, 296, 298, 301, 302, 326, 332, 334, 350, 356, 361

Offset: 1

Felice Russo

A115186 is a subsequence: A001222(A115186(n)) = A001222(A115186(n)+1) = n. - Reinhard Zumkeller, Jan 16 2006
Indices k such that A076191(k) = 0. - Ray Chandler, Dec 10 2008
A045939 is a subsequence. - Zak Seidov, Jul 02 2020
This sequence is infinite (Heath-Brown, 1984). - Amiram Eldar, Jul 11 2020

C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 250.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, A parity problem from sieve theory, Mathematika, Vol. 29, No. 1 (1982), pp. 1-6.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika, Vol. 31, No. 1 (1984), pp. 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific journal of mathematics, Vol. 129, No. 2 (1987), pp. 307-319.

Numbers m through m+k have the same number of prime divisors (with multiplicity): this sequence (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

import Data.List (elemIndices)
a045920 n = a045920_list !! (n-1)
a045920_list = map (+ 1) $ elemIndices 0 a076191_list
-- Reinhard Zumkeller, Mar 23 2012, Oct 11 2011

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};Do[If[f[n]==f[n+1],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
Transpose[Transpose[Select[Partition[Table[{n,PrimeOmega[n]},{n,400}], 2,1], #[[1,2]]==#[[2,2]]&]][[1]]][[1]] (* Harvey P. Dale, Feb 21 2012 *)
Position[Differences[PrimeOmega[Range[400]]], 0] // Flatten (* Zak Seidov, Oct 30 2012 *)

is(n)=bigomega(n)==bigomega(n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A278291(n) - 1. - Zak Seidov, Nov 17 2018

More terms from David W. Wilson

A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

Original entry on oeis.org

33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 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

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002

nonn

Each term is the beginning of a run of three 2-almost primes (semiprimes). No runs exist of length greater than three. For the same reason, each term must be odd: If k were even, then so would be k+2. In fact, one of k or k+2 would be divisible by 4, so must indeed be 4 to have only two prime factors. However, neither 2,3,4 nor 4,5,6 is such a run. - Rick L. Shepherd, May 27 2002
k+1, which is twice a prime, is in A086005. The primes are in A086006. - T. D. Noe, May 31 2006
The squarefree terms are listed in A039833. - Jianing Song, Nov 30 2021

			121 is in the sequence because 121 = 11^2, 122 = 2*61 and 123 = 3*41, each of which is the product of two primes.

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

Intersection of A070552 and A092207.

Cf. A045939, A039833, A086005, A086006, A123255.

f[n_] := Plus @@ Transpose[ FactorInteger[n]] [[2]]; Select[Range[10^4], f[ # ] == f[ # + 1] == f[ # + 2] == 2 & ]
Flatten[Position[Partition[PrimeOmega[Range[5000]],3,1],{2,2,2}]] (* Harvey P. Dale, Feb 15 2015 *)
SequencePosition[PrimeOmega[Range[5000]],{2,2,2}][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

forstep(n=1,5000,2, if(bigomega(n)==2 && bigomega(n+1)==2 && bigomega(n+2)==2, print1(n,",")))

is(n)=n%4==1 && isprime((n+1)/2) && bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Sep 08 2015

list(lim)=my(v=List(),t); forprime(p=2,(lim+1)\2, if(bigomega(t=2*p-1)==2 && bigomega(t+2)==2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Sep 08 2015

a(n) = A086005(n) - 1 = 2*A086006(n) - 1 = 4*A123255(n) + 1. - Jianing Song, Nov 30 2021

Edited and extended by Robert G. Wilson v, May 04 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

A045940 Numbers m such that the factorizations of m..m+3 have the same number of primes (including multiplicities).

Original entry on oeis.org

602, 603, 1083, 2012, 2091, 2522, 2523, 2524, 2634, 2763, 3243, 3355, 4023, 4202, 4203, 4921, 4922, 4923, 5034, 5035, 5132, 5203, 5282, 5283, 5785, 5882, 5954, 5972, 6092, 6212, 6476, 6962, 6985, 7314, 7730, 7731, 7945, 8393, 8825, 8956, 8972, 9162

Offset: 1

David W. Wilson

nonn

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..11998 (First 3356 terms from Zak Seidov)

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), this sequence (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

Cf. A045932 (similar, with omega).

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2]==f[n+3],AppendTo[lst,n]],{n,0,8!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
SequencePosition[PrimeOmega[Range[10000]],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 02 2020 *)

isok(n) = (bigomega(n) == bigomega(n+1)) && (bigomega(n+1) == bigomega(n+2)) && (bigomega(n+2) == bigomega(n+3)); \\ Michel Marcus, Jan 06 2015

A045941 Numbers m such that the factorizations of m..m+4 have the same number of primes (including multiplicities).

Original entry on oeis.org

602, 2522, 2523, 4202, 4921, 4922, 5034, 5282, 7730, 12122, 18241, 18242, 18571, 19129, 21931, 23161, 23305, 25203, 25553, 25554, 27290, 27291, 29233, 30354, 30793, 32035, 33843, 34561, 35124, 35714, 36001, 36835, 40313, 40314, 40394, 42182, 45265, 52854

Offset: 1

David W. Wilson

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), this sequence (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2]==f[n+3]==f[n+4],AppendTo[lst,n]],{n,0,9!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
Flatten[Position[Partition[Table[PrimeOmega[n],{n,55000}],5,1],?(Length[ Union[#]]==1&),{1},Heads->False]] (* _Harvey P. Dale, Nov 29 2015 *)

A045942 Numbers m such that the factorizations of m..m+5 have the same number of primes (including multiplicities).

Original entry on oeis.org

2522, 4921, 18241, 25553, 27290, 40313, 90834, 95513, 98282, 98705, 117002, 120962, 136073, 136865, 148682, 153794, 181441, 181554, 185825, 204323, 211673, 211674, 212401, 215034, 216361, 231002, 231665, 234641, 236041, 236634, 266282, 281402, 284344, 285410

Offset: 1

David W. Wilson

Donovan Johnson, Table of n, a(n) for n = 1..10000

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), this sequence (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2]==f[n+3]==f[n+4]==f[n+5],AppendTo[lst,n]],{n,0,10!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
SequencePosition[PrimeOmega[Range[300000]],{x_,x_,x_,x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Aug 29 2025 *)

A123103 Numbers m such that the factorizations of m..m+6 have the same number of primes (including multiplicities).

Original entry on oeis.org

211673, 298433, 355923, 381353, 460801, 506521, 540292, 568729, 690593, 705953, 737633, 741305, 921529, 1056529, 1088521, 1105553, 1141985, 1187121, 1362313, 1721522, 1811704, 1828070, 2016721, 2270633, 2369809, 2535721, 2590985

Offset: 1

Zak Seidov, Nov 05 2006

nonn

Subset of A045940, Numbers m such that the factorizations of m..m+3 have the same number of primes (including multiplicities).

			211673 = 7*11*2749, 211674 = 2*3*35279, 211675 = 5^2*8467, 211676 = 2^2*52919, 211677 = 3*37*1907, 211678 = 2*109*971, 211679 = 13*19*857 are all triprimes.
355923 = 3^2*71*557, 355924 = 2^2*101*881, 355925 = 5^2*23*619, 355926 = 2*3*137*433, 355927 = 11*13*19*131, 355928 = 2^3*44491, 355929 = 3*7*17*997 are all products of 4 primes (typo corrected _Zak Seidov_, Oct 24 2022).

Donovan Johnson, Table of n, a(n) for n = 1..10000

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), this sequence (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

c=0; p1=0; for(n=2, 10^8, p2=bigomega(n); if(p1==p2, c++; if(c>=6, print1(n-6 ",")), c=0; p1=p2)) /* Donovan Johnson, Mar 20 2013 */

a(14)-a(27) from Donovan Johnson, Mar 26 2010

A123201 Numbers m such that the factorizations of m..m+7 have the same number of primes (including multiplicities).

Original entry on oeis.org

3405122, 3405123, 6612470, 8360103, 8520321, 9306710, 10762407, 12788342, 12788343, 15212151, 15531110, 16890901, 17521382, 17521383, 21991382, 21991383, 22715270, 22715271, 22841702, 22841703, 22914722, 22914723

Offset: 1

Zak Seidov, Nov 05 2006

nonn

Note that because 3405130 = 2*5*167*2039 is also the product of 4 primes, 3405122 is the first m such that numbers m..m+8 are products of the same number k of primes (k=4).

			3405122 = 2*7*29*8387, 3405123 = 3^2*19*19913, 3405124 = 2^2*127*6703, 3405125 = 5^3*27241, 3405126 = 2*3*59*9619, 3405127 = 11*23*43*313, 3405128 = 2^3*425641, 3405129 = 3*7*13*12473 all products of 4 primes.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), this sequence (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

c=0; p1=0; for(n=2, 10^8, p2=bigomega(n); if(p1==p2, c++; if(c>=7, print1(n-7 ",")), c=0; p1=p2)) \\ Donovan Johnson, Mar 20 2013

a(7)-a(22) from Donovan Johnson, Apr 09 2010

A358017 Numbers m such that the factorizations of m..m+8 have the same number of primes (including multiplicities).

Original entry on oeis.org

3405122, 12788342, 17521382, 21991382, 22715270, 22841702, 22914722, 23553171, 27451669, 27793334, 49361762, 49799889, 49799890, 50727123, 51359029, 52154450, 53758502, 57379970, 60975410, 60975411, 75638644, 76502870, 76724630, 85432322

Offset: 1

Charles R Greathouse IV, Oct 24 2022

nonn

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

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), this sequence (k=8), A358018 (k=9), A358019 (k=10).

list(lim)=my(v=List(),ct,cur); forfactored(n=3405122,lim\1+8, my(t=bigomega(n)); if(t==cur, if(ct++>7, listput(v,n[1]-8)), cur=t; ct=0)); Vec(v)

A358018 Numbers m such that the factorizations of m..m+9 have the same number of primes (including multiplicities).

Original entry on oeis.org

49799889, 60975410, 92017202, 202536181, 202536182, 249221990, 284007602, 314623105, 326857970, 331212422, 405263521, 421980949, 476360643, 506580949, 520309427, 532896662, 572636822, 666966962, 703401061, 749908502, 816533270

Offset: 1

Charles R Greathouse IV, Oct 24 2022

nonn

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), this sequence (k=9), A358019 (k=10).

list(lim)=my(v=List(),ct,cur); forfactored(n=49799889,lim\1+9, my(t=bigomega(n)); if(t==cur, if(ct++>8, listput(v,n[1]-9)), cur=t; ct=0)); Vec(v)

cp's OEIS Frontend

A045920 Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).

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A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

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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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A045940 Numbers m such that the factorizations of m..m+3 have the same number of primes (including multiplicities).

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A045941 Numbers m such that the factorizations of m..m+4 have the same number of primes (including multiplicities).

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A045942 Numbers m such that the factorizations of m..m+5 have the same number of primes (including multiplicities).

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A123103 Numbers m such that the factorizations of m..m+6 have the same number of primes (including multiplicities).

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