A123201 -id:A123201 - 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

A045939 Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities).

Original entry on oeis.org

33, 85, 93, 121, 141, 170, 201, 213, 217, 244, 284, 301, 393, 428, 434, 445, 506, 602, 603, 604, 633, 637, 697, 841, 921, 962, 1041, 1074, 1083, 1084, 1130, 1137, 1244, 1261, 1274, 1309, 1345, 1401, 1412, 1430, 1434, 1448, 1490, 1532, 1556, 1586, 1604

Offset: 1

David W. Wilson

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), this sequence (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).
A056809 is a subsequence.
Cf. A006073. - Harvey P. Dale, Apr 19 2011

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2],AppendTo[lst,n]],{n,0,7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
pd2Q[n_]:=PrimeOmega[n]==PrimeOmega[n+1]==PrimeOmega[n+2]; Select[Range[1700],pd2Q]  (* Harvey P. Dale, Apr 19 2011 *)
SequencePosition[PrimeOmega[Range[1700]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Mar 08 2023 *)

is(n)=my(t=bigomega(n)); bigomega(n+1)==t && bigomega(n+2)==t \\ Charles R Greathouse IV, Sep 14 2015

list(lim)=my(v=List(),a=1,b=1,c); forfactored(n=4,lim\1+2,c=bigomega(n); if(a==b&&a==c, listput(v,n[1]-2)); a=b; b=c); Vec(v) \\ Charles R Greathouse IV, May 07 2020

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

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)

A358019 Numbers m such that the factorizations of m..m+10 have the same number of primes (including multiplicities).

Original entry on oeis.org

202536181, 913535284, 1124342785, 1443929905, 1587749041, 1688485665, 1733574769, 2090053141, 2308638625, 2403102228, 2751673525, 2841766801, 2898584161, 2936217602, 3195380868, 3195380869, 3324630612, 3423884341, 3520752468

Offset: 1

Charles R Greathouse IV, Oct 24 2022

nonn

a(111) = 21117216104 is the first term where the number of primes is 5. - Zak Seidov and Robert Israel, Jun 27 2024

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), A358018 (k=9), this sequence (k=10).

list(lim)=my(v=List(),ct,cur); forfactored(n=202536181,lim\1+10, my(t=bigomega(n)); if(t==cur, if(ct++>9, listput(v,n[1]-10)), cur=t; ct=0)); Vec(v)

A374023 Numbers m such that m .. m+11 all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

3195380868, 5208143601, 5208143602, 5327400945, 5604994082, 5604994083, 6940533603, 6940533604, 7109053186, 7112231268, 19355940562, 22180594465, 24073076004, 24155988484, 29495293764, 30997967601, 41999754228, 42322452483, 42322452484, 45479198003, 46553917683

Offset: 1

Zak Seidov and Robert Israel, Jun 25 2024

nonn

Since a(3) = a(2) + 1, a(6) = a(5) + 1 and a(8) = a(7) + 1, a(2) = 5208143601, a(5) = 5604994082 and a(7) = 6940533603 are the first three m such that m .. m+12 have the same number of prime factors, counted with multiplicity.

For n <= 12, A001222(a(n)) = 4. It must always be at least 4 because at least one of a(n) .. a(n)+11 is divisible by 8.

			5208143601 is a term because
  5208143601 = 3 * 139 * 2153 * 5801
  5208143602 = 2 * 47 * 4261 * 13003
  5208143603 = 13 * 103 * 419 * 9283
  5208143604 = 2^2 * 3 * 434011967
  5208143605 = 5 * 7^2 * 21257729
  5208143606 = 2 * 37 * 109 * 645691
  5208143607 = 3^2 * 647 * 894409
  5208143608 = 2^3 * 651017951
  5208143609 = 73^2 * 367 * 2663
  5208143610 = 2 * 3 * 5 * 173604787
  5208143611 = 11 * 29 * 1129 * 14461
  5208143612 = 2^2 * 7 * 186005129
all have 4 prime factors, counted with multiplicity.

Martin Ehrenstein, Table of n, a(n) for n = 1..10000

Subsequence of A033987.
Cf. A001222.
Numbers m through m+k have the same value of A001222: A045920 (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).

isok(m) = #Set(apply(bigomega, vector(11, i, m+i-1))) == 1; \\ Michel Marcus, Jul 11 2024

A001222(a(n)) = A001222(a(n)+1) = ... = A001222(a(n)+11).

Missing term inserted by, and more terms from Martin Ehrenstein, Jul 11 2024

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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A045939 Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities).

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

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

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A358019 Numbers m such that the factorizations of m..m+10 have the same number of primes (including multiplicities).

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