cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

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

Original entry on oeis.org

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

Views

Author

Felice Russo

Keywords

Comments

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

References

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

Links

Crossrefs

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).

Programs

  • Haskell
    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
  • Mathematica
    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 *)
  • PARI
    is(n)=bigomega(n)==bigomega(n+1) \\ Charles R Greathouse IV, Sep 14 2015

Formula

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

Extensions

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

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

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

Programs

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

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

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).

Programs

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

Views

Author

Zak Seidov, Nov 05 2006

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

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).

Programs

  • PARI
    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 */

Extensions

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

Views

Author

Zak Seidov, Nov 05 2006

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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).

Programs

  • PARI
    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

Extensions

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

Views

Author

Charles R Greathouse IV, Oct 24 2022

Keywords

Links

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Charles R Greathouse IV, Oct 24 2022

Keywords

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Charles R Greathouse IV, Oct 24 2022

Keywords

Comments

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

Crossrefs

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).

Programs

  • PARI
    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)

A045984 a(n) = smallest number m such that factorizations of n consecutive integers starting at m have same number of primes (counted with multiplicity).

Original entry on oeis.org

1, 2, 33, 602, 602, 2522, 211673, 3405122, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

a(16) > 10^13. a(16) must have at least 5 prime factors (counted with multiplicity) because one of the 16 consecutive numbers is divisible by 2^4. - Donovan Johnson, Apr 01 2013

Examples

			a(4) = 602 as 602 = 2 * 7 * 43, 603 = 3 * 3 * 67, 604 = 2 * 2 * 151, 605 = 5 * 11 * 11 so four consecutive positive integers have the same number of prime factors starting at 602, the first such number. - _David A. Corneth_, Feb 24 2024

Crossrefs

Cf. A001222, A045920, A045939, A045940, A045941, A045942, A077657, A117969, A356953.

Extensions

More terms from Vladeta Jovovic, Aug 06 2002
More terms from Martin Fuller, Nov 21 2006
Showing 1-10 of 15 results. Next

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