A045940 -id:A045940 - cp's OEIS frontend

A077657 Least number with exactly n consecutive successors, all having the same number of prime factors (counted with multiplicity).

1, 2, 33, 603, 602, 2522, 211673, 3405123, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465

Offset: 0

Reinhard Zumkeller, Nov 13 2002

A001222(a(n))=A001222(a(n)+k) for k<=n;

A077655(a(n))=n and A077655(k)

			a(0)=A077656(1)=1; a(1)=A045920(1)=2; a(2)=A045939(1)=33; a(3)=A045940(2)=603; a(4)=A045941(1)=602; a(5)=A045942(1)=2522.

Cf. A045984.

a(n)=A045984(n+1)+A077655(A045984(n+1))-n - Martin Fuller, Nov 21 2006

More terms from Martin Fuller, Nov 21 2006

A077655 Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Nov 13 2002

nonn

If a(n) > 0 then a(n+1) = a(n)-1.

			33=3*11 has only two successors also with two factors: 34=2*17 and 35=5*7 (whereas 33+3=36=2*2*3*3), therefore a(33)=2.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001222, A077656, A045920, A045939, A045940, A045941, A045942, A077657.

snpf[n_]:=Module[{f=PrimeOmega[n],k=0},While[f==PrimeOmega[n+k],k++];k]; Array[snpf,110]-1 (* Harvey P. Dale, Aug 01 2021 *)

A077655(n) = { my(k=n+1,w=bigomega(n)); while(bigomega(k)==w,k++); (k-n)-1; }; \\ Antti Karttunen, Jan 22 2020

A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

Original entry on oeis.org

602, 603, 1083, 2012, 2091, 2522, 2523, 2524, 2634, 2763, 3243, 3355, 4202, 4203, 4921, 4922, 4923, 5034, 5035, 5132, 5203, 5282, 5283, 5785, 5882, 5954, 5972, 6092, 6212, 6476, 6962, 6985, 7730, 7731, 7945, 8393, 8825, 8956, 8972, 9188, 9482, 10011

Offset: 1

Jonathan Vos Post, Nov 03 2006

n such that n, n+1, n+2 and n+3 are 3-almost primes. Subset of A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes. A067813 has some runs of up to 7 consecutive 3-almost primes (i.e. starting 211673). But there cannot be 8 consecutive 3-almost primes, as every run of 8 consecutive positive integers contains exactly one multiple of 8 = 2^3 and only 8 of all positive multiples of 8 is a 3-almost prime (i.e. all larger multiples have at least 4 prime factors, with multiplicity).

A subset of A045940. - Zak Seidov, Nov 05 2006

			a(1) = 602 because 602 = 2 * 7 * 43 and 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 are all 3-almost primes.
a(2) = 603 because 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 and 606 = 2 * 3 * 101 are all 3-almost primes.
a(3) = 1083 because 1083 = 3 * 19^2 and 1084 = 2^2 * 271 and 1085 = 5 * 7 * 31 and 1086 = 2 * 3 * 181 are all 3-almost primes.
a(4) = 2012 because 2012 = 2^2 * 503, 2013 = 3 * 11 * 61, 2014 = 2 * 19 * 53, 2015 = 5 * 13 * 31.
a(5) = 2091 because 2091 = 3 * 17 * 41, 2092 = 2^2 * 523, 2093 = 7 * 13 * 23, 2094 = 2 * 3 * 349.

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

Cf. A000040, A014612, A056809, A067813, A113789, A045940.

with(numtheory): a:=proc(n) if bigomega(n)=3 and bigomega(n+1)=3 and bigomega(n+2)=3 and bigomega(n+3)=3 then n else fi end: seq(a(n),n=1..15000); # Emeric Deutsch, Nov 07 2006

okQ[{a_,b_,c_,d_}]:=Union[{a,b,c,d}]=={3}; Flatten[Position[Partition[ PrimeOmega[ Range[11000]],4,1],?(okQ)]] (* _Harvey P. Dale, Sep 23 2012 *)

is(n)=if(!isprime((n+3)\4), return(0)); for(k=n,n+3, if(bigomega(k)!=3, return(0))); 1 \\ Charles R Greathouse IV, Feb 05 2017

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q), listput(u,t*r)))); u=Set(u); for(i=4,#u, if(u[i]-u[i-3]==3, listput(v,u[i-3]))); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

n, n+1, n+2 and n+3 are all elements of A014612. n and n+1 are elements of A113789.

More terms from Zak Seidov, Nov 05 2006

More terms from Emeric Deutsch, Nov 07 2006

A124729 Numbers k such that k, k+1, k+2 and k+3 are products of 5 primes.

Original entry on oeis.org

57967, 491875, 543303, 584647, 632148, 632149, 715374, 824523, 878875, 914823, 930123, 931623, 955448, 964143, 995874, 1021110, 1053351, 1070223, 1076535, 1099374, 1251963, 1289223, 1337355, 1380246, 1380247, 1436694, 1507623, 1517282, 1539873, 1669380, 1895222

Offset: 1

Zak Seidov, Nov 05 2006

nonn

Subset of A045940 Numbers m such that factorizations of m through m+3 have same number of primes (including multiplicities).
There are no numbers k such that k, k+1, k+2 and k+3 are products of exactly 6 primes(?).
First counterexample: 8706123. - Charles R Greathouse IV, Jan 31 2017

			57967=7^3*13^2, 57968=2^4*3623, 57969=3^3*19*113, 57970=2*5*11*17*31 (all product of 5 primes, including multiplicities).
632148 is the first number such that n through n+4 are 5-almost primes.

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

Cf. A045940.

Cf. A124057, A124728 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3,4 primes.

SequencePosition[Table[If[PrimeOmega[n]==5,1,0],{n,19*10^5}],{1,1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 03 2019 *)

isok(n) = (bigomega(n) == 5) && (bigomega(n+1) == 5) && (bigomega(n+2) == 5) && (bigomega(n+3) == 5); \\ Michel Marcus, Oct 11 2013

More terms from Michel Marcus, Oct 11 2013

A338454 Starts of runs of 4 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

242, 947767, 1041607, 2545015, 3275463, 8170983, 15720871, 21532430, 23752181, 25135885, 25595913, 27981703, 28226983, 30505142, 30962767, 33364805, 37264493, 49002661, 49766629, 52910454, 53408456, 57917191, 57952016, 58331576, 59230454, 60014053, 60723111, 63378005

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2) = A093653(k+3).

			242 is a term since A093653(242) = A093653(243) = A093653(244) = A093653(245) = 18.

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

Cf. A093653.
Subsequence of A338452 and A338453.
Similar sequences: A006601, A045932, A045940.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 4; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 10^7}]; s

A124728 Numbers k such that k, k+1, k+2 and k+3 are products of 4 primes.

Original entry on oeis.org

4023, 7314, 9162, 12122, 12123, 16674, 19434, 19940, 23874, 24723, 29094, 33234, 35124, 35125, 39234, 42182, 42183, 44163, 45175, 46988, 49147, 51793, 52854, 52855, 54584, 54585, 54663, 58375, 63594, 64074, 64075, 64323, 64491, 64712

Offset: 1

Zak Seidov, Nov 05 2006

nonn

Subset of A045940 Numbers m such that factorizations of m through m+3 have same number of primes (including multiplicities). Cf. A124057, A124729 Numbers k such that k, k+1, k+2 and k+3 are products of exactly 3,5 primes. There are no numbers k such that k, k+1, k+2 and k+3 are products of exactly 6 primes(?)

			4023=3^3*149, 4024=2^3*503, 4025=5^2*7*23, 4026=2*3*11*61 (all products of 4 primes).

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

Cf. A045940, A124057, A124729.

Transpose[Select[Partition[Range[65000],4,1],Union[PrimeOmega[#]] == {4}&]] [[1]] (* Harvey P. Dale, Nov 01 2011 *)

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

A375160 Square array T(n, k), n >= 2 and k >= 1, read by antidiagonals in ascending order, give the smallest number that starts a sequence of exactly k consecutive numbers each having exactly n prime factors (counted with multiplicity), or -1 if no such number exists.

Original entry on oeis.org

4, 8, 9, 16, 27, 33, 32, 135, 170, -1, 64, 944, 1274, 603, -1, 128, 5264, 15470, 4023, 602, -1, 256, 29888, 33614, 57967, 12122, 2522, -1, 512, 50624, 3145310, 8706123, 632148, 204323, 211673, -1

Offset: 2

Jean-Marc Rebert, Aug 09 2024

All positive terms are composite.

			T(2,3) = 33 = 3*11, because both 34 and 35 have the same number of prime factors. Thus, 33 is the starting number of a run of 3 numbers that each have 2 prime factors (counted with multiplicity). No lesser number has this property, so T(2,3) = 33.
Table begins (upper left corner = T(2,1)):
   4        9       33      -1 ...
   8       27      170     603 ...
  16      135     1274    4023 ...
  32      944    15470   57967 ...
  ...     ...      ...     ... ...

Cf. A002808, A062502.

Cf. 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), A358019 (k=10).

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cp's OEIS Frontend

A077657 Least number with exactly n consecutive successors, all having the same number of prime factors (counted with multiplicity).

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A077655 Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).

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A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

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A124729 Numbers k such that k, k+1, k+2 and k+3 are products of 5 primes.

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A338454 Starts of runs of 4 consecutive numbers with the same total binary weight of their divisors (A093653).

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A124728 Numbers k such that k, k+1, k+2 and k+3 are products of 4 primes.

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Mathematica

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

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