A072875 -id:A072875 - cp's OEIS frontend

A201220 Numbers m such that m, m-1, m-2 and m-3 are 1,2,3,4-almost primes respectively.

107, 263, 347, 479, 863, 887, 1019, 2063, 2447, 3023, 3167, 3623, 5387, 5399, 5879, 6599, 6983, 7079, 8423, 8699, 9743, 9887, 10463, 11807, 12263, 12347, 14207, 15383, 15767, 18959, 20663, 22343, 23039, 23567, 24239, 27239, 32183, 33647, 33767, 37799

Offset: 1

Antonio Roldán, Nov 28 2011

nonn

Following a suggestion of Claudio Meller.

m is of the form 12k-1, so m-2 is a multiple of 3 and m-3 is a multiple of 4.

			6599 is prime, 6598=2*3299 is semiprime, 6597=3*3*733 is 3-almost prime, 6596=2*2*17*97 is 4-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1700

Subsequence of A005385 and of A201147.

Cf. A005383, A112998, A113000, A113008, A072875, A093552.

primeCount[n_] := Plus @@ Transpose[FactorInteger[n]][[2]]; Select[Range[40000], primeCount[#] == 1 && primeCount[#-1] == 2 && primeCount[#-2] == 3 && primeCount[#-3] == 4 &] (* T. D. Noe, Nov 28 2011 *)
Select[Range[40000],PrimeOmega[Range[#,#+3]]=={4,3,2,1}&]+3 (* Harvey P. Dale, Dec 10 2011 *)
SequencePosition[PrimeOmega[Range[40000]],{4,3,2,1}][[;;,2]] (* Harvey P. Dale, Oct 08 2023 *)

list(lim)=my(v=List(), L=(lim-2)\3, t); forprime(p=3, L\3, forprime(q=3, min(p, L\p), t=3*p*q+2; if(isprime(t) && isprime((t-1)/2) && bigomega(t-3)==4, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 02 2017

A214343 a(n) is the smallest integer j such that the numbers of prime factors (counting multiplicity) in j, j+1, ... , j+n-1 are the full set {1,2,...,n}.

Original entry on oeis.org

2, 3, 6, 15, 77, 726, 6318, 189375, 755968, 871593371, 33714015615

Offset: 1

Jake Foster, Jul 13 2012

nonn

Next term a(10) > 5*10^7. Joerg Arndt, Jul 14 2012

			a(4)=15 because 15 has two prime factors, 16 has four, 17 has one and 18 has three (and 15 is the smallest number with this property).
a(5) = 77 because 77, 78, 79, 80 and 81 have 2, 3, 1, 5 and 4 prime factors.

Cf. A072875, A001222.

A214343 := proc(n)
    refs := {seq(i,i=1..n)} ;
    for j from 1 do
        pf := {} ;
        for k from 0 to n-1 do
            pf := pf union {numtheory[bigomega](j+k)} ;
            if nops(pf) < k+1 then
                break;
            end if;
        end do:
        if pf = refs then
            return j;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 13 2012

f[n_] := f[n] = FactorInteger[n][[All, 2]] // Total;
n = 1;
i = 2;
While[True,
  While[Union[Table[f[j], {j, i, i + n - 1}]] != Range[n],
   i += 1; f[i] =.
   ];
  Print[i]; n += 1;
  ];

a(10)-a(11) from Donovan Johnson, Jul 15 2012

A279518 Start of first run of n successive numbers in which the sum of aliquot parts of the i-th number has exactly i prime factors, for i = 1..n.

Original entry on oeis.org

4, 8, 8, 1909, 558031, 783975, 185363811, 1584002413

Offset: 1

Paolo P. Lava, Dec 14 2016

			sigma(1909) - 1909 = 107 that is a prime number;
sigma(1910) - 1910 = 1546 = 2*773;
sigma(1911) - 1911 = 1281 = 3*7*61;
sigma(1912) - 1912 = 1688 = 2*2*2*211.
No other number < 1909 has this property and therefore a(4) = 1909.

Cf. A001065, A086560, A072875, A279520.

with(numtheory): P:=proc(q) local a,b,d,i,j,k,ok,n; d:=1;
for k from 1 to q do for n from d to q do ok:=1; for j from 1 to k do
b:=ifactors(sigma(n+j-1)-n-j+1)[2]; if add(b[i][2],i=1..nops(b))<>j then ok:=0; break; fi; od;
if ok=1 then d:=n; print(n); break; fi; od; od; end: P(10^12);

a(7)-a(8) from Giovanni Resta, Dec 14 2016

A279520 Start of first run of n successive numbers in which the arithmetic derivative of the i-th number has exactly i prime factors, for i = 1..n.

Original entry on oeis.org

6, 105, 1001, 2945, 240485, 1671414, 22551962, 22551962

Offset: 1

Paolo P. Lava, Dec 14 2016

			2945' = 839 that is a prime number;
2946' = 2461 = 23*107;
2947' = 428 = 2*2*107;
2948' = 3260 = 2*2*5*163.
No other number < 2945 has this property and therefore a(4) = 2945.

Cf. A003415, A086560, A072875, A279518.

with(numtheory): P:=proc(q) local a,b,d,i,j,k,ok,n; d:=1;
for k from 1 to q do for n from d to q do ok:=1; for j from 1 to k do
b:=ifactors(sigma(n+j-1))[2]; if add(b[i][2],i=1..nops(b))<>j then ok:=0; break; fi; od;
if ok=1 then d:=n; print(n); break; fi; od; od; end: P(10^12);

a(7)-a(8) from Giovanni Resta, Dec 14 2016

A373618 Least prime starting a run of n consecutive primes p_i, i=1..n, such that p_i + 1 is squarefree and p_(n+1) + 1 is not squarefree.

Original entry on oeis.org

2, 37, 397, 389, 11617, 11597, 11593, 2048509, 2772409, 5193997, 33933701, 125624813, 125624809, 432787781, 432787777, 4762221193, 4762221181, 182839149373, 547414016069, 551900822513

Offset: 1

Jean-Marc Rebert, Jun 11 2024

			a(1) = 2, because 2 is the least prime starting a run of 1 prime such that 2+1 is squarefree and 3+1 = 4 = 2^2 is not squarefree.
For n=4 the first run of 4 squarefree p+1 starts at a(4) = 389, and no run of n=3 so a(3) = 397 is the ending 3 of this run.
  p              = 389, 397, 401, 409, 419
  p+1 squarefree = yes  yes  yes  yes  no
  n=4 run          \----------------/
  n=3 run               \-----------/

Cf. A005117, A072875, A086560, A369097.

a[n_]:=Module[{k=1}, While[pr=Product[Boole[SquareFreeQ[Prime[k+i-1]+1]], {i, n}]==0||pr&& Boole[SquareFreeQ[Prime[k+n]+1]]==1, k++]; Prime[k]]; Array[a, 8] (* Stefano Spezia, Jun 11 2024 *)

A373626 Least prime of a run of n consecutive primes p_i, i = 1..n, such that bigomega(p_i + 1) = omega(p_i + 1) + i and bigomega(p_(n+1) + 1) <> omega(p_(n+1) + 1) + n + 1, or -1 if no such prime exists.

Original entry on oeis.org

3, 19, 739, 76913, 4510333, 746264059, 290623032907

Offset: 1

Jean-Marc Rebert, Jun 11 2024

			19 starts a run of 2 consecutive primes 19 and 23, bigomega(19+1) = 2 = omega(19+1) + 1, bigomega(23+1) = 4 = omega(23+1) + 2 and bigomega(29+1) = 3 <> omega(29+1) + 3. So a(2) = 19.
Let i, p, b and w be the indices, the primes p_i, bigomega(p_i + 1) and omega(p_i + 1).
i: [ 1  2  3]
p: [19 23 29]
b: [ 3  4  3]
w: [ 2  2  3]
a(2) = 19
i: [  1   2   3   4]
p: [739 743 751 757]
b: [  4   5   5   2]
w: [  3   3   2   2]
a(3) = 739

Cf. A001221, A001222, A072875, A086560, A369097, A373618.

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A201220 Numbers m such that m, m-1, m-2 and m-3 are 1,2,3,4-almost primes respectively.

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A214343 a(n) is the smallest integer j such that the numbers of prime factors (counting multiplicity) in j, j+1, ... , j+n-1 are the full set {1,2,...,n}.

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A279518 Start of first run of n successive numbers in which the sum of aliquot parts of the i-th number has exactly i prime factors, for i = 1..n.

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A279520 Start of first run of n successive numbers in which the arithmetic derivative of the i-th number has exactly i prime factors, for i = 1..n.

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A373618 Least prime starting a run of n consecutive primes p_i, i=1..n, such that p_i + 1 is squarefree and p_(n+1) + 1 is not squarefree.

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A373626 Least prime of a run of n consecutive primes p_i, i = 1..n, such that bigomega(p_i + 1) = omega(p_i + 1) + i and bigomega(p_(n+1) + 1) <> omega(p_(n+1) + 1) + n + 1, or -1 if no such prime exists.

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