A093552 -id:A093552 - cp's OEIS frontend

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

2, 3, 61, 193, 15121, 838561, 807905281, 19896463921, 3059220303001, 3931520917431241

Offset: 1

Rick L. Shepherd, Jun 30 2002 and Jens Kruse Andersen, Jul 28 2002

By definition, each term of this sequence is prime.
a(11) <= 1452591346605212407096281241 (Frederick Schneider), see primepuzzles link. - sent by amd64(AT)vipmail.hu, Dec 21 2007
Prime factors are counted with multiplicity. - Harvey P. Dale, Mar 09 2021

			a(3)=61 because 61 (prime), 62 (=2*31), 63 (=3*3*7) have exactly 1, 2, 3 prime factors respectively, and this is the smallest solution;
a(6)=807905281: 807905281 is prime; 807905281+1=2*403952641;
807905281+2=3*15733*17117; 807905281+3=2*2*1871*107951;
807905281+4=5*11*43*211*1619; 807905281+5=2*3*3*3*37*404357;
807905281+6=7*7*7*7*29*41*283; 807905281 is the smallest number m such that m+k is product of k+1 primes for k=0,1,2,3,4,5,6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 61, p. 22, Ellipses, Paris 2008.

alt.math.recreational thread, Consecutive numbers with counting prime factors.
Carlos Rivera, Puzzle 425. Consecutive numbers, increasing quantity of prime factors, The Prime Puzzles & Problems Connection.

Cf. A001222, A093552, A093550, A086560, A124592.

a(1) = A000040(1), a(2) = A005383(1), a(3) = A112998(1), a(4) = A113000(1), a(5) = A113008(1), a(6) = A113150(1).

(* This program is not suitable to compute a large number of terms. *) nmax = 6; kmax = 10^6; a[1] = 2; a[n_] := a[n] = For[k = a[n-1]+n-1, k <= kmax, k++, If[AllTrue[Range[0, n-1], PrimeOmega[k+#] == #+1&], Return[k] ] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Sep 06 2017 *)

a(7) found by Mark W. Lewis
a(8) and a(9) found by Jens Kruse Andersen
a(10) found by Jens Kruse Andersen; probably a(11) > 10^20. - Aug 24 2002
Entry revised by N. J. A. Sloane, Jan 26 2007
Cross-references and editing by Charles R Greathouse IV, Apr 20 2010

A093850 Triangle T(n,k) = 10^(n-1) -1 + kfloor(910^(n-1)/(n+1)), with 1 <= r <= n, read by rows.

Original entry on oeis.org

4, 39, 69, 324, 549, 774, 2799, 4599, 6399, 8199, 24999, 39999, 54999, 69999, 84999, 228570, 357141, 485712, 614283, 742854, 871425, 2124999, 3249999, 4374999, 5499999, 6624999, 7749999, 8874999, 19999999, 29999999, 39999999, 49999999, 59999999, 69999999, 79999999, 89999999

Offset: 1

Amarnath Murthy, Apr 18 2004

The n-th row of this triangle contains n uniformly located n-digit numbers, i.e., n terms of an arithmetic progression with 10^(n-1)-1 as the term preceding the first term and (n+1)-th term is the largest possible n-digit term.
Starting with n=2, the n-th row of this triangle can be obtained by deleting the least significant digit, 9, from terms ending in 9 in the (n+1)-th row, and ignoring the main diagonal terms, of the triangle in A093846.
Floor(A093846(4,1)/10) = T(3,1) = 324, but floor(A093846(2,1)/10) = 5 and T(1,1) = 4, floor(A093846(7,1)/10) = 228571 and T(6,1) = 228570, etc. - Michael De Vlieger, Jul 18 2016

			Triangle begins with:
      4;
     39,    69;
    324,   549,   774;
   2799,  4599,  6399,  8199;
  24999, 39999, 54999, 69999, 84999;
  ....

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A093846, A093847, A061772, A093451, A093552.

Cf. A093852.

[[10^(n-1) -1 +k*Floor(9*10^(n-1)/(n+1)): k in [1..n]]: n in [1..8]]; // G. C. Greubel, Mar 21 2019

A093850 := proc(n,r)
        10^(n-1)-1+r*floor(9*10^(n-1)/(n+1)) ;
end proc:
seq(seq(A093850(n,r),r=1..n),n=1..14) ; # R. J. Mathar, Sep 28 2011

Table[# -1 +r*Floor[9*#/(n+1)] &[10^(n-1)], {n, 8}, {r, n}]//Flatten (* Michael De Vlieger, Jul 18 2016 *)

{T(n,k) = 10^(n-1) -1 +k*floor(9*10^(n-1)/(n+1))}; \\ G. C. Greubel, Mar 21 2019

[[10^(n-1) -1 +k*floor(9*10^(n-1)/(n+1)) for k in (1..n)] for n in (1..8)] # G. C. Greubel, Mar 21 2019

Second comment clarified by Michael De Vlieger, Jul 18 2016

Edited by G. C. Greubel, Mar 21 2019

A201147 Numbers m such that m, m-1 and m-2 are 1,2,3-almost primes respectively.

Original entry on oeis.org

47, 107, 167, 263, 347, 359, 467, 479, 563, 863, 887, 983, 1019, 1187, 1283, 1907, 2039, 2063, 2099, 2447, 2819, 2879, 3023, 3167, 3203, 3623, 3803, 3947, 4139, 4919, 5387, 5399, 5507, 5879, 6599, 6659, 6983, 7079, 7187, 7523, 7559, 7703, 8423, 8699, 8963

Offset: 1

Antonio Roldán, Nov 27 2011

nonn

m-2 is multiple of 3.
m is  of the form 12k-1.
This sequence is subset of A005385.
Following a suggestion of Claudio Meller.

			2099 is prime, 2098=2*1049 is semiprime, 2097=3*3*233 is 3-almost prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..8400

Cf. A005385, A005383, A112998, A113000, A113008, A072875, A093552.

primeCount[n_] := Plus @@ Transpose[FactorInteger[n]][[2]]; Select[Range[10000], primeCount[#] == 1 && primeCount[#-1] == 2 && primeCount[#-2] == 3 &] (* T. D. Noe, Nov 28 2011 *)
Select[Range[10000],PrimeOmega[Range[#,#+2]]=={3,2,1}&]+2 (* Harvey P. Dale, Dec 10 2011 *)

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), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

A093851 a(n) = A002283(n-1) + floor(A052268(n)/(1+n)).

Original entry on oeis.org

4, 39, 324, 2799, 24999, 228570, 2124999, 19999999, 189999999, 1818181817, 17499999999, 169230769229, 1642857142856, 15999999999999, 156249999999999, 1529411764705881, 14999999999999999, 147368421052631577, 1449999999999999999, 14285714285714285713

Offset: 1

Amarnath Murthy, Apr 18 2004

The first column r=1 of a triangle defined by T(n,r) = 10^(n-1) -1 + r*floor(9*10^(n-1)/(n+1)).

A row starts with a (virtual) 0th column of a rep-9-digit and fills the remainder with n+1 numbers in arithmetic progression with the largest step such that all numbers in the n-th row are n-digit numbers.

			The triangle starts in row n=1 as
4 9 # -1, -1+5, -1+2*5
39 69 99 # 9,9+30,9+2*30
324 549 774 999 # 99, 99+225, 99+2*225, 99+3*225
2799 4599 6399 8199 9999 # 999, 999+1800, 999+2*1800,..
...
The sequence contains the first column.

G. C. Greubel, Table of n, a(n) for n = 1..995

Cf. A061772, A093846, A093847, A093450, A093552.

[10^(n-1) -1 +Floor(9*10^(n-1)/(n+1)): n in [1..20]]; // G. C. Greubel, Apr 02 2019

A093851 := proc(n) 10^(n-1)-1+floor(9*10^(n-1)/(n+1)) ; end proc: seq(A093851(n),n=1..20) ; # R. J. Mathar, Oct 14 2010

Table[10^(n-1) -1 +Floor[9*10^(n-1)/(n+1)], {n, 1, 20}] (* G. C. Greubel, Apr 02 2019 *)

{a(n) = 10^(n-1) -1 +floor(9*10^(n-1)/(n+1))}; \\ G. C. Greubel, Apr 02 2019

[10^(n-1) -1 +floor(9*10^(n-1)/(n+1)) for n in (1..20)] # G. C. Greubel, Apr 02 2019

a(n) = 10^(n-1) -1 + floor(9*10^(n-1)/(n+1)). - G. C. Greubel, Apr 02 2019

More terms from R. J. Mathar, Oct 14 2010

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

Original entry on oeis.org

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

A285433 Integers m such that m-k is the product of k+1 primes for k=0..6.

Original entry on oeis.org

305778479, 306748679, 1067113823, 2837965199, 4533292679, 8345667119, 12120181079, 12148200719, 13765945199, 13949792159, 14404208279, 16237621679, 18147459479, 18780179879, 19542401339, 19662679679, 20045705819, 20383699199, 22383737879, 24039703967, 24405534719

Offset: 1

Zak Seidov, Apr 18 2017

nonn

a(1) = 305778479 = A093552(7). All terms are congruent to 5 mod 6.

a(21) = 24405534719 = A093552(8).

Cf. A093552, A113008.

isok(m) = for (k=0, 6, if (bigomega(m-k) != k+1, return(0));); return(1); \\ Michel Marcus, Nov 20 2022

cp's OEIS Frontend

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

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A093850 Triangle T(n,k) = 10^(n-1) -1 + k*floor(9*10^(n-1)/(n+1)), with 1 <= r <= n, read by rows.

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A201147 Numbers m such that m, m-1 and m-2 are 1,2,3-almost primes respectively.

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A093851 a(n) = A002283(n-1) + floor(A052268(n)/(1+n)).

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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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Mathematica

PARI

A285433 Integers m such that m-k is the product of k+1 primes for k=0..6.

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PARI

A093850 Triangle T(n,k) = 10^(n-1) -1 + kfloor(910^(n-1)/(n+1)), with 1 <= r <= n, read by rows.