A072875 -id:A072875 - cp's OEIS frontend

A124592 Ends of the runs of numbers described in A072875.

Original entry on oeis.org

2, 4, 63, 196, 15125, 838566, 807905287, 19896463928, 3059220303009, 3931520917431250

Offset: 1

N. J. A. Sloane, Jan 28 2007

nonn

a(n) = A072875(n)+n-1.

A093551 Erroneous version of A072875.

Original entry on oeis.org

2, 3, 61, 193, 15121, 838561, 807905281, 23066039641

Offset: 1

dead

A086560 Start of first run of n successive numbers in which i-th number has exactly i distinct prime divisors for i = 1..n.

Original entry on oeis.org

2, 5, 64, 1867, 491851, 17681491, 35565206671, 43194825904693

Offset: 1

Amarnath Murthy, Aug 30 2003

			a(3) = 64, as 64 has 1, 65 has 2 and 66 has 3 prime divisors: 64 = 2^6, 65 = 5*13 and 66 = 2*3*11.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 64, p. 23, Ellipses, Paris 2008.

Cf. A001221, A072875.

More terms from Don Reble, Sep 13 2003
a(7) from Donovan Johnson, Mar 06 2008
a(8) from Donovan Johnson, Jul 19 2011

A093552 a(n) is the smallest number m such that m-k is product of k+1 primes for k=0,1,2,...,n.

Original entry on oeis.org

2, 5, 47, 107, 71999, 392279, 305778479, 24405534719, 899133243623

Offset: 0

Farideh Firoozbakht, Apr 14 2004

Each term of this sequence is prime.

			a(5)=392279 because:
392279 is prime;
392279-1 = 2*196139;
392279-2 = 3*229*571;
392279-3 = 2*2*281*349;
392279-4 = 5*5*13*17*71;
392279-5 = 2*3*3*19*31*37;
392279 is the smallest number m such that m-k is product of k+1 primes for k=0,1,2,3,4 & 5.

Cf. A072664, A072875, A086560.

Table[m = 1; While[Table[Plus @@ Last /@ FactorInteger[Prime[m] - k], {k, 0, n}] != Range[1, n + 1], m++]; Prime[m], {n, 0, 5}] (* Robert Price, Jun 09 2019 *)

a(6) corrected by Mike Oakes, Jun 20 2012
a(7) from Zak Seidov, Apr 23 2017
a(8) from Giovanni Resta, Apr 27 2017

A369097 Least starting prime of exactly n consecutive primes p_i (i = 1..n) such that bigomega(p_i + 1) = 1 + i.

Original entry on oeis.org

3, 5, 541, 997, 328753, 5385217, 1287133, 9483302497, 107887226353

Offset: 1

Jean-Marc Rebert, Jun 07 2024

			a(1) = 3, because bigomega(3+1) = 2 and no lesser number has this property.
a(2) = 5, because bigomega(5+1) = 2 and bigomega(7+1) = 3, and no lesser number has this property.
a(3) = 541, because bigomega(541+1) = 2, bigomega(547+1) = 3, bigomega(557+1) = 4 and no lesser number has this property.

Cf. A001222, A072875.

isok(p, n) = if (bigomega(p+1) != 2, return(0)); for (i=1, n-1, p = nextprime(p+1); if (bigomega(p+1) != i+2, return(0))); if (bigomega(nextprime(p+1)+1) == n+2, return(0)); return(1);
a(n) = my(p=2); while (!isok(p, n), p = nextprime(p+1)); p; \\ Michel Marcus, Jun 07 2024

A093553 a(n) is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,n.

Original entry on oeis.org

2, 3, 13, 12721, 19441, 5516281, 5516281, 7321991041, 363500177041, 2394196081201, 3163427380990801, 22755817971366481, 3788978012188649281, 2918756139031688155201

Offset: 1

Farideh Firoozbakht, Apr 14 2004

This sequence is A074200(n) + 1. See that entry for more information. - N. J. A. Sloane, May 04 2009
It is obvious that this sequence is increasing and each term is prime. If n > 3 then a(n) == 1 (mod 10).
From Jean-Christophe Hervé, Sep 14 2014: (Start)
a(n) == 1 (mod 120) for all n > 3 (see A163573).
a(4) = 12721 is a quite remarkable number: it is a palindromic prime, its 5 (prime) digits sum to 13, still a prime number (and the preceding element in this sequence, among other things), and as the fourth element of this sequence, it is the smallest prime such that (p-1)/2, (p-2)/3 and (p-3)/4 are also prime, and many other properties. (End)

			a(9)=363500177041 because all the nine numbers 363500177041,
(363500177041+1)/2, (363500177041+2)/3, (363500177041+3)/4,
(363500177041+4)/5, (363500177041+5)/6, (363500177041+6)/7,
(363500177041+7)/8 and (363500177041+8)/9 are primes and
363500177041 is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,9.

Walter Nissen, Calculation without Words : Doric Columns of Primes, Up for the Count !

Cf. A072875.

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

A093852 a(n) = 10^(n-1) - 1 + nfloor(910^(n-1)/(n+1)).

Original entry on oeis.org

4, 69, 774, 8199, 84999, 871425, 8874999, 89999999, 909999999, 9181818179, 92499999999, 930769230759, 9357142857140, 93999999999999, 943749999999999, 9470588235294111, 94999999999999999, 952631578947368403, 9549999999999999999, 95714285714285714279

Offset: 1

Amarnath Murthy, Apr 18 2004

This sequence is the main diagonal of A093850.

			n-th row of the following 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.
Given the triangle defined in A093850:
...4;
..39   69;
.324  549  774;
2799 4599 6399 8199.....
then this sequence is the leading diagonal.

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

Cf. A093846, A093847, A061772, A093450, A072875.

[10^(n-1) -1 +n*Floor(9*10^(n-1)/(n+1)): n in [1..25]]; // G. C. Greubel, Mar 21 2019

A093852 := proc(n)
        r := n ;
        10^(n-1)-1+r*floor(9*10^(n-1)/(n+1)) ;
end proc:
seq(A093852(n),n=1..50) ; # R. J. Mathar, Oct 01 2011

Table[10^(n-1) -1 +n*Floor[9*10^(n-1)/(n+1)], {n,25}] (* G. C. Greubel, Mar 21 2019 *)

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

[10^(n-1) -1 +n*floor(9*10^(n-1)/(n+1)) for n in (1..25)] # G. C. Greubel, Mar 21 2019

A255092 Least prime p such that p+n is product of (n+1) primes (with multiplicity).

Original entry on oeis.org

2, 3, 43, 13, 239, 59, 171869, 569, 32797, 2551, 649529, 6133, 1708984363, 57331, 103630981, 65521, 301327031, 262127, 82244873046857, 11943917, 38354628391, 26214379, 679922958173, 37748713, 584125518798828101, 553648103, 7625597484961, 2281701349, 882592301503097, 8153726947

Offset: 0

Zak Seidov, Feb 14 2015

For n>0, terms with odd indices 3, 13, 59, 569... are much smaller than neighbor terms with even indices.

For n > 0, a(n) >= A053669(n)^(n+1) - n. - Robert Israel, Sep 25 2024

			2+0=2(prime), 3+1=4=2*2, 43+2=45=3*3*5, 13+3=16=2^4, 239+4=243=3^5,59+5=64=2^6,171869+6=171875=5^6*11,569+7=574=2^6*3^2,
32797+8=32805=3^5*5, 2551+9=2590=2^9*5, 649529+10=649539=3^10*11, 6133+11=6143=2^11*3.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A053669, A072875, A112998, A113000, A113008.

f:= proc(n)
    uses priqueue;
      local pq, t, v, p,w,i;
      initialize(pq);
      p:= 2;
      while n mod p = 0 do p:= nextprime(p) od;
      insert([-p^(n+1),[p$(n+1)]],pq);
      do
        t:= extract(pq);
        v:= -t[1]; w:= t[2];
        if isprime(v-n) then return v-n fi;
        p:= nextprime(w[-1]);
      while n mod p = 0 do p:= nextprime(p) od:
       for i from n+1 to 1 by -1 while w[i] = w[n+1] do
        insert([t[1]*(p/w[n+1])^(n+2-i),[op(w[1..i-1]),p$(n+2-i)]],pq);
     od od
end proc:
f(0):= 2:
map(f, [$0..40]); # Robert Israel, Sep 25 2024

More terms from Robert Israel, Sep 25 2024

A136106 a(n) is the smallest prime p such that in the sequence of n numbers p, p+1, p+2, ..., p+n-1, the i-th term has exactly i distinct prime factors, for i = 1, ..., n.

Original entry on oeis.org

2, 5, 103, 1867, 491851, 17681491, 35565206671, 43194825904693

Offset: 1

Enoch Haga, Dec 14 2007

			a(4) = 1867 because it begins with the prime 1867 followed by 1868 with two distinct prime factors, 2 and 467; then 1869 with three distinct prime factors, 3, 7 and 89; then 1870 with four distinct prime factors, 2, 5, 11 and 17.

Cf. A001221, A072875, A086560.

Table[First[Select[Prime@Range@100000,(n=1; k=#; While[Length[First/@FactorInteger@k]==n, k++; n++]; n-1==t)&]],{t, 5}]  (* Giorgos Kalogeropoulos, May 07 2019 *)

/* a brute force program */ a136106(st,ed,ct)={ forprime(x=st,ed, if ((x%6)!=1,next); goodFlag = 1; c = 1; while(goodFlag, if (!(c%2) && isprime(x+c), goodFlag=0, v = factor(x+c); if (length(v[,2]) == c+1, c+=1; if (c > ct,print("Level = ",c," at ",x+c-1,"=",v); ct+=1), goodFlag = 0 ) ) ) ); } \\ Fred Schneider, Dec 18 2007

a(n) >= A086560(n). - R. J. Mathar, Feb 05 2008

A001221(a(n)+k) = k+1 for 0 <= k <= n-1. - Pontus von Brömssen, Jan 09 2023

Edited by N. J. A. Sloane, Dec 23 2007
a(5)-a(6) from Fred Schneider, Dec 18 2007
a(7) from Donovan Johnson, Sep 19 2009
a(8) from Donovan Johnson, Jul 19 2011
Name clarified by Pontus von Brömssen, Jan 09 2023

cp's OEIS Frontend

A124592 Ends of the runs of numbers described in A072875.

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A093551 Erroneous version of A072875.

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A086560 Start of first run of n successive numbers in which i-th number has exactly i distinct prime divisors for i = 1..n.

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A093552 a(n) is the smallest number m such that m-k is product of k+1 primes for k=0,1,2,...,n.

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A369097 Least starting prime of exactly n consecutive primes p_i (i = 1..n) such that bigomega(p_i + 1) = 1 + i.

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PARI

A093553 a(n) is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,n.

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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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A093852 a(n) = 10^(n-1) - 1 + n*floor(9*10^(n-1)/(n+1)).

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A255092 Least prime p such that p+n is product of (n+1) primes (with multiplicity).

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A093852 a(n) = 10^(n-1) - 1 + nfloor(910^(n-1)/(n+1)).