A072875
Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.
Original entry on oeis.org
2, 3, 61, 193, 15121, 838561, 807905281, 19896463921, 3059220303001, 3931520917431241
Offset: 1
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.
-
(* 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
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
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.
-
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 *)
A072664
End of smallest run of n consecutive integers with n, n-1, ..., 1 distinct prime factors in that order.
Original entry on oeis.org
2, 7, 107, 2187, 578309, 12239309, 45640671719, 29591195200067
Offset: 1
a(1)=2 because 2 is prime and therefore the smallest integer with exactly one distinct prime factor. a(2)=7 because 6=2*3 and 7 (prime) is the smallest run of consecutive integers with exactly 2 and 1 distinct prime factors in that order. a(3)=107 because 105=3*5*7, 106=2*53 and 107 (prime) is the smallest run with exactly 3, 2 and 1 distinct prime factors in that order. Note that a(1), a(2), a(3), a(5) and a(6) are prime but that a(4)=2187=3^7 is not.
Cf.
A086560 (smallest start with run pattern 1, 2, ..., n),
A072665 (center with run pattern n+1, n, ..., 2, 1, 2, ..., n, n+1),
A068069 (run order not specified),
A001221 (omega(n)).
A072665
Center of smallest run of 2n+1 consecutive numbers with exactly n+1,n,...,2,1,2,...,n,n+1 distinct prime factors, respectively.
Original entry on oeis.org
a(0) = 2 (prime) is the smallest number with one prime factor. a(1) = 11 as 10 (=2*5), 11 (prime) and 12 (=2^2*3) have 2,1,2 distinct prime factors (A001221), respectively and there is no smaller center of such a run. a(2) = 2917 as 2915 (=5*11*53), 2916 (=2^2*3^6), 2917 (prime), 2918 (=2*1459) and 2919 (=3*7*139) have 3,2,1,2,3 distinct prime factors and there is no smaller such run.
Cf.
A072664 (smallest finish with run pattern n, ..., 2, 1),
A086560 (smallest start with run pattern 1, 2, ..., n),
A001221 (omega).
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
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.
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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
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
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.
-
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);
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
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.
-
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);
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
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 \-----------/
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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 *)
A373533
Least starting prime of exactly n consecutive primes p_i (i = 1..n) such that omega(p_i + 1) = 1 + i.
Original entry on oeis.org
5, 23, 499, 13093, 501343, 162598021, 25296334003
Offset: 1
a(1) = 5, because omega(5+1) = 2, and no lesser number has this property.
For n=3, the primes starting at a(3) = 499 are as follows and are a run of exactly 3 omega = i+1,
i = 1 2 3
p = 499, 503, 509, 521
omega(p+1) = 2, 3, 4, 3
\-----------/
-
a[n_]:=Module[{k=1},While[Product[Boole[PrimeNu[Prime[k+i-1]+1]==1+i],{i,n}]==0, k++]; Prime[k]]; Array[a,5] (* Stefano Spezia, Jun 10 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
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
Showing 1-10 of 10 results.
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