A115186
Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
2, 9, 27, 135, 944, 5264, 29888, 50624, 203391, 3290624, 6082047, 32535999, 326481920, 3274208000, 6929459199, 72523096064, 37694578688, 471672487935, 11557226700800, 54386217385983, 50624737509375, 275892612890624, 4870020829413375, 68091093855502335, 2280241934368767, 809386931759611904, 519017301463269375
Offset: 1
a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
A001222(3290624) = A001222(3290625) = 10.
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.
-
f:= proc(n) uses priqueue; local t,x,p,i;
initialize(pq);
insert([-3^n, 3$n], pq);
do
t:= extract(pq);
x:= -t[1];
if numtheory:-bigomega(x-1)=n then return x-1
elif numtheory:-bigomega(x+1)=n then return x
fi;
p:= nextprime(t[-1]);
for i from n+1 to 2 by -1 while t[i] = t[-1] do
insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
od;
od
end proc:
seq(f(i),i=1..27); # Robert Israel, Sep 30 2024
A356893
a(n) is the smallest number m such that m, m+1, m+2 and m+3 each have exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
602, 4023, 57967, 8706123, 296299374, 4109290623, 1691788490622, 59004826709373
Offset: 3
4023 = 3^3*149, 4024 = 2^3*503, 4025=5^2*7*23, 4026=2*3*11*61 (all products of four prime factors).
A356855
a(n) is the least number m such that u defined by u(i) = bigomega(m + 2i) satisfies u(i) = u(0) for 0 <= i < n and u(n) != u(0), or -1 if no such number exists.
Original entry on oeis.org
1, 4, 3, 215, 213, 1383, 3091, 8129, 151403, 151401, 2560187, 33396293, 33396291, 56735777, 1156217487, 2514196079
Offset: 1
Let u be defined by u(i) = bigomega(3 + 2i). u(i) = 1 for 0 <= i < 3 and u(3) = 2 != 1, and 3 is the smallest such number, hence a(3) = 3.
Let u be defined by u(i) = bigomega(4 + 2i). u(i) = 2 for 0 <= i < 2 and u(3) = 3 != 2 , and 4 is the smallest such number, hence a(2) = 4.
Let u be defined by u(i) = bigomega(151403 + 2i). u(i) = 3 for 0 <= i < 9 and u(9) = 2 != 3, and 151403 is the smallest such number, hence a(9) = 151403.
-
u(m,i)=bigomega(m+2*i)
card(m)=my(k=u(m,0),c=0);while(u(m,c)==k,c++);c
a(n)=my(c=0);for(m=1,+oo,c=card(m);if(c==n,return(m)))
A356953
Least nonzero starting number in the first run of exactly n consecutive numbers having the same number of prime factors counted with multiplicity, or -1 if no such number exists.
Original entry on oeis.org
1, 2, 33, 1083, 602, 2522, 211673, 6612470, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465, 361385490681003, 441826936079342
Offset: 1
2 and 3 are 2 consecutive numbers and have the same number of prime factors, and 2 is the smallest such number, hence a(2) = 2.
-
card(m)=my(c=0,k=bigomega(m));if(bigomega(m-1)!=k,while(bigomega(m)==k,c++;m++));c
a(n)=if(n==1,return(1));for(m=2,+oo,if(card(m)==n,return(m)))
A334583
Numbers m such that m, m + 1 and m + 2 each have exactly eight prime factors, not necessarily distinct.
Original entry on oeis.org
40909374, 71410624, 87278750, 126237375, 152439488, 161590624, 166450624, 209140623, 227929624, 243409374, 267308990, 267639470, 290696768, 291513248, 292088510, 295644734, 307885374, 310314158, 319874750, 321890750, 331690624, 336958622, 343030624, 352749248, 354109374, 356269374, 366681248, 391390623, 401375168, 407590623
Offset: 1
40909374 = 2 * 3^4 * 11^2 * 2087, 40909375 = 5^5 * 13 * 19 * 53, and 40909376 = 2^6 * 179 * 3571.
-
list(lim)=my(v=List(), k, o); forfactored(n=40909374, lim\1+2, o=bigomega(n); if(o==8, if(k++>2, listput(v, n[1]-2)), k=0)); Vec(v) \\ Charles R Greathouse IV, May 07 2020
A374392
a(n) is the least number k such that k, k + 2 and k + 4 all have exactly n prime factors, counted with multiplicity.
Original entry on oeis.org
3, 91, 66, 340, 2548, 30940, 67228, 6290620, 81818748, 1336727934, 19729482496, 358398854656, 1934923637500, 115877891562496
Offset: 1
a(3) = 66 because 66 = 2 * 3 * 11, 68 = 2^2 * 17 and 70 = 2 * 5 * 7 all have 3 prime factors, counted with multiplicity, and 66 is the least number that works.
-
f:= proc(m) uses priqueue;
local S, pq, T, v, TP, q, p, j;
S:= {-10,-9,-8,-7};
initialize(pq);
insert([-2^m,2$m],pq);
do
T:= extract(pq); v:= -T[1];
if {v-2,v-4} subset S then return v-4 fi;
S:= (S minus {min(S)}) union {v};
q:= T[-1];
p:= nextprime(q);
for j from m+1 to 2 by -1 do
if T[j] <> q then break fi;
TP:= [T[1]*(p/q)^(m+2-j),op(T[2..j-1]),p$(m+2-j)];
insert(TP, pq)
od od;
end proc:
map(f, [$1..11]);
A374449
Triangle read by rows: T(m,k) is the first number that starts a sequence of exactly k consecutive numbers with m prime factors, counted with multiplicity, if such a sequence is possible.
Original entry on oeis.org
5, 2, 4, 9, 33, 8, 27, 170, 1083, 602, 2522, 211673, 16, 135, 1274, 4023, 12122, 204323, 355923, 6612470, 3405122, 49799889, 202536181, 3195380868, 5208143601
Offset: 1
Triangle starts
5 2
4 9 33
8 27 170 1083 603 3533 211673
T(3,2) = 27 because 27 = 3^3 and 28 = 2^2 * 7 each have 3 prime factors (counted with multiplicity) while 26 = 2*13 and 29 (prime) do not.
-
f:= proc(n)
uses priqueue;
local V,L, count, T, v, j, q, p, TP;
V:= Vector(2^n-1); count:= 0;
L:= [(-1)$(2^n),2^n];
initialize(pq);
insert([-2^(n),2$n],pq);
while count < 2^n-1 do
T:= extract(pq); v:= -T[1];
if L[-1] <> v-1 then
for j from 1 while L[-1]-L[-j] = j-1 do
if L[-j]-L[-j-1] <> 1 and V[j] = 0 then
V[j]:= L[-j]; count:= count+1;
fi od fi;
L:= [op(L[2..-1]),v];
q:= T[-1];
p:= nextprime(q);
for j from n+1 to 2 by -1 do
if T[j] <> q then break fi;
TP:= [T[1]*(p/q)^(n+2-j), op(T[2..j-1]), p$(n+2-j)];
insert(TP,pq);
od od;
op(convert(V,list));
end proc:
f(1):= 5,2:
seq(f(i),i=1..3);
Showing 1-7 of 7 results.
Comments