A375422
a(n) is the maximum number of points from the set {(k, prime(k)), k = 1..n} belonging to a straight line passing through the point (n, prime(n)) (where prime(k) denotes the k-th prime number).
Original entry on oeis.org
1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 5, 6, 3, 7, 4, 5, 8, 6, 7, 3, 3, 3, 4, 4, 3, 4, 4, 4, 5, 3, 4, 5, 6, 5, 7, 4, 4, 6, 5, 5, 8, 4, 5, 6, 6, 4, 4, 4, 5, 4, 5, 6, 7, 5, 4, 5, 4, 5, 5, 8, 5, 4, 6, 7, 7, 8, 9, 10, 9, 10, 11, 11, 12, 12, 13, 13, 8, 14, 9, 15
Offset: 1
The first terms, alongside an appropriate set of points, are:
n a(n) Points
-- ---- --------------------------------------------------
1 1 (1,2)
2 2 (1,2), (2,3)
3 2 (1,2), (3,5)
4 3 (2,3), (3,5), (4,7)
5 2 (1,2), (5,11)
6 2 (1,2), (6,13)
7 3 (3,5), (5,11), (7,17)
8 3 (2,3), (5,11), (8,19)
9 4 (3,5), (5,11), (7,17), (9,23)
10 3 (6,13), (7,17), (10,29)
11 3 (8,19), (9,23), (11,31)
12 4 (6,13), (7,17), (10,29), (12,37)
13 5 (6,13), (7,17), (10,29), (12,37), (13,41)
14 4 (8,19), (9,23), (11,31), (14,43)
15 5 (8,19), (9,23), (11,31), (14,43), (15,47)
16 6 (6,13), (7,17), (10,29), (12,37), (13,41), (16,53)
A375511
a(n) is the common difference in the longest arithmetic progression of semiprimes ending in the n-th semiprime. If there is more than one such arithmetic progression, the smallest difference is chosen.
Original entry on oeis.org
2, 3, 1, 4, 1, 6, 8, 3, 11, 12, 12, 1, 12, 1, 12, 14, 13, 17, 11, 12, 24, 7, 18, 35, 19, 24, 8, 24, 18, 29, 8, 36, 1, 24, 17, 30, 12, 4, 3, 48, 4, 36, 11, 48, 23, 24, 1, 30, 12, 13, 12, 36, 42, 24, 14, 16, 36, 14, 8, 32, 36, 7, 60, 42, 60, 60, 3, 4, 36, 46, 4, 12, 32, 4, 60, 16, 18, 44, 36, 16
Offset: 2
The 5th semiprime is 14, A373887(5) = 3, and there are two arithmetic progressions of semiprimes of length 3 ending in 14, namely 6, 10, 14 with common difference 4 and 4, 9, 14 with common difference 5. Therefore a(5) = min(4, 5) = 4.
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S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
f:= proc(n) local s, i, m, d, j, dm;
m:= 1;
s:= S[n];
for i from n-1 to 1 by -1 do
d:= s - S[i];
if s - m*d < 4 then return dm fi;
for j from 2 while ListTools:-BinarySearch(S, s-j*d) <> 0 do od;
if j > m then m:= j; dm:= d fi;
od;
dm;
end proc:
map(f, [$2..200]);
A376115
Least common differences in the arithmetic progressions corresponding to A376109.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 4, 2, 4, 1, 1, 6, 6, 6, 2, 6, 8, 6, 8, 3, 11, 7, 8, 6, 2, 12, 1, 12, 12, 1, 12, 6, 12, 1, 4, 12, 12, 12, 16, 1, 12, 18, 16, 14, 5, 13, 22, 12, 14, 17, 16, 11, 12, 6, 4, 24, 24, 18, 1, 7, 24, 24, 24, 18, 2, 12, 24, 6, 35, 5, 13, 19, 33, 6, 8, 21, 24, 12, 24, 8, 24
Offset: 1
a(7) = 2 because the arithmetic progression 3, 5, 7 of A376109(7) = 3 primes ending in 7 has common difference of 5 - 3 = 7 - 5 = 2.
There are two arithmetic progressions of semiprimes of A376109(14) = 3 ending in 14, namely 6, 10, 14 with common difference 4 and 4, 9, 14 with common difference 5, so a(14) = 4.
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M:= Array(1..10):
for n from 2 to 100 do
v:= numtheory:-bigomega(n);
if M[v] = 0 then M[v]:= n else M[v]:= M[v], n fi;
od:
for i from 1 to 10 do M[i]:= [M[i]] od:
f:= proc(s) local n,i,m,d,v,j,dm;
m:= 1; dm:= 1;
v:= numtheory:-bigomega(s);
member(s,M[v],n);
for i from n-1 to 1 by -1 do
d:= s - M[v][i];
if s - m*d < M[v][1] then return dm fi;
for j from 2 while ListTools:-BinarySearch(M[v],s-j*d) <> 0 do od:
if j > m then m:= j; dm:= d fi;
od;
dm;
end proc:
f(1):= 1:
map(f, [$1..100]);
Showing 1-3 of 3 results.
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