Original entry on oeis.org
0, 0, 0, 4, 12, 20, 18, 50, 34, 66, 80, 70, 120, 76, 90, 198, 140, 70, 110, 156, 100, 164, 10, -6, 198, 66, 258, 280, 732, 390, 594, 342, 270, 314, 210, 402, 252, 246, -28, -204, -224, 234, 14, 528, 500, 850, 1110, 942, 1014, 1542, 1906, 1416, 1034, 1454, 970, 982, 1518
Offset: 1
a(6) = A079848(6) - A062294(6) = 37 - 17 = 20.
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from collections import deque
from itertools import islice
from sympy import nextprime
def A133096_gen(): # generator of terms
aset2, alist, bset2, blist, aqueue, bqueue, k = set(), [], set(), [], deque(), deque(), 1
while (k:=nextprime(k)):
cset2 = set()
for a in alist:
if (m:=k-a) in aset2:
break
cset2.add(m)
else:
aqueue.append(k)
alist.append(k)
aset2.update(cset2)
cset2 = set()
for b in blist:
if (m:=b+k) in bset2:
break
cset2.add(m)
else:
bqueue.append(k)
blist.append(k)
bset2.update(cset2)
if len(aqueue) > 0 and len(bqueue) > 0:
yield aqueue.popleft()-bqueue.popleft()
A133096_list = list(islice(A133096_gen(),30)) # Chai Wah Wu, Sep 11 2023
Original entry on oeis.org
1, 2, 6, 12, 14, 10, 50, 4, 48, 62, 22, 120, 30, 104, 154, 116, 40, 122, 178, 66, 218, 28, 72, 296, 220, 290, 130, 552, 80, 322, 158, 400, 302, 528, 520, 170, 268, 236, 576, 454, 600, 470, 766, 224, 1090, 666, 180, 764, 868, 606, 572, 294, 738, 70, 918, 1122, 1196
Offset: 1
-
from itertools import islice
from sympy import nextprime
def A079849_gen(): # generator of terms
aset2, alist, k, m = set(), [2], 2, 2
while (k:=nextprime(k)):
bset2 = set()
for a in alist:
if (b:=k-a) in aset2:
break
bset2.add(b)
else:
yield k-m
alist.append(m:=k)
aset2.update(bset2)
A079849_list = list(islice(A079849_gen(),30)) # Chai Wah Wu, Sep 11 2023
A062294
A B_2 sequence: a(n) is the smallest prime such that the pairwise sums of distinct elements are all distinct.
Original entry on oeis.org
2, 3, 5, 7, 11, 17, 29, 47, 67, 83, 131, 163, 233, 307, 397, 443, 617, 727, 809, 941, 1063, 1217, 1399, 1487, 1579, 1931, 2029, 2137, 2237, 2659, 2777, 3187, 3659, 3917, 4549, 4877, 5197, 5471, 5981, 6733, 7207, 7349, 8039, 8291, 8543, 9283, 9689, 10037
Offset: 1
-
from itertools import islice
from sympy import nextprime
def A062294_gen(): # generator of terms
aset2, alist, k = set(), [], 0
while (k:=nextprime(k)):
bset2 = set()
for a in alist:
if (b:=a+k) in aset2:
break
bset2.add(b)
else:
yield k
alist.append(k)
aset2.update(bset2)
A062294_list = list(islice(A062294_gen(),30)) # Chai Wah Wu, Sep 11 2023
A135257
Smallest semiprimes such that a(j) - a(k) are all different.
Original entry on oeis.org
4, 6, 9, 10, 21, 34, 55, 65, 74, 94, 141, 177, 203, 219, 298, 321, 395, 403, 529, 581, 635, 662, 731, 934, 982, 1073, 1145, 1317, 1418, 1762, 1769, 1849, 1915, 2066, 2165, 2305, 2327, 2746, 2809, 3127, 3409, 3859, 3983, 4138, 4285, 4559, 4742, 5186, 5299, 5854
Offset: 1
a(1) = 4, the smallest semiprime (A001358(1)).
a(2) = 6 = A001358(2); the difference set so far is {2}.
a(3) = 9 = A001358(3); the difference set so far is {2,3,5}.
a(4) = 10 = A001358(4); the difference set so far is {1,2,3,4,5,6}.
a(5) can't be 14 = A001358(5) because 14-10 =4, in the difference set through a(4); can't be 15 = A001358(6) because 15-10 =5, in the difference set through a(4)...
a(5) = 21 = A001358(7); the difference set so far is {1,2,3,4,5,6,11,12,15,17}.
a(6) = 34 = A001358(12); the difference set so far is {1,2,3,4,5,6,11,12,13,15,17,24,25,28,30}.
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A135257 := proc(nmax) local a,anext,diffset,good,an ; a := [4,6] ; diffset := {2} ; while nops(a) < nmax do for anext from op(-1,a)+1 do if numtheory[bigomega](anext) = 2 then good := true ; for an in a do if ( anext-an) in diffset then good := false ; break ; fi ; od; if good then for an from 1 to nops(a) do diffset := diffset union { anext-op(an,a) } ; od; a := [op(a),anext] ; break ; fi ; fi ; od ; od: a ; end: A135257(60) ; # R. J. Mathar, Jan 08 2008
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from itertools import count, islice
from sympy import factorint
def A135257_gen(): # generator of terms
aset2, alist = set(), []
for k in count(0):
if sum(factorint(k).values()) == 2:
bset2 = set()
for a in alist:
if (b:=k-a) in aset2:
break
bset2.add(b)
else:
yield k
alist.append(k)
aset2.update(bset2)
A135257_list = list(islice(A135257_gen(),30)) # Chai Wah Wu, Sep 11 2023
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