A298465
The first of two consecutive heptagonal numbers the sum of which is equal to the sum of two consecutive primes.
Original entry on oeis.org
1, 18, 403, 16281, 24354, 167314, 172528, 183196, 191407, 223054, 413512, 446688, 476767, 507826, 512343, 791578, 926289, 994456, 1032658, 1248562, 1284147, 2221708, 2278630, 2453716, 2604571, 2738952, 2770443, 3207523, 3333330, 4203577, 4400332, 4628761
Offset: 1
18 is in the sequence because 18+34 (consecutive heptagonal numbers) = 52 = 23+29 (consecutive primes).
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chcpQ[{a_,b_}]:=Module[{c=(a+b)/2},NextPrime[c]+ NextPrime[c,-1] ==a+b]; Select[ Partition[PolygonalNumber[7,Range[2000]],2,1],chcpQ][[;;,1]] (* Harvey P. Dale, Mar 14 2023 *)
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L=List(); forprime(p=2, 6000000, q=nextprime(p+1); t=p+q; if(issquare(20*t-16, &sq) && (sq-2)%10==0, u=(sq-2)\10; listput(L, (5*u^2-3*u)/2))); Vec(L)
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from sympy import prevprime, nextprime
A298465_list, n, m = [], 1 ,8
while len(A298465_list) < 10000:
k = prevprime(m//2)
if k + nextprime(k) == m:
A298465_list.append(n*(5*n-3)//2)
n += 1
m += 10*n-3 # Chai Wah Wu, Jan 19 2018
A298466
The first of two consecutive primes the sum of which is equal to the sum of two consecutive heptagonal numbers.
Original entry on oeis.org
3, 23, 433, 16481, 24593, 167953, 173183, 183871, 192097, 223781, 414521, 447743, 477857, 508951, 513473, 792983, 927803, 996019, 1034251, 1250309, 1285937, 2224063, 2281003, 2456191, 2607109, 2741561, 2773073, 3210353, 3336209, 4206817, 4403647, 4632161
Offset: 1
23 is in the sequence because 23+29 (consecutive primes) = 52 = 18+34 (consecutive heptagonal numbers).
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Module[{hep=Total/@Partition[PolygonalNumber[7,Range[1500]],2,1]},Select[ Partition[Prime[Range[PrimePi[Max[hep]/2]]],2,1],MemberQ[hep,Total[#]]&]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 04 2019 *)
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L=List(); forprime(p=2, 6000000, q=nextprime(p+1); t=p+q; if(issquare(20*t-16, &sq) && (sq-2)%10==0, u=(sq-2)\10; listput(L, p))); Vec(L)
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from sympy import prevprime, nextprime
A298466_list, n, m = [], 1 ,8
while len(A298466_list) < 10000:
k = prevprime(m//2)
if k + nextprime(k) == m:
A298466_list.append(k)
n += 1
m += 10*n-3 # Chai Wah Wu, Jan 19 2018
A206281
Smallest of five consecutive primes whose sum is a square.
Original entry on oeis.org
181, 199, 317, 3529, 3733, 4177, 4663, 9049, 15329, 15991, 19577, 24907, 43607, 47017, 58073, 84223, 86843, 146191, 152417, 156623, 175543, 217559, 227671, 288461, 308999, 323077, 331249, 333323, 354301, 390289, 397037, 407249, 474923, 476137, 491059, 520339
Offset: 1
a(4) = 3529. The next four primes are 3533, 3539, 3541, and 3547, and the sum of all five primes = 17689 = 133^2.
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count:= 0: Res:= NULL:
for y from 10 while count < 100 do
target:= y^2;
t:= prevprime(ceil(target/5));
s:= prevprime(t);
r:= prevprime(s);
q:= prevprime(r);
p:= prevprime(q);
u:= p+q+r+s+t;
while u < target do
p:= q; q:= r; r:= s; s:= t; t:= nextprime(t);
u:= p+q+r+s+t;
od;
if u = target then
count:= count+1; Res:= Res, p;
fi
od:
Res; # Robert Israel, Oct 20 2020
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Transpose[Select[Partition[Prime[Range[80000]],5,1],IntegerQ[Sqrt[ Total[#]]]&]][[1]]
A225077
Smaller of the two consecutive primes whose sum is a triangular number.
Original entry on oeis.org
17, 37, 59, 103, 137, 149, 313, 467, 491, 883, 911, 1277, 1423, 1619, 1783, 2137, 2473, 2729, 4127, 4933, 5437, 5507, 6043, 6359, 10039, 10453, 11717, 13397, 15809, 17489, 20807, 21821, 23027, 27631, 28307, 28813, 29669, 33029, 36947, 39103, 44203, 48281
Offset: 1
Cf.
A175132 (numbers n such that sum of two consecutive primes is triangular(n)).
Cf.
A181902 and
A154634 (average of two consecutive primes is a triangular number).
Cf.
A075190 and
A225195 (average of two consecutive primes is a square).
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f:= proc(n) local m,p,q;
m:= n*(n+1)/2;
p:= prevprime(ceil(m/2));
q:= nextprime(p);
if p+q=m then p fi
end proc:
map(f, [$3..500]); # Robert Israel, May 04 2020
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tri[n_] := IntegerQ[Sqrt[1 + 8 n]]; t = {}; p1 = 2; While[Length[t] < 50, p2 = NextPrime[p1]; If[tri[p1 + p2], AppendTo[t, p1]]; p1 = p2]; t (* T. D. Noe, May 28 2013 *)
A247302
Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).
Original entry on oeis.org
0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 2, 2, 2, 4, 4, 4, 8, 6, 8, 12, 12, 12, 24, 20, 24, 40, 36, 40, 72, 64, 72, 128, 112, 128, 224, 200, 224, 400, 352, 400, 704, 624, 704, 1248, 1104, 1248, 2208, 1952, 2208, 3904, 3456, 3904, 6912, 6112, 6912, 12224, 10816, 12224
Offset: 0
First 10 columns:
0 .. 1 .. 1 .. 2 .. 4 .. 6 .. 12 .. 20 .. 36 .. 64
1 .. 0 .. 2 .. 2 .. 4 .. 8 .. 12 .. 24 .. 40 .. 72
0 .. 1 .. 0 .. 2 .. 2 .. 4 .. 8 ... 12 .. 24 .. 40
T(4,1) counts these 4 paths, given as vector sums applied to (0,0):
(1,1) + (1,-1) + (1,1) + (1,-1);
(1,1) + (1,-1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,1) + (1,-1).
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t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0;
t[1, 0] = 1; t[1, 1] = 0; t[1, 2] = 1;
t[2, 0] = 0; t[2, 1] = 2; t[2, 2] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2] + t[n - 2, 0];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 2, 1];
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247302 *)
A118591
Larger of two consecutive primes whose sum is a square.
Original entry on oeis.org
19, 53, 73, 293, 883, 1153, 1931, 2593, 3529, 4051, 6053, 7207, 7451, 15139, 20809, 21647, 24203, 26921, 28807, 34849, 46819, 53147, 56453, 69193, 74507, 83233, 84053, 98573, 103067, 103969, 109517, 110459, 112339, 136247, 149059, 151253
Offset: 1
17 and 19 are consecutive primes whose sum is 36, a square, so 19 is a term.
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Transpose[Select[Partition[Prime[Range[14000]],2,1],IntegerQ[ Sqrt[ Total[#]]]&]] [[2]] (* Harvey P. Dale, May 03 2012 *)
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g(n) = for(x=2,n,if(issquare(prime(x)+prime(x-1)),print1(prime(x)",")))
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lista(pmax) = {my(p1 = 2); forprime(p2 = 3, pmax, if(issquare(p1 + p2), print1(p2, ", ")); p1 = p2);} \\ Amiram Eldar, Jul 12 2024