A048448 -id:A048448 - cp's OEIS frontend

A047844 Patrick De Geest's "Generations" array read by antidiagonals: a(n,1) = n-th prime, a(1,k+1) = a(2,k), a(n,k+1) = a(n-1,k) + a(n+1,k).

2, 3, 3, 5, 7, 7, 7, 10, 13, 13, 11, 16, 23, 30, 30, 13, 20, 30, 43, 56, 56, 17, 28, 44, 67, 97, 127, 127, 19, 32, 52, 82, 125, 181, 237, 237, 23, 40, 68, 112, 179, 276, 403, 530, 530, 29, 48, 80, 132, 214, 339, 520, 757, 994, 994, 31, 54, 94, 162, 274

Offset: 1

N. J. A. Sloane

			Array begins:
  2,    3,    7,   13,   30,   56,  127,  237,  530, ...
  3,    7,   13,   30,   56,  127,  237,  530,  994, ...
  5,   10,   23,   43,   97,  181,  403,  757, 1662, ...
  7,   16,   30,   67,  125,  276,  520, 1132, 2156, ...

Nathaniel Johnston, Table of n, a(n) for n = 1..16653

Columns give A000040, A048448-A048455. See also A048456-A048466.

A047844:=proc(n,k)global a:if(type(a[n,k],integer))then return a[n,k]:elif(k=1)then a[n,k]:=ithprime(n):elif(n=1)then a[n,k]:=A047844(2,k-1):else a[n,k]:=A047844(n-1,k-1)+A047844(n+1,k-1):fi:return a[n,k]:end:
for d from 1 to 8 do for m from 1 to d do print(A047844(d-m+1,m)):od:od: # Nathaniel Johnston, Apr 11 2011

a[n_, 1] := a[n, 1] = Prime[n]; a[1, k_] := a[1, k] = a[2, k-1]; a[n_, k_] := a[n, k] = a[n-1, k-1] + a[n+1, k-1]; Table[a[n-k+1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 26 2013 *)

a(n,k)=if(k==1,prime(n),n==1,a(2,k-1),a(n-1,k-1)+a(n+1,k-1))
for(s=2,9,for(k=1,s-1,print1(a(s-k,k)", "))) \\ Charles R Greathouse IV, Nov 26 2013

a(46)-a(60) from Nathaniel Johnston, Apr 11 2011

A048455 Starting from generation 8 add previous and next term yielding generation 9.

Original entry on oeis.org

530, 994, 1662, 2156, 2970, 3598, 4491, 5335, 6231, 7278, 8178, 9308, 10290, 11382, 12502, 13530, 14738, 15768, 16960, 18082, 19228, 20462, 21608, 22870, 24050, 25238, 26522, 27600, 29148, 30120, 32004, 32880, 34930, 35772, 37724, 38646

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 8 is A048454. See A048448-A048466. See also A047844.

A048466 Total number of primes in "generation" n.

Original entry on oeis.org

2, 3, 3, 2, 2, 2, 2, 0, 2, 0, 1, 1, 1, 3, 4, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 4, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 2, 0, 0, 0, 2, 1, 1, 0, 2, 0, 0, 0, 4, 1, 2, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 2, 0

Offset: 2

Patrick De Geest, May 15 1999

nonn

For "Generations" see A048448-A048455. See also A047844.

Offset corrected and more terms from Sean A. Irvine, Jun 18 2021

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Original entry on oeis.org

6, 8, 10, 10, 12, 14, 14, 16, 18, 22, 16, 18, 20, 24, 26, 20, 22, 24, 28, 30, 34, 22, 24, 26, 30, 32, 36, 38, 26, 28, 30, 34, 36, 40, 42, 46, 32, 34, 36, 40, 42, 46, 48, 52, 58, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74, 44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82

Offset: 1

Reinhard Zumkeller, Feb 06 2006

T(n,k) = 2*A065305(n,k) = A065342(n+1,k+1);
Row sums give A116367; central terms give A116368;
T(n,1) = A113935(n+1);
T(n,n-2) = A048448(n) for n>2;
T(n,n-1) = A001043(n) for n>1;
T(n,n) = A001747(n+2) = A100484(n+1).

			Triangle begins:
  6;
  8,  10;
  10, 12, 14;
  14, 16, 18, 22;
  16, 18, 20, 24, 26;
  20, 22, 24, 28, 30, 34;
  22, 24, 26, 30, 32, 36, 38;
  26, 28, 30, 34, 36, 40, 42, 46;
  32, 34, 36, 40, 42, 46, 48, 52, 58;
  34, 36, 38, 42, 44, 48, 50, 54, 60, 62;
  40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74;
  44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82; etc. - _Bruno Berselli_, Aug 16 2013

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A001043, A001747, A048448, A065305, A065342, A100484, A113935, A116367, A116368.

[NthPrime(n+1)+NthPrime(k+1): k in [1..n], n in [1..15]]; // Bruno Berselli, Aug 16 2013

Table[Prime[n+1] + Prime[k+1], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, May 12 2019 *)

{T(n,k) = prime(n+1) + prime(k+1)}; \\ G. C. Greubel, May 12 2019

[[nth_prime(n+1) + nth_prime(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 12 2019

T(n,k) = prime(n+1) + prime(k+1), 1 <= k <= n.

A336381 Primes p(n) such that gcd(n, prime(n-1)+prime(n+1)) > 1.

Original entry on oeis.org

7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 61, 71, 73, 79, 89, 97, 101, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 181, 193, 199, 223, 229, 233, 239, 251, 263, 269, 271, 281, 293, 311, 317, 337, 349, 359, 373, 379, 383, 397, 409, 421, 433, 443, 449

Offset: 1

Clark Kimberling, Oct 25 2020

nonn

			In the following table, P(n) = A000040(n) = prime(n).
  n    P(n)   P(n-1)+P(n+1)   gcd
  2     3          7           1
  3     5         10           1
  4     7         16           4
  5    11         20           5
  6    13         28           2
2 and 3 are in A336378; 4 and 5 are in A336379; 3 and 5 are in A336380; 7 and 11 are in A336381.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A048448, A336366, A336378, A336379, A336380.

q:= 2: r:= 3:
R:= NULL: count:= 0:
for n from 2 while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if igcd(n,p+r) > 1 then count:= count+1; R:= R, q; fi
od:
R; # Robert Israel, Dec 08 2020

p[n_] := Prime[n];
u = Select[Range[2, 200], GCD[#, p[# - 1] + p[# + 1]] == 1 &]  (* A336378 *)
v = Select[Range[2, 200], GCD[#, p[# - 1] + p[# + 1]] > 1 &]   (* A336379 *)
Prime[u]  (* A336380 *)
Prime[v]  (* A336381 *)
Select[Partition[Prime[Range[100]],3,1],GCD[PrimePi[#[[2]]],#[[1]]+#[[3]]]>1&][[All,2]] (* Harvey P. Dale, Dec 07 2022 *)

for(n=2,200,if(gcd(n,prime(n-1)+prime(n+1))>1,print1(prime(n),", "))) \\ Derek Orr, Nov 23 2020

Offset changed by Robert Israel, Dec 08 2020

A048460 Total of odd numbers in the generations from 2 onwards.

Original entry on oeis.org

2, 3, 3, 3, 4, 6, 5, 3, 4, 6, 6, 6, 8, 12, 9, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 17, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 33, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16

Offset: 2

Patrick De Geest, May 15 1999

nonn

			a(7)=6 because in generation 7 there are six odd numbers: 127,237,403,729,879,1109.

For "Generations" see A048448-A048455. See also A047844.

Cf. A220466.

A048460 := proc(nmax) local par, c, r, prevc, prevl, cpar; par := [[],[1,1]] ; for c from 3 to nmax do prevc := op(-1,par) ; prevl := nops(prevc) ; if nops(prevc) < 2 then cpar := [0] ; else cpar := [op(2,prevc)] ; end if; for r from 2 to prevl-1 do cpar := [op(cpar),( op(r-1,prevc) + op(r+1,prevc)) mod 2] ; end do: cpar := [op(cpar), op(prevl-1,prevc),1] ; par := [op(par),cpar] ; end do: cpar := [] ; for c from 2 to nops(par) do add(r,r=op(c,par)) ; cpar := [op(cpar),%] ; end do: cpar ; end proc: A048460(120) ; # R. J. Mathar, Aug 07 2010
nmax := 86: A001316 := n -> if n <=- 1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to nmax/(p+2) do a((2*n-3)*2^p) := (2^(p-1)+1)*A001316(n-2) od: od: seq(a(n), n=2..nmax); # Johannes W. Meijer, Jan 22 2013

A105321[n_] := Sum[Binomial[1, n-k] Mod[Binomial[k, j], 2], {k, 0, n}, {j, 0, k}];
a[n_] := A105321[n]/2;
Table[a[n], {n, 2, 86}] (* Jean-François Alcover, Oct 25 2023 *)

def A048460(n): return (1<>1 # Chai Wah Wu, Jul 30 2025

It appears that a(n) = A105321(n)/2. - Omar E. Pol, May 29 2010. Proof from Nathaniel Johnston, Nov 07 2010: If you remove every 2nd row from Pascal's triangle then the rule for constructing the parity of the next row from the current row is the same as the rule for constructing generation n+1 of the primes from generation n: add up the previous and next term in the current row.

a((2*n-3)*2^p) = (2^(p-1)+1)*A001316(n-2), p >= 0 and n >= 2. - Johannes W. Meijer, Jan 22 2013

More terms from R. J. Mathar, Aug 07 2010

A048449 Starting from generation 2 add previous and next term yielding generation 3.

Original entry on oeis.org

7, 13, 23, 30, 44, 52, 68, 80, 94, 114, 126, 146, 160, 176, 194, 210, 232, 246, 266, 282, 296, 318, 336, 358, 380, 398, 412, 428, 448, 464, 500, 514, 550, 560, 592, 604, 630, 650, 670, 690, 716, 728, 758, 766, 796, 814, 846, 874, 898, 920, 934, 958, 972, 1004

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 2 is A048448. See A048448-A048466. See also A047844.

A048457 Last odd terms from generation 2 onwards.

Original entry on oeis.org

7, 23, 67, 179, 453, 1109, 2653, 6231, 14409, 32877, 74137, 165429, 365691, 801747, 1745331, 3776605, 8130401, 17427659, 37217597, 79224121, 168170537, 356107787, 752453861, 1586875049, 3340696135, 7021048691, 14731810645

Offset: 1

Patrick De Geest, May 15 1999

nonn

For "Generations" see A048448-A048455. See also A047844.

A048458 'Prime last odd terms' from generation 2 onwards.

Original entry on oeis.org

7, 23, 67, 179, 1109, 17427659, 1586875049, 7021048691, 1104052140838673681, 80729882077782801781, 49474191359283212247841, 152695677551802424534973144788818335406948326813, 50258816309715893690860594601285860231033059311672877749

Offset: 1

Patrick De Geest, May 15 1999

Intersection of A000040 and A048457.

Nathaniel Johnston, Table of n, a(n) for n = 1..20

For "Generations" see A048448-A048455. Cf. A048459. See also A047844.

a(9) - a(13) from Nathaniel Johnston, Apr 11 2011

A048450 Starting from generation 3 add previous and next term yielding generation 4.

Original entry on oeis.org

13, 30, 43, 67, 82, 112, 132, 162, 194, 220, 260, 286, 322, 354, 386, 426, 456, 498, 528, 562, 600, 632, 676, 716, 756, 792, 826, 860, 892, 948, 978, 1050, 1074, 1142, 1164, 1222, 1254, 1300, 1340, 1386, 1418, 1474, 1494, 1554, 1580, 1642, 1688, 1744

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 3 is A048449. See A048448-A048466. See also A047844.

cp's OEIS Frontend

A047844 Patrick De Geest's "Generations" array read by antidiagonals: a(n,1) = n-th prime, a(1,k+1) = a(2,k), a(n,k+1) = a(n-1,k) + a(n+1,k).

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A048455 Starting from generation 8 add previous and next term yielding generation 9.

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A048466 Total number of primes in "generation" n.

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A116366 Triangle read by rows: even numbers as sums of two odd primes.

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A336381 Primes p(n) such that gcd(n, prime(n-1)+prime(n+1)) > 1.

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A048460 Total of odd numbers in the generations from 2 onwards.

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A048449 Starting from generation 2 add previous and next term yielding generation 3.

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A048457 Last odd terms from generation 2 onwards.

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A048458 'Prime last odd terms' from generation 2 onwards.

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