A048448 -id:A048448 - cp's OEIS frontend

A048451 Starting from generation 4 add previous and next term yielding generation 5.

30, 56, 97, 125, 179, 214, 274, 326, 382, 454, 506, 582, 640, 708, 780, 842, 924, 984, 1060, 1128, 1194, 1276, 1348, 1432, 1508, 1582, 1652, 1718, 1808, 1870, 1998, 2052, 2192, 2238, 2364, 2418, 2522, 2594, 2686, 2758, 2860, 2912, 3028, 3074, 3196, 3268

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 4 is A048450. See A048448-A048466. See also A047844.

A048452 Starting from generation 5 add previous and next term yielding generation 6.

Original entry on oeis.org

56, 127, 181, 276, 339, 453, 540, 656, 780, 888, 1036, 1146, 1290, 1420, 1550, 1704, 1826, 1984, 2112, 2254, 2404, 2542, 2708, 2856, 3014, 3160, 3300, 3460, 3588, 3806, 3922, 4190, 4290, 4556, 4656, 4886, 5012, 5208, 5352, 5546, 5670, 5888, 5986, 6224

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 5 is A048451. See A048448-A048466. See also A047844.

A048453 Starting from generation 6 add previous and next term yielding generation 7.

Original entry on oeis.org

127, 237, 403, 520, 729, 879, 1109, 1320, 1544, 1816, 2034, 2326, 2566, 2840, 3124, 3376, 3688, 3938, 4238, 4516, 4796, 5112, 5398, 5722, 6016, 6314, 6620, 6888, 7266, 7510, 7996, 8212, 8746, 8946, 9442, 9668, 10094, 10364, 10754, 11022, 11434, 11656

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 6 is A048452. See A048448-A048466. See also A047844.

A048454 Starting from generation 7 add previous and next term yielding generation 8.

Original entry on oeis.org

237, 530, 757, 1132, 1399, 1838, 2199, 2653, 3136, 3578, 4142, 4600, 5166, 5690, 6216, 6812, 7314, 7926, 8454, 9034, 9628, 10194, 10834, 11414, 12036, 12636, 13202, 13886, 14398, 15262, 15722, 16742, 17158, 18188, 18614, 19536, 20032, 20848

Offset: 0

Patrick De Geest, May 15 1999

nonn

Generation 7 is A048453. See A048448-A048466. See also A047844.

Missing a(5) inserted by Sean A. Irvine, Jun 16 2021

A048456 First terms from generation 1 onwards.

Original entry on oeis.org

2, 3, 7, 13, 30, 56, 127, 237, 530, 994, 2192, 4144, 9016, 17192, 36949, 71023, 151024, 292340, 616032, 1199556, 2508660, 4909028, 10201810, 20044004, 41437560, 81683022, 168134730, 332321130, 681582636, 1350091410, 2760703019

Offset: 0

Patrick De Geest, May 15 1999

nonn

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

A048459 Generation indices to 'Prime last odd terms' of sequence A048458.

Original entry on oeis.org

2, 3, 4, 5, 7, 19, 25, 27, 53, 59, 68, 148, 176, 241, 347, 441, 444, 509, 844, 990, 1607, 1823, 2745, 3244, 3517, 3695

Offset: 1

Patrick De Geest, May 15 1999

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

a(9)-a(21) from Nathaniel Johnston, Apr 11 2011

a(22)-a(26) from Sean A. Irvine, Jun 18 2021

A048461 Numbers in sorted order taken from all generations.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 16, 17, 19, 20, 23, 28, 29, 30, 31, 32, 37, 40, 41, 43, 44, 47, 48, 52, 53, 54, 56, 59, 61, 66, 67, 68, 71, 72, 73, 79, 80, 82, 83, 88, 89, 94, 96, 97, 101, 103, 106, 107, 109, 112, 113, 114, 125, 126, 127, 131, 132, 137, 139, 140, 146, 149

Offset: 0

Patrick De Geest, May 15 1999

nonn

Cf. A048448-A048455. See also A047844.

A260989 Integers n such that prime(n-1) + prime(n+1) is a multiple of n.

Original entry on oeis.org

4, 5, 8, 11, 12, 18, 20, 70, 72, 1053, 4116, 6459, 6460, 40083, 63328, 251742, 399924, 637320, 637322, 637330, 2582288, 2582436, 2582488, 10553828, 16899042, 69709721, 179992913, 179992922, 465769813, 749973302, 749973314, 1208198617, 1208198629

Offset: 1

Zak Seidov, Aug 06 2015

nonn

			n=4: prime(n-1) + prime(n+1) = 5 + 11 = 16 = 4*n,
n=20: 67 + 73 = 140 = 7*n,
n=16899042: 312632263 + 312632291 = 625264554 = 37*n,
n=69709721: 1394194387 + 1394194453 = 2788388840 = 40*n.

Zak Seidov, Table of n, (c=prime(n-1)+prime(n-1), c/n

Cf. A000040, A004648, A048448, A045924, A023143, A092044, A092045, A156149.

[n: n in [2..7*10^3], k in [2..7*10^3] | (NthPrime(n-1) + NthPrime(n+1)) eq n*k]; // Vincenzo Librandi, Aug 07 2015

Select[Range[2, 100000], Mod[Prime[# - 1] + Prime[# + 1], #] == 0 &] (* Michael De Vlieger, Aug 07 2015 *)

a=2;b=5;for(n=2,10^8,c=a+b;if(c%n<1,print1(n", "));a=nextprime(a+1);b=nextprime(b+1))

p=2;q=3;n=1; forprime(r=5,1e9, if((p+r)%n++==0, print1(n", "));p=q;q=r) \\ Charles R Greathouse IV, Aug 10 2015

a(27)-a(33) from Charles R Greathouse IV, Aug 10 2015

A342993 Numbers of the form prime(i-1)+prime(i+1) that are the average of a twin prime pair.

Original entry on oeis.org

72, 150, 180, 270, 1032, 1062, 1452, 1608, 2028, 2082, 2130, 2592, 2790, 3120, 3258, 3300, 3360, 3930, 4020, 4272, 4650, 4722, 4788, 4932, 5442, 5880, 6702, 7128, 7332, 7878, 8388, 8430, 8862, 9240, 9342, 9678, 10008, 10140, 10272, 10890, 11490, 11940, 12072, 12162, 12918, 13002, 13218, 13932

Offset: 1

J. M. Bergot and Robert Israel, Apr 01 2021

nonn

			a(3) = 180 is a term because 180 = 83 + 97 = prime(23)+prime(25) and (180-1 = 179, 180+1 = 181) is a twin prime pair.

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

Intersection of A048448 and A014574.

Cf. A000040.

count:= 0: R:= NULL:
for i from 1 while count < 100 do
  x:= ithprime(i)+ithprime(i+2);
  if isprime(x-1) and isprime(x+1) then R:= R, x; fi
od:
R;

lista(nn) = {for (n=2, nn, my(x=prime(n)+prime(n+2)); if (isprime(x-1) && isprime(x+1), print1(n, ", ")););} \\ Michel Marcus, Apr 02 2021

A354430 First diagonal of an array, generated from the sequence of the nonprimes.

Original entry on oeis.org

1, 7, 22, 58, 142, 334, 766, 1726, 3837, 8435, 18364, 39646, 84986, 181117, 384160, 811676, 1709425, 3590213, 7522354, 15728427, 32827027, 68405533, 142344708, 295824870, 614046159, 1273068141, 2636250146, 5452584131, 11264148401, 23242423457, 47903544728

Offset: 1

Tamas Sandor Nagy, May 27 2022

Mirroring the idea in A048457, here with nonprimes, and including 1 of the first generation.

We write down the sequence of the nonprimes 1, 4, 6, ... in the first row of the array. Nonprime(k) + nonprime(k+2) will generate the second row. Thereafter we generate the further rows in a similar manner. The leftmost diagonal gives the sequence.

			1    4    6    8    9   10   12   14   15   16   18   20   21 ...
     7   12   15   18   21   24   27   30   33   36   39 ...
         22   30   36   42   48   54   60   66   72 ...
              58   72   84   96  108  120  132 ...
                  142  168  192  216  240 ...
                       334  384  432 ...
                            766 ...

Michael S. Branicky, Table of n, a(n) for n = 1..3311

Cf. A001787, A018252, A048457, A048448, A099862.

from sympy import composite
from functools import lru_cache
@lru_cache(maxsize=None)
def T(r, k):
    if r == 1: return 1 if k == 1 else composite(k-1)
    return T(r-1, k) + T(r-1, k+2)
def a(n): return T(n, 1)
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, May 28 2022

a(8) and beyond from Michael S. Branicky, May 28 2022

cp's OEIS Frontend

A048451 Starting from generation 4 add previous and next term yielding generation 5.

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A048452 Starting from generation 5 add previous and next term yielding generation 6.

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A048453 Starting from generation 6 add previous and next term yielding generation 7.

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A048454 Starting from generation 7 add previous and next term yielding generation 8.

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A048456 First terms from generation 1 onwards.

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A048459 Generation indices to 'Prime last odd terms' of sequence A048458.

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A048461 Numbers in sorted order taken from all generations.

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A260989 Integers n such that prime(n-1) + prime(n+1) is a multiple of n.

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Magma

Mathematica

PARI

PARI

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A342993 Numbers of the form prime(i-1)+prime(i+1) that are the average of a twin prime pair.

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Maple

PARI

A354430 First diagonal of an array, generated from the sequence of the nonprimes.

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Python

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