A100705 -id:A100705 - cp's OEIS frontend

A127483 Numbers n such that A100705(n) = n^3 + (n+1)^2 is prime.

1, 2, 3, 4, 8, 9, 13, 14, 15, 17, 22, 23, 24, 25, 30, 32, 34, 35, 38, 39, 42, 45, 50, 58, 60, 64, 65, 79, 83, 85, 88, 90, 92, 94, 98, 99, 100, 102, 113, 115, 122, 125, 127, 130, 133, 134, 137, 140, 144, 147, 148, 153, 154, 157, 164, 167, 170, 178, 179, 184, 190, 193

Offset: 1

Alexander Adamchuk, Jan 16 2007

nonn

Corresponding primes of the form n^3 + (n+1)^2 are listed in A100662(n) = {5, 17, 43, 89, 593, 829, 2393, 2969, 3631, 5237, ...}.
Note that there are many consecutive twins, triples and quadruplets in a(n). For example: (1,2,3,4), {8,9}, {13,14,15}, {22,23,24,25}, {34,35}, {38,39}, {64,65}, {98,99,100}.
Twins start with n = {1,2,3,8,13,14,22,23,24,34,38,64,98,99,,...} = A127484, or numbers n such that a(n) = a(n+1) - 1.
Triplets start with n = {1,2,13,22,23,98,253,343,573,638,702,...} = A127485, or numbers n such that a(n) = a(n+1) - 1 = a(n+2) - 2.
Quadruplets start with n = {1,22,13077,14267,16092,16267,162,...} = A127486.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A100705, A100662, A127484, A127485, A127486.

Select[Range[1000],PrimeQ[ #^3+(#+1)^2]&]

A268261 T(n,k)=Number of length-(n+1) 0..k arrays with new repeated values introduced in sequential order starting with zero.

Original entry on oeis.org

3, 7, 5, 13, 17, 9, 21, 43, 42, 17, 31, 89, 143, 106, 33, 43, 161, 378, 479, 273, 65, 57, 265, 837, 1610, 1616, 717, 129, 73, 407, 1634, 4357, 6877, 5492, 1918, 257, 91, 593, 2907, 10082, 22710, 29461, 18804, 5218, 513, 111, 829, 4818, 20771, 62249, 118530, 126591

Offset: 1

R. H. Hardin, Jan 29 2016

Table starts
....3.....7.....13.......21.......31........43.........57..........73
....5....17.....43.......89......161.......265........407.........593
....9....42....143......378......837......1634.......2907........4818
...17...106....479.....1610.....4357.....10082......20771.......39154
...33...273...1616.....6877....22710.....62249.....148468......318261
...65...717...5492....29461...118530....384605....1061632.....2587557
..129..1918..18804...126591...619490...2377935....7594224....21042479
..257..5218..64869...545627..3242265..14712729...54345509...171161319
..513.14413.225483..2359152.16993552..91096234..389060724..1392571084
.1025.40349.789747.10233188.89197862.564452368.2786424182.11332701236

			Some solutions for n=5 k=4
..0....4....1....2....0....3....2....1....0....1....4....1....1....1....3....2
..0....3....4....4....0....2....4....0....3....4....0....3....0....2....4....0
..4....4....0....0....4....3....3....0....2....1....4....4....4....3....0....0
..3....0....2....0....0....4....4....4....0....4....1....0....3....2....3....4
..1....0....3....1....2....0....1....3....3....0....0....3....1....3....0....3
..3....3....2....4....0....0....4....0....2....3....3....1....3....0....4....0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000051.
Row 1 is A002061(n+1).
Row 2 is A100705.

Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2)
k=2: a(n) = 7*a(n-1) -15*a(n-2) +7*a(n-3) +6*a(n-4)
k=3: a(n) = 13*a(n-1) -60*a(n-2) +105*a(n-3) -11*a(n-4) -94*a(n-5) -24*a(n-6)
k=4: [order 8]
k=5: [order 10]
k=6: [order 12]
k=7: [order 14]
Empirical for row n:
n=1: a(n) = n^2 + n + 1
n=2: a(n) = n^3 + n^2 + 2*n + 1
n=3: a(n) = n^4 + n^3 + 3*n^2 + 2*n + 2
n=4: a(n) = n^5 + n^4 + 4*n^3 + 3*n^2 + 6*n + 2
n=5: a(n) = n^6 + n^5 + 5*n^4 + 4*n^3 + 12*n^2 + 6*n + 5 for n>1
n=6: a(n) = n^7 + n^6 + 6*n^5 + 5*n^4 + 20*n^3 + 12*n^2 + 20*n + 5 for n>1
n=7: a(n) = n^8 + n^7 + 7*n^6 + 6*n^5 + 30*n^4 + 20*n^3 + 50*n^2 + 20*n + 15 for n>2

A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 3, 1, 1, 5, 7, 7, 4, 1, 1, 6, 9, 13, 9, 4, 1, 1, 7, 11, 21, 16, 11, 4, 1, 1, 8, 13, 31, 25, 22, 13, 4, 1, 1, 9, 15, 43, 36, 37, 28, 15, 5, 1, 1, 10, 17, 57, 49, 56, 49, 40, 17, 5, 1, 1, 11, 19, 73, 64, 79, 76, 85, 43, 19, 5, 1, 1, 12, 21, 91, 81, 106, 109, 156, 89, 49, 21, 5, 1

Offset: 1

Omar E. Pol, Mar 05 2015

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
-------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A070941: 1, 2, 3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5,   5,   5
A005408: 1, 3, 5,  7,  9, 11, 13,  15,  17,  19,  21,  23,  25,  27,  29
A151788: 1, 4, 7, 13, 16, 22, 28,  40,  43,  49,  55,  67,  73,  85,  97
A147562: 1, 5, 9, 21, 25, 37, 49,  85,  89, 101, 113, 149, 161, 197, 233
A151790: 1, 6,11, 31, 36, 56, 76, 156, 161, 181, 201, 281, 301, 381, 461
A151781: 1, 7,13, 43, 49, 79,109, 259, 265, 295, 325, 475, 505, 655, 805
A151792: 1, 8,15, 57, 64,106,148, 400, 407, 449, 491, 743, 785,1037,1289
A151793: 1, 9,17, 73, 81,137,193, 585, 593, 649, 705,1097,1153,1545,1937
A255764: 1,10,19, 91,100,172,244, 820, 829, 901, 973,1549,1621,2197,2773
A255765: 1,11,21,111,121,211,301,1111,1121,1211,1301,2111,2201,3011,3821
A255766: 1,12,23,133,144,254,364,1464,1475,1585,1695,2795,2905,4005,5105
...

Index entries for sequences related to cellular automata

Rows 1-12: A000012, A070941, A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255764, A255765, A255766.
Columns 1-10: A000012, A000027, A005408, A002061, A000290, A084849, A056107, A053698, A100705, A100104.
Cf. A000120, A255740.

A100662 Primes of the form k^3 + (k+1)^2.

Original entry on oeis.org

5, 17, 43, 89, 593, 829, 2393, 2969, 3631, 5237, 11177, 12743, 14449, 16301, 27961, 33857, 40529, 44171, 56393, 60919, 75937, 93241, 127601, 198593, 219721, 266369, 278981, 499439, 578843, 621521, 689393, 737281, 787337, 839609, 950993, 980299, 1010201, 1071817

Offset: 1

Giovanni Teofilatto, Jan 03 2005

			a(1) = 5 because 5 = 1^3 + 2^2.
a(2) = 17 because 17 = 2^3 + 3^2.
a(3) = 43 because 43 = 3^3 + 4^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Intersection of A100705 and A000040.

Cf. A127483.

[ a: n in [0..100] | IsPrime(a) where a is n^3 + (n+1)^2 ]; // Vincenzo Librandi, Jul 18 2012

Select[Table[n^3+(n+1)^2,{n,200}],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

list(lim)=my(v=List(),t); for(n=1,lim, t=n^3+(n+1)^2; if(t>lim, break); if(isprime(t), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Dec 23 2016

a(n) = A100705(A127483(n)). - Elmo R. Oliveira, Apr 19 2025

More terms from Mark Hudson, Jan 04 2005

A127484 Numbers k such that A127483(k) = A127483(k+1) - 1.

Original entry on oeis.org

1, 2, 3, 8, 13, 14, 22, 23, 24, 34, 38, 64, 98, 99, 133, 147, 153, 178, 232, 253, 254, 297, 328, 343, 344, 367, 407, 498, 573, 574, 582, 587, 624, 638, 639, 653, 668, 679, 702, 703, 759, 772, 793, 797, 849, 874, 944, 958, 1023, 1058, 1067, 1087, 1203, 1212, 1322

Offset: 1

Alexander Adamchuk, Jan 16 2007

nonn

A127483(n) = {1,2,3,4,8,9,13,14,15,17,22,23,24,25,30,32,34,35,38,39,42,45,50,...} are the numbers n such that A100705(n) = n^3 + (n+1)^2 is prime.
Corresponding primes of the form n^3 + (n+1)^2 are listed in A100662(n) = {5, 17, 43, 89, 593, 829, 2393, 2969, 3631, 5237, ...}.
Note that there are many consecutive twins, triples and quadruplets in A127483(n). For example: (1,2,3,4), {8,9}, {13,14,15}, {22,23,24,25}, {34,35}, {38,39}, {64,65}, {98,99,100}. Twins in A127483(k) start with numbers k = a(n). Triplets in A127483(k) start with k = {1,2,13,22,23,98,253,343,573,638,702,...} = A127485, or numbers n such that a(k) = a(k+1) - 1 = a(k+2) - 2. Quadruplets in A127483(k) start with k = {1,22,13077,14267,16092,16267,16282,36387,47012,51912,54662,...} = A127486.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A100705, A100662, A127483, A127485, A127486.

[k:k in [1..1350]|IsPrime(k^3+(k+1)^2) and IsPrime((k+1)^3+(k+2)^2)]; // Marius A. Burtea, Jan 20 2020

Select[Range[3000],PrimeQ[ #^3+(#+1)^2]&&PrimeQ[(#+1)^3+(#+2)^2]&]

A127485 Numbers k such that A127483(k) = A127483(k+1) - 1 = A127483(k+2) - 2.

Original entry on oeis.org

1, 2, 13, 22, 23, 98, 253, 343, 573, 638, 702, 1322, 1862, 2543, 2638, 2758, 2792, 2912, 3093, 3158, 3242, 3578, 3968, 4382, 5013, 6503, 7048, 7877, 8372, 8912, 9022, 9207, 10298, 10443, 11538, 12482, 13077, 13078, 13868, 14267, 14268, 14323, 14783

Offset: 1

Alexander Adamchuk, Jan 16 2007

nonn

A127483(n) = {1,2,3,4,8,9,13,14,15,17,22,23,24,25,30,32,34,35,38,39,42,45,50,...} are the numbers n such that A100705(n) = n^3 + (n+1)^2 is prime. Corresponding primes of the form n^3 + (n+1)^2 are listed in A100662(n) = {5, 17, 43, 89, 593, 829, 2393, 2969, 3631, 5237, ...}. Note that there are many consecutive twins, triples and quadruplets in A127483(n). For example: (1,2,3,4), {8,9}, {13,14,15}, {22,23,24,25}, {34,35}, {38,39}, {64,65}, {98,99,100}. Twins in A127483(k) start with k = {1,2,3,8,13,14,22,23,24,34,38,64,98,99,133,147,153,178,232,253,254,297,328,343, 344,367,407,498,...} = A127484. Triplets in A127483(k) start with numbers k = a(n). Quadruplets in A127483(k) start with k = {1,22,13077,14267,16092,16267,16282,36387,47012,51912,54662,...} = A127486.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A100705, A100662, A127483, A127484, A127486.

f:=func; [k:k in [1..15000]|f(k) and f(k+1) and f(k+2)]; // Marius A. Burtea, Jan 20 2020

Select[Range[30000],PrimeQ[ #^3+(#+1)^2]&&PrimeQ[(#+1)^3+(#+2)^2]&&PrimeQ[(#+2)^3+(#+3)^2]&]

A127486 Numbers k such that A127483(k) = A127483(k+1) - 1 = A127483(k+2) - 2 = A127483(k+3) - 3.

Original entry on oeis.org

1, 22, 13077, 14267, 16092, 16267, 16282, 36387, 47012, 51912, 54662, 144487, 181777, 205897, 210022, 213982, 250517, 263717, 344092, 409697, 454607, 568777, 643677, 665917, 702947, 749967, 824167, 858187, 887677, 888427, 997787, 1075537

Offset: 1

Alexander Adamchuk, Jan 16 2007

A127483(n) = {1,2,3,4,8,9,13,14,15,17,22,23,24,25,30,32,34,35,38,39,42,45,50,...} are the numbers n such that A100705(n) = n^3 + (n+1)^2 is prime. Corresponding primes of the form n^3 + (n+1)^2 are listed in A100662(n) = {5, 17, 43, 89, 593, 829, 2393, 2969, 3631, 5237, ...}. Note that there are many consecutive twins, triples and quadruplets in A127483(n). For example: (1,2,3,4), {8,9}, {13,14,15}, {22,23,24,25}, {34,35}, {38,39}, {64,65}, {98,99,100}. Twins in A127483(k) start with k = {1,2,3,8,13,14,22,23,24,34,38,64,98,99,133,147,153,178,232,253,254,297,328,343, 344,367,407,498,...} = A127484. Triplets in A127483(k) start with k = {1,2,13,22,23,98,253,343,573,638,702,...} = A127485. Quadruplets in A127483(k) start with numbers k = a(n).

For n>1 the final digit of all listed terms of a(n) is 2 or 7. - Alexander Adamchuk, Jan 16 2007

Amiram Eldar, Table of n, a(n) for n = 1..3000

Cf. A100705, A100662, A127483, A127484, A127485.

f[s_]:=s^3+(s+1)^2; Do[If[PrimeQ[f[n]]&&PrimeQ[f[n+1]]&&PrimeQ[f[n+2]]&&PrimeQ[f[n+3]],Print[n]],{n,1,100000}]

More terms from Alexander Adamchuk, Jan 16 2007

A304159 a(n) = 2n^3 - 4n^2 + 6*n - 2 (n>=1).

Original entry on oeis.org

2, 10, 34, 86, 178, 322, 530, 814, 1186, 1658, 2242, 2950, 3794, 4786, 5938, 7262, 8770, 10474, 12386, 14518, 16882, 19490, 22354, 25486, 28898, 32602, 36610, 40934, 45586, 50578, 55922, 61630, 67714, 74186, 81058, 88342, 96050, 104194, 112786, 121838, 131362, 141370, 151874, 162886, 174418, 186482, 199090

Offset: 1

Emeric Deutsch, May 09 2018

a(n) is the first Zagreb index of the Barbell graph B(n) (n>=3).
The Barbell graph B(n) is defined as two copies of the complete graph K(n) (n>=3), connected by a bridge.
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of the Barbell graph B(n) is M(B(n),x,y) = (n-1)*(n-2)*x^{n-1}*y^{n-1} + 2*(n-1)*x^{n-1}*y^n + x^n*y^n.

Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavžar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Eric Weisstein's World of Mathematics, Barbell Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002943, A014105, A033431, A100705, A304160.

Maple
```
seq(2*n^3-4*n^2+6*n-2, n = 1 .. 40);
```

Table[2n^3-4n^2+6n-2 ,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,10,34,86},50] (* Harvey P. Dale, Mar 05 2023 *)

Vec(2*x*(1 + x + 3*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, May 09 2018

a(n) = 2*n^3-4*n^2+6*n-2; \\ Altug Alkan, May 09 2018

a(n) = 2 * A100705(n-1).
From Colin Barker, May 09 2018: (Start)
G.f.: 2*x*(1 + x + 3*x^2 + x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. (End)
a(n) = A033431(n) - A002943(n-1) = A033431(n) - 2*A014105(n-1). - Omar E. Pol, May 09 2018

cp's OEIS Frontend

A127483 Numbers n such that A100705(n) = n^3 + (n+1)^2 is prime.

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A268261 T(n,k)=Number of length-(n+1) 0..k arrays with new repeated values introduced in sequential order starting with zero.

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A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

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A100662 Primes of the form k^3 + (k+1)^2.

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A127484 Numbers k such that A127483(k) = A127483(k+1) - 1.

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A127485 Numbers k such that A127483(k) = A127483(k+1) - 1 = A127483(k+2) - 2.

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A127486 Numbers k such that A127483(k) = A127483(k+1) - 1 = A127483(k+2) - 2 = A127483(k+3) - 3.

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A304159 a(n) = 2*n^3 - 4*n^2 + 6*n - 2 (n>=1).

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A304159 a(n) = 2n^3 - 4n^2 + 6*n - 2 (n>=1).