A049984 -id:A049984 - cp's OEIS frontend

A049989 a(n) is the number of arithmetic progressions of positive integers, nondecreasing with sum <= n.

1, 3, 6, 10, 14, 21, 26, 33, 42, 51, 58, 72, 80, 91, 107, 120, 130, 150, 161, 178, 199, 215, 228, 255, 272, 290, 316, 338, 354, 389, 406, 429, 460, 483, 508, 549, 569, 594, 630, 663, 685, 731, 754, 785, 833, 863, 888, 940, 969, 1007, 1054, 1090, 1118, 1175, 1212, 1253, 1305, 1342, 1373, 1444, 1476, 1515, 1577, 1621

Offset: 1

Clark Kimberling

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A047966, A049982, A049983, A049984, A049986, A049987, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

seq(n)={my(w=(sqrtint(8*n+1)-1)\2+1); Vec(x/(1-x)^2 + sum(k=2, n, x^k/(1 - if(k<=w, x^(k*(k-1)/2)))/(1-x^k) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Sep 28 2019

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049988(k). [Note that the offset of A049988 is 0.]
G.f.: (-1 + g.f. of A049988)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 28 2019

A049432 Numbers k such that k! - (k-1)! + 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 13, 20, 24, 46, 59, 150, 152, 198, 683, 880, 1135, 1907, 6617, 10243, 12016

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

nonn

a(21) > 12000. [From Donovan Johnson, Dec 18 2009]

			6! - (6-1)! + 1 = 601 is prime.

Index entries for sequences related to factorial numbers

Cf. A049984, A049433, A090703.

a(n) = A090703(n) + 1. - Michael S. Branicky, Jun 11 2025

a(15)-a(16) from Farideh Firoozbakht, Jul 18 2003
a(17)-a(20) from Donovan Johnson, Dec 18 2009
a(21) from Michael S. Branicky, Jun 11 2025

A188914 a(n) = n*n! + 1 = (n+1)! - n! + 1.

Original entry on oeis.org

1, 2, 5, 19, 97, 601, 4321, 35281, 322561, 3265921, 36288001, 439084801, 5748019201, 80951270401, 1220496076801, 19615115520001, 334764638208001, 6046686277632001, 115242726703104001, 2311256907767808001, 48658040163532800001, 1072909785605898240001

Offset: 0

John M. Campbell, Apr 17 2011

It is unknown if all numbers of the form n*n!+1 are squarefree. n*n!+1 is squarefree for 0 < n < 52. It is unknown if there exist infinitely many primes of the form n*n!+1. For primes in this sequence, see A049984.

Cf. A001563, A049984, A094258.

Mathematica
```
Table[(n*Factorial[n])+1,{n,0,30}]
```

a(n) = n*n! + 1; \\ Michel Marcus, Aug 03 2022

E.g.f.: exp(x) + x/(1 - x)^2. - Stefano Spezia, Aug 03 2022

a(0)=1 prepended by Alois P. Heinz, Aug 03 2022

A245495 Primes of the form n! - (n+1)! + (n+2)! + 1.

Original entry on oeis.org

103, 4441, 36650881, 5787936001, 19702293811201, 1075342687614074880001, 8547762518578406446202880000001, 59043709472234119545920159524322926688993280000000001, 698533028148544417308552639358841460358000936394290829866303488000000000001

Offset: 1

K. D. Bajpai, Jul 24 2014

nonn

The next term a(10) has 95 digits which is too large to show in data section.
a(16) has 1181 digits, hence not included in b-file.
Primes for indices 3, 5, 9, 11, 14, 20, 27, 41, 54, 65, 81, 83, 105, 315, 323, 515, ... - Robert G. Wilson v, Aug 07 2014

			m = 3: m! - (m+1)! + (m+2)! + 1 = 103, which is prime, hence appears in the sequence.
m = 5: m! - (m+1)! + (m+2)! + 1 = 4441, which is prime, hence appears in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..15

Cf. A000040, A049984, A049432.

Select[Table[n! - (n + 1)! + (n + 2)! + 1, {n, 200}], PrimeQ[#] &]
Select[#[[1]]-#[[2]]+#[[3]]+1&/@Partition[Range[70]!,3,1],PrimeQ] (* Harvey P. Dale, Aug 20 2021 *)

a(n) = p=n!-(n+1)!+(n+2)!+1;if(ispseudoprime(p),return(p))
n=1;while(n<100,if(a(n),print1(a(n),", "));n++) \\ Derek Orr, Jul 27 2014

A245528 Primes of the form n! - (n + 1)! + (n + 2)! - 1.

Original entry on oeis.org

19, 101, 35999, 327599, 3306239, 81430271999, 24779106953279078399999, 10501089199335077511167999999, 1372369506422963989169318155460666934165503999999999, 117024364553755119629556890816711613171571359743999999999

Offset: 1

K. D. Bajpai, Jul 25 2014

The term a(11) has 129 digits which is too large to show in data section.
a(15) has 1081 digits, hence not included in b-file.
The first 20 primes are for n = 2, 3, 6, 7, 8, 12, 21, 25, 40, 43, 83, 107, 132, 139, 478, 505, 931, 1516, 1739, 5208. - Jens Kruse Andersen, Aug 10 2014

			m = 2: m! - (m + 1)! + (m + 2)! - 1 = 19 which is prime, hence appears in the sequence.
m = 6: m! - (m + 1)! + (m + 2)! - 1 = 35999 which is prime, hence appears in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..14

Cf. A000040, A049984, A049432.

[a: n in [0..100] | IsPrime(a) where a is Factorial(n) - Factorial(n + 1) + Factorial(n + 2) - 1 ]; // Vincenzo Librandi, Aug 11 2014

Select[Table[n! - (n + 1)! + (n + 2)! - 1, {n, 200}], PrimeQ[#] &]

for(n=1,200,s=n!-(n+1)!+(n+2)!-1;if(ispseudoprime(s),print1(s,", "))) \\ Derek Orr, Aug 10 2014

cp's OEIS Frontend

A049989 a(n) is the number of arithmetic progressions of positive integers, nondecreasing with sum <= n.

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A049432 Numbers k such that k! - (k-1)! + 1 is prime.

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A188914 a(n) = n*n! + 1 = (n+1)! - n! + 1.

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A245495 Primes of the form n! - (n+1)! + (n+2)! + 1.

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Mathematica

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A245528 Primes of the form n! - (n + 1)! + (n + 2)! - 1.

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Magma

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