A033286 -id:A033286 - cp's OEIS frontend

A090942 n-th arithmetic mean = prime(n).

2, 4, 9, 13, 27, 23, 41, 33, 55, 83, 51, 103, 89, 69, 103, 143, 155, 95, 175, 147, 113, 205, 171, 227, 289, 201, 155, 215, 165, 229, 547, 255, 329, 205, 489, 221, 373, 385, 319, 407, 419, 263, 611, 279, 373, 289, 763, 787, 419, 327, 433, 545, 345, 781, 581, 593

Offset: 1

Amarnath Murthy, Dec 29 2003

nonn

Partial sums give A033286. - Omar E. Pol, Apr 20 2015

In other words, this is the unique sequence such that for all n >= 1, (1/n) * Sum_{k=1..n} a(k) = prime(n). - Antti Karttunen, Apr 30 2015

			From _Omar E. Pol_, Apr 20 2015: (Start)
Illustration of initial terms:
Consider a diagram in the first quadrant of the square grid in which the length of the n-th horizontal line segment is equal to the n-th prime, as shown below:
.   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  51|
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _          83|   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _      55|           |   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  33|       |           |   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _      41|   |       |           |   |
.  |_ _ _ _ _ _ _ _ _ _ _  23|       |   |       |           |   |
.  |_ _ _ _ _ _ _      27|   |       |   |       |           |   |
.  |_ _ _ _ _  13|       |   |       |   |       |           |   |
.  |_ _ _   9|   |       |   |       |   |       |           |   |
.  |_ _ 4|   |   |       |   |       |   |       |           |   |
.  |_2_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|
.
a(n) is also the area (or the number of cells) in the n-th region of the diagram. For example: a(4) = 7 + 6 = 13.
(End)

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A007504, A033286, A033287, A141042, A152535.

[n le 1 select 2 else n*NthPrime(n) - (n-1)*NthPrime(n-1): n in [1..60]]; // G. C. Greubel, Feb 04 2019

Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* Michael De Vlieger, Apr 20 2015 *)

vector(60, n, if(n==1,2, n*prime(n) -(n-1)*prime(n-1))) \\ G. C. Greubel, Feb 04 2019

[2] + [n*nth_prime(n) - (n-1)*nth_prime(n-1) for n in (2..60)] # G. C. Greubel, Feb 04 2019

a(n) = n*prime(n) - (n-1)*prime(n-1).
a(n) = A033287(n-2), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n) + A141042(n-1), n >=2. - Omar E. Pol, Apr 20 2015

Corrected and extended by Ray Chandler, Dec 31 2003

A272211 Product of n-th prime and the sum of the divisors of n.

Original entry on oeis.org

2, 9, 20, 49, 66, 156, 136, 285, 299, 522, 372, 1036, 574, 1032, 1128, 1643, 1062, 2379, 1340, 2982, 2336, 2844, 1992, 5340, 3007, 4242, 4120, 5992, 3270, 8136, 4064, 8253, 6576, 7506, 7152, 13741, 5966, 9780, 9352, 15570, 7518, 17376, 8404, 16212, 15366, 14328, 10128, 27652, 12939, 21297, 16776, 23422

Offset: 1

Omar E. Pol, Apr 26 2016

nonn

			For n = 9 the 9th prime is 23, and the sum of the divisors of 9 is 1 + 3 + 9 = 13, and 23*13 = 299, so a(9) = 299.
On the other hand 9*23 = 207 and the sum of the divisors of 207 is 1 + 3 + 9 + 23 + 69 + 207 = 312 and 312 - 13 = 299, so a(9) = 299.

Main diagonal of A272214.

Cf. A000040, A000203, A033286, A272173, A272400.

Table[DivisorSigma[1, n]*Prime[n], {n, 1, 50}] (* G. C. Greubel, Apr 27 2016 *)

a(n) = prime(n)*sigma(n); \\ Michel Marcus, Apr 27 2016

a(n) = prime(n)*sigma(n) = A000040(n)*A000203(n).

a(n) = sigma(n*prime(n)) - sigma(n) = A000203(n*A000040(n)) - A000203(n) = A000203(A033286(n)) - A000203(n) = A272173(n) - A000203(n).

A340128 a(n) = (n*prime(n)) mod prime(n+1).

Original entry on oeis.org

2, 1, 1, 6, 3, 10, 5, 14, 4, 11, 8, 34, 17, 38, 16, 22, 27, 26, 66, 33, 32, 78, 40, 2, 1, 51, 106, 53, 110, 88, 7, 82, 73, 107, 81, 98, 104, 15, 112, 118, 99, 153, 107, 21, 109, 81, 105, 35, 131, 33, 172, 137, 223, 190, 196, 202, 157, 206, 45, 163, 269, 53

Offset: 1

Simon Strandgaard, Jan 15 2021

			a(1) = (prime(1) * 1) mod prime(1+1) =  2 * 1 mod  3 = 2,
a(2) = (prime(2) * 2) mod prime(2+1) =  3 * 2 mod  5 = 1,
a(3) = (prime(3) * 3) mod prime(3+1) =  5 * 3 mod  7 = 1,
a(4) = (prime(4) * 4) mod prime(4+1) =  7 * 4 mod 11 = 6,
a(5) = (prime(5) * 5) mod prime(5+1) = 11 * 5 mod 13 = 3.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A033286, A340649.

Table[Mod[n*Prime[n],Prime[n+1]],{n,62}] (* Stefano Spezia, Jan 17 2021 *)

a(n) = (prime(n)*n) % prime(n+1); \\ Michel Marcus, Jan 20 2021

require 'prime'
values = []
primes = Prime.first(20)
primes.each_index do |n|
    next if n < 1
    values << (primes[n-1] * n) % primes[n]
end
p values

A356492 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

Original entry on oeis.org

1, 2, 5, 51, 264, 19532, -11904, 1261296, -2052864, 70621632, 24618221568, 3996020736, 743171562496, 24567175118848, -1257930752000, 864893030400, 12289833785344000, 1099483729459478528, 100515455071223808, 757166323365314560, 6294658173770137600, 7801939905505132544

Offset: 0

Stefano Spezia, Aug 09 2022

sign

Conjecture: a(n) is prime only for n = 1 and 2.

Conjecture is true because a(n) is even for n >= 4. This is because all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023

			For n = 1 the matrix M(1) is
    2
with determinant a(1) = 2.
For n = 2 the matrix M(2) is
    3, 2
    2, 3
with determinant a(2) = 5.
For n = 3 the matrix M(3) is
    5, 3, 2
    3, 5, 3
    2, 3, 5
with determinant a(3) = 51.

Robert Israel, Table of n, a(n) for n = 0..532
Mathematics Stack Exchange, Determinant of a Toeplitz matrix
Wikipedia, Toeplitz Matrix

Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356493 (permanent of the matrix M(n)).

Cf. A350955, A350956.

f:=proc(n) uses LinearAlgebra; local i;
 Determinant(ToeplitzMatrix([seq(ithprime(i),i=n..1,-1)],symmetric));
end proc:
q(0):= 1:
map(q, [$0..25]); # Robert Israel, Oct 13 2023

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,21}]]

a(n) = matdet(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022

A350955(n) <= a(n) <= A350956(n).

A008507 Number of odd composite numbers less than n-th odd prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11, 13, 13, 15, 16, 16, 18, 19, 21, 24, 25, 25, 26, 26, 27, 33, 34, 36, 36, 40, 40, 42, 44, 45, 47, 49, 49, 53, 53, 54, 54, 59, 64, 65, 65, 66, 68, 68, 72, 74, 76, 78, 78, 80, 81, 81, 85, 91, 92, 92, 93, 99, 101, 105, 105, 106, 108, 111, 113, 115, 116, 118

Offset: 1

Gary Findley (chfindley(AT)alpha.nlu.edu)

nonn

a(n) = A067076(n) - n + 1. - Vincenzo Librandi, Feb 02 2013

For n>=4, a(n) = k+1, where A000217(j) is the smallest triangular number such that A000217(j) - A033286(n+1) also is a triangular number, i.e., A000217(k). Example n=29, a(29) = 27: A033286(30) = 3390, A000217(86) = 3741. 3741-3390 = 351 = A000217(26); k=26, 26+1 = 27. - Bob Selcoe, Apr 12 2016

Cf. A067076.
Cf. A000040 (prime numbers), A000217 (triangular numbers), A033286 (n*prime(n)).
Partial sums of A100820.

[(Odd Primes - 1)/2] - n for n > 0, or A005097(n) - A000027(n). For example, A005097(1) - A000027(1) = 1 - 1 = 0, A005097(2) - A000027(2) = 2 - 2 = 0, A005097(9) - A000027(9) = 14 - 9 = 5. - William A. Tedeschi, Apr 25 2008

More terms from David W. Wilson, Jan 13 2000

A071228 a(n) = n*(n-th composite number).

Original entry on oeis.org

4, 12, 24, 36, 50, 72, 98, 120, 144, 180, 220, 252, 286, 336, 375, 416, 459, 504, 570, 640, 693, 748, 805, 864, 950, 1014, 1080, 1176, 1276, 1350, 1426, 1536, 1617, 1700, 1785, 1872, 1998, 2090, 2184, 2280, 2378, 2520, 2666, 2772, 2880, 2990, 3102

Offset: 1

Amarnath Murthy, May 17 2002

nonn

Complement of A171521. - Jaroslav Krizek, Dec 13 2009

			a(1)= 1*4, a(2)= 2*6, a(3)= 3*9, a(4) = 4*10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002808, A033286, A171521.

Partial sums of A092930.

count := 1: for i from 2 to 100 do if isprime(i) then else printf(`%d,`,i*count); count := count+1 fi: od: # James Sellers, May 28 2002

Module[{cn=Select[Range[70],CompositeQ]},Times@@@Thread[{cn, Range[ Length[ cn]]}]] (* Harvey P. Dale, Nov 13 2014 *)

lista(nn) = {my(nc = 0); forcomposite(c=1, nn, nc ++; print1(c*nc, ", "););} \\ Michel Marcus, Aug 31 2019

a(n) = n * A002808(n). - Jaroslav Krizek, Dec 13 2009

More terms from James Sellers, May 28 2002

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A189892 a(n) = n*prime(n) - sum_{i=1..n-1} prime(i).

Original entry on oeis.org

2, 4, 10, 18, 38, 50, 78, 94, 130, 190, 212, 284, 336, 364, 424, 520, 622, 658, 772, 852, 894, 1026, 1118, 1262, 1462, 1566, 1620, 1732, 1790, 1910, 2344, 2472, 2670, 2738, 3088, 3160, 3382, 3610, 3766, 4006, 4252, 4336, 4766, 4854, 5034, 5126, 5690

Offset: 1

Bruno Berselli, Apr 30 2011

			a(4) = 4*prime(4) - (prime(3) + prime(2) + prime(1)) = 4*7 - (5 + 3 + 2) = 18.

Bruno Berselli, Table of n, a(n) for n = 1..5000

Cf. A152535, A166448.

[2] cat [n*NthPrime(n)-(&+[NthPrime(k): k in [1..n-1]]): n in [2..47]];

nn=50;Join[{2},With[{prs=Accumulate[Prime[Range[nn]]]},Table[n*Prime[n]-prs[[n-1]],{n,2,nn}]]] (* Harvey P. Dale, Nov 21 2014 *)

v=vector(10000);s=n=i=0;forprime(p=2,1e9,v[i++]=n++*p-s;if(i==#v,return,s+=p)) \\ Charles R Greathouse IV, May 01 2011

a(n) = A033286(n+1) - A007504(n) for n>1.

A268465 Primes of the form mprime(m) + (m + 1)prime(m + 1) + (m + 2)*prime(m + 2).

Original entry on oeis.org

23, 349, 1579, 4691, 5783, 7187, 9547, 11519, 15377, 45779, 52289, 353359, 361787, 384277, 510227, 678413, 710599, 1007861, 1218709, 1301617, 1484449, 1567567, 1839469, 2073989, 2264959, 2409163, 2438377, 2520779, 2735281, 2882653, 2998867, 3100271, 3211751

Offset: 1

Zak Seidov, Feb 05 2016

nonn

Primes arising in A105455.

Primes of the form A033286(m)+A033286(m+1)+A033286(m+2).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A033286.

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    m, p, q, r = 1, 2, 3, 5
    while True:
        t = m*p + (m+1)*q + (m+2)*r
        if isprime(t): yield t
        m, p, q, r = m+1, q, r, nextprime(r)
print(list(islice(agen(), 33))) # Michael S. Branicky, May 17 2022

Typo in a(28) fixed by Seth A. Troisi, May 17 2022

a(29) and beyond from Michael S. Branicky, May 17 2022

A352028 a(n) = Product p_{n*i}^e_i if the prime factorization of n is Product p_i^e_i.

Original entry on oeis.org

1, 3, 13, 49, 47, 481, 107, 6859, 3721, 3277, 257, 121841, 397, 11309, 22261, 7890481, 653, 1390861, 881, 1416521, 78373, 47479, 1279, 157208087, 143641, 92011, 15813251, 7018237, 1889, 14701639, 2293, 38579489651, 309709, 207527, 461939, 2938615681, 3119

Offset: 1

Alois P. Heinz, Mar 01 2022

nonn

Or replace prime(i) in n by prime(n*i).

All terms are odd.

			a(1) = 1 because 1 is the empty product.
a(2) = 3 = prime(2) = prime(2*1) because 2 = prime(1).
a(3) = 13 = prime(6) = prime(3*2) because 3 = prime(2).
a(4) = 49 = 7^2 = prime(4)^2 = prime(4*1)^2 because 4 = prime(1)^2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Main diagonal of A352001.

Cf A000040, A033286, A228529.

a:= n-> mul(ithprime(n*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..45);

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(n*primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Mar 02 2022

a(n) = A352001(n,n).

a(prime(n)) = A228529(n) = A000040(A033286(n)).

A356493 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

Original entry on oeis.org

1, 2, 13, 271, 12030, 1346758, 214022024, 51763672608, 16088934953136, 6611717516842608, 4412314619046451200, 3533754988232088933120, 3506189715435673999194112, 4444138735439968822425464576, 5893766827264238066914528545792, 8502284313901016361834901076874240, 15350799440394462109333953415858960384

Offset: 0

Stefano Spezia, Aug 09 2022

nonn

Conjecture: a(n) is prime only for n = 1, 2, and 3.

Conjecture is true because a(n) is even for n >= 4. This is because a(n) == A356492(n) (mod 2), and all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    3, 2
    2, 3
with permanent a(2) = 13.
For n = 3 the matrix M(3) is
    5, 3, 2
    3, 5, 3
    2, 3, 5
with permanent a(3) = 271.

Mathematics Stack Exchange, Determinant of a Toeplitz matrix
Wikipedia, Toeplitz Matrix

Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356492 (determinant of the matrix M(n)).

Cf. A351021, A351022.

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Permanent[M[n]]; PrimeQ[Join[{1},Table[a[n],{n,16}]]]

a(n) = matpermanent(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022

A351021(n) <= a(n) <= A351022(n).

cp's OEIS Frontend

A090942 n-th arithmetic mean = prime(n).

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A272211 Product of n-th prime and the sum of the divisors of n.

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A340128 a(n) = (n*prime(n)) mod prime(n+1).

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A356492 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

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Maple

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A008507 Number of odd composite numbers less than n-th odd prime.

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A071228 a(n) = n*(n-th composite number).

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Maple

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A189892 a(n) = n*prime(n) - sum_{i=1..n-1} prime(i).

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A268465 Primes of the form mprime(m) + (m + 1)prime(m + 1) + (m + 2)*prime(m + 2).