A033286 -id:A033286 - cp's OEIS frontend

A005145 n copies of n-th prime.

2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31

Offset: 1

Russ Cox

Seen as a triangle read by rows: T(n,k) = A000040(n), 1 <= k <= n; row sums = A033286; central terms = A031368. - Reinhard Zumkeller, Aug 05 2009

Seen as a square array read by antidiagonals, a subtable of the binary operation multiplication tables A297845, A306697 and A329329. - Peter Munn, Jan 15 2020

			Triangle begins:
  2;
  3, 3;
  5, 5, 5;
  7, 7, 7, 7;
  ...

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought", Basic Books, 1995.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Sequences with similar definitions: A002024, A175944.
Cf. A000040 (range of values), A003961, A031368 (main diagonal), A033286 (row sums), A097906.
Subtable of A297845, A306697, A329329.

a005145 n k = a005145_tabl !! (n-1) !! (k-1)
a005145_row n = a005145_tabl !! (n-1)
a005145_tabl = zipWith ($) (map replicate [1..]) a000040_list
a005145_list = concat a005145_tabl
-- Reinhard Zumkeller, Jul 12 2014, Mar 18 2011, Oct 17 2010

[NthPrime(Round(Sqrt(2*n))): n in [1..60]]; // Vincenzo Librandi, Jan 18 2020

Table[Prime[Floor[1/2 + Sqrt[2*n]]], {n, 1, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Flatten[Table[Table[Prime[n], {n}], {n, 12}]] (* Alonso del Arte, Jan 18 2012 *)
Table[PadRight[{},n,Prime[n]],{n,15}]//Flatten (* Harvey P. Dale, Feb 29 2024 *)

a(n) = prime(round(sqrt(2*n))) \\ Charles R Greathouse IV, Oct 23 2015

from sympy import primerange
a = []; [a.extend([pn]*n) for n, pn in enumerate(primerange(1, 32), 1)]
print(a) # Michael S. Branicky, Jul 13 2022

from math import isqrt
from sympy import prime
def A005145(n): return prime(isqrt(n<<3)+1>>1) # Chai Wah Wu, Jun 08 2025

From Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006: (Start)
a(n) = prime(floor(1/2 + sqrt(2*n))).
a(n) = A000040(A002024(n)). (End)
From Peter Munn, Jan 15 2020: (Start)
When viewed as a square array A(n,k), the following hold for n >= 1, k >= 1:
A(n,k) = prime(n+k-1).
A(n,1) = A(1,n) = prime(n), where prime(n) = A000040(n).
A(n+1,k) = A(n,k+1) = A003961(A(n,k)).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)) = A329329(A(n,1), A(1,k)).
(End)
Sum_{n>=1} 1/a(n)^2 = A097906. - Amiram Eldar, Aug 16 2022

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 10, 9, 16, 11, 14, 15, 12, 32, 13, 22, 21, 20, 18, 64, 17, 26, 33, 28, 25, 24, 128, 19, 34, 39, 44, 35, 30, 27, 256, 23, 38, 51, 52, 55, 42, 40, 36, 512, 29, 46, 57, 68, 65, 66, 49, 45, 48, 1024, 31, 58, 69, 76, 85, 78, 77, 56, 50, 54, 2048, 37, 62, 87, 92

Offset: 1

Leroy Quet, Jan 27 2007

This sequence is a permutation of the integers >= 2.
Since the table has been entered by rising instead of falling antidiagonals, the sequence represents the transpose, with columns instead of rows: cf. the "table" link, section "infinite square array". - M. F. Hasler, Oct 22 2019
Start with table headed by primes in the first row, then list beneath each prime(k) the ordered prime(k)-smooth numbers. Read the table by falling antidiagonals to get the terms of this sequence. - David James Sycamore, Jun 23 2024

			Array begins: (rows here appear as columns in the "table" display of the sequence)
   2,  4,  8, 16, 32, 64, 128, 256, 512, ... (A000079)
   3,  6,  9, 12, 18, 24,  27,  36,  48, ... (A065119)
   5, 10, 15, 20, 25, 30,  40,  45,  50, ... (A080193)
   7, 14, 21, 28, 35, 42,  49,  56,  63, ... (A080194)
  11, 22, 33, 44, 55, 66,  77,  88,  99, ... (A080195)
  13, 26, 39, 52, 65, 78,  91, 104, 117, ... (A080196)
The 3rd row, for example, contains the positive integers where the 3rd prime, 5, is the largest prime divisor. That is, each integer in this row is divisible by 5 and may be divisible by 2 or 3 as well, but none of the integers in this row are divisible by primes larger than 5. (So for example, 35 = 5*7 is excluded from the 3rd row.)

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A083140, A006530, A000040 (1st col), A033286 (main diag), A077320.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681.

lpf[n_] := FactorInteger[n][[ -1, 1]];f[n_, m_] := f[n, m] = Block[{k},k = If[m == 1, Prime[n], f[n, m - 1] + 1];While[lpf[k] != Prime[n], k++ ];k];Table[f[ d - m + 1, m], {d, 12}, {m, d}] // Flatten (* Ray Chandler, Feb 09 2007 *)

T=List(); r=c=1; for(n=1,99, #TT[r][1], ); print1(T[r][c]","); r-- && c++ || r=c+c=1) \\ M. F. Hasler, Oct 22 2019

Extended by Ray Chandler, Feb 09 2007

A057855 Greatest k such that (k-th prime) <= (n times n-th prime).

Original entry on oeis.org

1, 3, 6, 9, 16, 21, 30, 36, 46, 61, 68, 86, 99, 110, 126, 146, 168, 184, 205, 223, 242, 270, 292, 321, 360, 381, 404, 429, 446, 477, 546, 574, 614, 637, 693, 717, 762, 804, 842, 890, 935, 965, 1029, 1059, 1105, 1134, 1222, 1304, 1348, 1381, 1423, 1483

Offset: 1

Henry Bottomley, Nov 13 2000

nonn

Might be roughly n^2/2 (seems to be marginally more at least for small n).

			a(4)=9 since 4th prime is 7, 4*7=28, greatest prime less than or equal to 28 is 23 which is the 9th prime.

Michel Marcus, Table of n, a(n) for n = 1..5000

Cf. A020900, A020901, A020934-A020940, A033286 (n*prime(n)).

a:= n-> numtheory[pi](n*ithprime(n)):
seq(a(n), n=1..61);  # Alois P. Heinz, Aug 30 2019

Mathematica
```
Table[PrimePi[w*Prime[w]], {w, 1, 100}]
```

a(n) = primepi(n*prime(n)); \\ Michel Marcus, Aug 30 2019

a(n) = pi(n*prime(n)).

A117495 Product of a prime number p and the number of primes smaller than p.

Original entry on oeis.org

0, 3, 10, 21, 44, 65, 102, 133, 184, 261, 310, 407, 492, 559, 658, 795, 944, 1037, 1206, 1349, 1460, 1659, 1826, 2047, 2328, 2525, 2678, 2889, 3052, 3277, 3810, 4061, 4384, 4587, 5066, 5285, 5652, 6031, 6346, 6747, 7160, 7421, 8022, 8299, 8668, 8955, 9706

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 25 2006

			a(9) = 184 because (1) the 9th prime number is 23, (2) there are 8 primes smaller than 23 and (3) 23*8 = 184.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000720, A033286, A036689.

with(numtheory):a:=n->sum(ithprime(n), j=2..n):seq(a(n), n=1..47); # Zerinvary Lajos, Aug 24 2008

Table[(n - 1)Prime[n], {n, 60}] (* Zak Seidov, Aug 15 2010 *)

a(n)=prime(n)*(n-1) \\ Charles R Greathouse IV, Sep 15 2014

a(n) = (n-1)*prime(n). - Zak Seidov, Aug 15 2010

a(31) corrected by Jens Kruse Andersen, Sep 15 2014

A124012 Decimal expansion of Sum_{k>=1} 1/(k*prime(k)).

Original entry on oeis.org

8, 4, 8, 9, 6, 9, 0, 3, 4, 0, 4, 3

Offset: 0

Pierre CAMI, Nov 02 2006

From Robert Price, Jul 14 2010: (Start)
This series converges very slowly. I could not find any transform that converges faster, so I did this by brute force using 256 bits of precision.
After k=596765000000 terms (p(k)=17581469834441) the partial sum is 0.848 969 034 043 245 206 069 544 346 415 327 714...
The next two digits are either 29 or 30. (End)
The table in the Example section shows, for increasing values of j, the results of computing the partial sum s(j) = Sum_{k=1..j} 1/(k*prime(k)) and adding to it an approximate value for the tail (i.e., the sum for all the terms k > j). See the Links entry for an explanation of the method used in approximating the size of the tail of the summation beyond the j-th prime. - Jon E. Schoenfield, Jan 20 2019

			0.848969034043...
From _Jon E. Schoenfield_, Jan 14 2019: (Start)
We can obtain prime(2^d) for d = 0..57 from the b-file for A033844. Given the above result from _Robert Price_, and letting j_RP = 596765000000, the partial sum through
   prime(j_RP) = 17581469834441
is
   s(j_RP) = Sum_{k=1..j_RP} 1/(k*prime(k))
           = 0.848969034043245206069544346415327714...;
adding to this actual partial sum s(j_RP) the approximate tail value
   t(j_RP) =
         h'(prime(j_RP), prime(2^40))
       + (Sum_{d=41..57} h'(prime(2^(d-1)), prime(2^d)))
       + lim_{x->infinity} h(prime(2^57), x)
(see the Links entry for an explanation) gives the result 0.84896903404330021273712255895762255... (which seems likely to be correct to at least 20 significant digits).
The table below gives, for j = 2^16, 2^17, ..., 2^32, and j_RP, the actual partial sum s(j) and the sum s(j) + t(j) where t(j) is the approximate tail value beyond prime(j).
.
   j             s(j)                s(j) + t(j)
  ====  ======================  ======================
  2^16  0.84896790758922908159  0.84896903393397518971
  2^17  0.84896850050492294891  0.84896903400552099072
  2^18  0.84896878057566843770  0.84896903404214147367
  2^19  0.84896891330602605081  0.84896903404317536927
  2^20  0.84896897639243509768  0.84896903404350431035
  2^21  0.84896900645590169648  0.84896903404376063663
  2^22  0.84896902081581006534  0.84896903404343742139
  2^23  0.84896902768965496764  0.84896903404337393698
  2^24  0.84896903098637626311  0.84896903404331189996
  2^25  0.84896903257029535468  0.84896903404329806633
  2^26  0.84896903333252861584  0.84896903404330030271
  2^27  0.84896903369988697984  0.84896903404330084536
  2^28  0.84896903387717904236  0.84896903404330042023
  2^29  0.84896903396285181513  0.84896903404330024036
  2^30  0.84896903400430044877  0.84896903404330021861
  2^31  0.84896903402437548991  0.84896903404330021472
  2^32  0.84896903403410856545  0.84896903404330021655
  ...            ...                     ...
  j_RP  0.84896903404324520607  0.84896903404330021274
(End)

Jon E. Schoenfield, Notes on approximating the size of the summation's tail beyond the j-th prime
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A033286, A085548, A209329, A210473.

Offset and leading zero corrected by R. J. Mathar, Jan 31 2009
Four more terms (4,0,4,3) from Robert Price, Jul 14 2010
Title and example edited by M. F. Hasler, Jan 13 2015

A272173 Product of the sum of the divisors of n and the sum of the divisors of n-th prime.

Original entry on oeis.org

3, 12, 24, 56, 72, 168, 144, 300, 312, 540, 384, 1064, 588, 1056, 1152, 1674, 1080, 2418, 1360, 3024, 2368, 2880, 2016, 5400, 3038, 4284, 4160, 6048, 3300, 8208, 4096, 8316, 6624, 7560, 7200, 13832, 6004, 9840, 9408, 15660, 7560, 17472, 8448, 16296, 15444, 14400, 10176, 27776

Offset: 1

Omar E. Pol, Apr 26 2016

nonn

Numbers that occur twice in the sequence include 7560, 816000, 2709504, 31752000. Are there infinitely many? Does any number occur more than twice? - Robert Israel, Sep 12 2018

			For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13, and the 9th prime is 23, and the sum of the divisors of 23 is 1 + 23 = 24, and 13*24 = 312, so a(9) = 312.
On the other hand 9*23 = 207, and the sum of the divisors of 207 is 1 + 3 + 9 + 23 + 69 + 207 = 312, so a(9) = 312.

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

Cf. A000040, A000203, A008864, A033286, A272211.

[SumOfDivisors(n)*SumOfDivisors(NthPrime(n)): n in [1..50]]; // Vincenzo Librandi, Sep 13 2018

f:= n -> numtheory:-sigma(n)*(1+ithprime(n)):
map(f, [$1..100]); # Robert Israel, Sep 12 2018

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

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

a(n) = sigma(n)*sigma(prime(n)) = sigma(n)*(1 + prime(n)) = A000203(n)*(1 + A000040(n)) = A000203(n)*A008864(n).
a(n) = sigma(n*prime(n)) = A000203(n*A000040(n)) = A000203(A033286(n)).
a(n) = A000203(n) + A272211(n).

A080697 a(n) = n * prime(prime(n)).

Original entry on oeis.org

3, 10, 33, 68, 155, 246, 413, 536, 747, 1090, 1397, 1884, 2327, 2674, 3165, 3856, 4709, 5094, 6289, 7060, 7707, 8822, 9913, 11064, 12725, 14222, 15201, 16436, 17371, 18510, 21979, 23648, 25509, 27098, 30065, 31572, 34003, 36746, 38649, 41240

Offset: 1

Cino Hilliard, Mar 04 2003

Previous name was: "Product of prime-index-primes and the index of their prime index".

Sum of reciprocals converges to about 1/2. [More accurately, 0.50056748... . - Amiram Eldar, Jul 13 2024]

			a(1) = 1*prime(prime(1)) = 1*prime(2) = 1*3 = 3.
a(2) = 2*prime(prime(2)) = 2*prime(3) = 2*5 = 10.

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

Cf. A000040, A006450, A033286.

[n*NthPrime(NthPrime(n)): n in [1..40]]; // Vincenzo Librandi, Jun 09 2016

Table[n Prime[Prime[n]], {n, 1, 40}] (* Vincenzo Librandi, Jun 09 2016 *)

pipxindex(n) = {sr=0; pr=1; for(x=1,n, y=prime(prime(x)); pr=x*y; print1(pr" "); sr+=1.0/pr; ); print(); print(sr) }

a(n) = n*A006450(n). - Omar E. Pol, Oct 21 2013

New name from Omar E. Pol, Oct 21 2013

A306192 a(n) = (n - 1)*prime(n + 1).

Original entry on oeis.org

0, 5, 14, 33, 52, 85, 114, 161, 232, 279, 370, 451, 516, 611, 742, 885, 976, 1139, 1278, 1387, 1580, 1743, 1958, 2231, 2424, 2575, 2782, 2943, 3164, 3683, 3930, 4247, 4448, 4917, 5134, 5495, 5868, 6179, 6574, 6981, 7240, 7831, 8106, 8471, 8756, 9495, 10258

Offset: 1

Stefano Spezia, Jan 28 2019

For n > 1, a(n) is the subdiagonal sum of the matrix M(n) whose determinant is A318173(n).

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000040, A033286, A318173.

Magma
```
[(n-1)*NthPrime(n+1): n in [1..100]];
```

a := n -> (n-1)*ithprime(n+1): seq(a(n), n = 1 .. 100);

Mathematica
```
a[n_]:=(n-1)*Prime[n+1]; Array[a,100]
```
PARI
```
a(n) = (n-1)*prime(n+1);
```

from sympy import prime
[(n-1)*prime(n+1) for n in range(1,100)]

a(n) = A033286(n + 1) - 2*A000040(n + 1).

a(n) = (n - 1)/(n + 1)*A033286(n + 1).

A077320 Triangle in which n-th row contains n smallest multiples of the n-th prime.

Original entry on oeis.org

2, 3, 6, 5, 10, 15, 7, 14, 21, 28, 11, 22, 33, 44, 55, 13, 26, 39, 52, 65, 78, 17, 34, 51, 68, 85, 102, 119, 19, 38, 57, 76, 95, 114, 133, 152, 23, 46, 69, 92, 115, 138, 161, 184, 207, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290

Offset: 1

Amarnath Murthy, Nov 04 2002

A000040 (primes) gives initial terms of rows.
A033286 contains the final terms of rows.
Sum of the n-th row = prime(n)*A000217(n), by definition.
a(A000217(n) + 1) = prime(n+1), by definition.

			From _Bruno Berselli_, Sep 05 2017: (Start)
Triangle begins:
   2;
   3, 6;
   5, 10,  15;
   7, 14,  21,  28;
  11, 22,  33,  44,  55;
  13, 26,  39,  52,  65,  78;
  17, 34,  51,  68,  85, 102, 119;
  19, 38,  57,  76,  95, 114, 133, 152;
  23, 46,  69,  92, 115, 138, 161, 184, 207;
  29, 58,  87, 116, 145, 174, 203, 232, 261, 290;
  31, 62,  93, 124, 155, 186, 217, 248, 279, 310, 341;
  37, 74, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444;
  41, 82, 123, 164, 205, 246, 287, 328, 369, 410, 451, 492, 533;
  43, 86, 129, 172, 215, 258, 301, 344, 387, 430, 473, 516, 559, 602, etc.
(End)

Ivan Neretin, Table of n, a(n) for n = 1..8128

Cf. A000040, A000217, A033286.

Row sums give A196421. - Omar E. Pol, Mar 12 2012

Table[Prime[n]*Range[n], {n, 10}] // Flatten (* Ivan Neretin, May 02 2019 *)

T(n,k) = k*prime(n) with 1 <= k <= n. - Bruno Berselli, Sep 05 2017

A084295 n is such that pi(n*prime(n))/n is an integer.

Original entry on oeis.org

1, 3, 47, 88, 200, 547, 12182, 15335, 39104, 58122, 73282, 150740, 480886

Offset: 1

Labos Elemer, May 27 2003

a(9) > 30000. - Michel Marcus, Sep 02 2019

a(13) > 200000. - Giovanni Resta, Sep 02 2019

Cf. A000040, A000720, A033286, A057855, A084297, A084298.

Do[s=PrimePi[n*Prime[n]]/n; If[IntegerQ[s], Print[n]], {n, 1, 100000}]
Select[Range[16000],IntegerQ[PrimePi[#*Prime[#]]/#]&] (* Harvey P. Dale, Jul 23 2015 *)

isok(n) = denominator(primepi(n*prime(n))/n) == 1; \\ Michel Marcus, Sep 02 2019

Corrected by Harvey P. Dale, Jul 23 2015
a(9)-a(12) from Giovanni Resta, Sep 02 2019
a(13) from Chai Wah Wu, May 14 2020

cp's OEIS Frontend

A005145 n copies of n-th prime.

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A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

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A057855 Greatest k such that (k-th prime) <= (n times n-th prime).

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Maple

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A117495 Product of a prime number p and the number of primes smaller than p.

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Maple

Mathematica

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Extensions

A124012 Decimal expansion of Sum_{k>=1} 1/(k*prime(k)).

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Extensions

A272173 Product of the sum of the divisors of n and the sum of the divisors of n-th prime.

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Programs

Magma

Maple

Mathematica

PARI

Formula

A080697 a(n) = n * prime(prime(n)).