A005145 -id:A005145 - cp's OEIS frontend

A002024 k appears k times; a(n) = floor(sqrt(2n) + 1/2).

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13

Offset: 1

N. J. A. Sloane

Integer inverse function of the triangular numbers A000217. The function trinv(n) = floor((1+sqrt(1+8n))/2), n >= 0, gives the values 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, ..., that is, the same sequence with offset 0. - N. J. A. Sloane, Jun 21 2009
Array T(k,n) = n+k-1 read by antidiagonals.
Eigensequence of the triangle = A001563. - Gary W. Adamson, Dec 29 2008
Can apparently also be defined via a(n+1)=b(n) for n >= 2 where b(0)=b(1)=1 and b(n) = b(n-b(n-2))+1. Tested to be correct for all n <= 150000. - José María Grau Ribas, Jun 10 2011
For any n >= 0, a(n+1) is the least integer m such that A000217(m)=m(m+1)/2 is larger than n. This is useful when enumerating representations of n as difference of triangular numbers; see also A234813. - M. F. Hasler, Apr 19 2014
Number of binary digits of A023758, i.e., a(n) = ceiling(log_2(A023758(n+2))). - Andres Cicuttin, Apr 29 2016
a(n) and A002260(n) give respectively the x(n) and y(n) coordinates of the sorted sequence of points in the integer lattice such that x(n) > 0, 0 < y(n) <= x(n), and min(x(n), y(n)) < max(x(n+1), y(n+1)) for n > 0. - Andres Cicuttin, Dec 25 2016
Partial sums (A060432) are given by S(n) = (-a(n)^3 + a(n)*(1+6n))/6. - Daniel Cieslinski, Oct 23 2017
As an array, T(k,n) is the number of digits columns used in carryless multiplication between a k-digit number and an n-digit number. - Stefano Spezia, Sep 24 2022
a(n) is the maximum number of possible solutions to an n-statement Knights and Knaves Puzzle, where each statement is of the form "x of us are knights" for some 1 <= x <= n, knights can only tell the truth and knaves can only lie. - Taisha Charles and Brittany Ohlinger, Jul 29 2023

			From _Clark Kimberling_, Sep 16 2008: (Start)
As a rectangular array, a northwest corner:
  1 2 3 4 5 6
  2 3 4 5 6 7
  3 4 5 6 7 8
  4 5 6 7 8 9
This is the weight array (cf. A144112) of A107985 (formatted as a rectangular array). (End)
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^9 + 4*x^9 + 4*x^10 + ...

Edward S. Barbeau, Murray S. Klamkin, and William O. J. Moser, Five Hundred Mathematical Challenges, Prob. 441, pp. 41, 194. MAA 1995.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97.
K. Hardy and K. S. Williams, The Green Book of Mathematical Problems, p. 59, Solution to Prob. 14, Dover NY, 1985
R. Honsberger, Mathematical Morsels, pp. 133-134, MAA 1978.
J. F. Hurley, Litton's Problematical Recreations, pp. 152; 313-4 Prob. 22, VNR Co., NY, 1971.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 43.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5050
Jaegug Bae and Sungjin Choi, A generalization of a subset-sum-distinct sequence, J. Korean Math. Soc. 40 (2003), no. 5, 757--768. MR1996839 (2004d:05198). See b(n).
Jonathan H. B. Deane and Guido Gentile, A diluted version of the problem of the existence of the Hofstadter sequence, arXiv:2311.13854 [math.NT], 2023. See p. 10.
Nathan Fox, Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences, arXiv:2203.09340 [math.CO], 2022.
H. T. Freitag and H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187.
H. T. Freitag (Proposer) and H. W. Gould (Solver), Problem 571: An Ordering of the Rationals, Math. Mag., 38 (1965), 185-187 [Annotated scanned copy]
Mikel Garcia-de-Andoin, Álvaro Saiz, Pedro Pérez-Fernández, Lucas Lamata, Izaskun Oregi, and Mikel Sanz, Digital-Analog Quantum Computation with Arbitrary Two-Body Hamiltonians, arXiv:2307.00966 [quant-ph], 2023.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, A combinatorial approach for solving certain nested recursions with non-slow solutions, arXiv:1202.0276 [math.CO], 2012.
Stanley Wu-Wei Liu, Closed form expressions for A002024(n).
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564.
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Raphael Schumacher, Extension of Summation Formulas involving Stirling series, arXiv:1605.09204 [math.NT], 2016.
Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.
Raphael Schumacher, The Self-Counting Flow, Fibonacci Quart. 60 (2022), no. 5, 324-343.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970. (note that A1148 has now become A005282)
Michael Somos, Sequences used for indexing triangular or square arrays.
L. J. Upton, Letter to N. J. A. Sloane, May 22 1991.
Eric Weisstein's World of Mathematics, Self-Counting Sequence.
Index entries for Hofstadter-type sequences

a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
Cf. A001462, A002262, A003881, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums).
A123578 is an essentially identical sequence.

a002024 n k = a002024_tabl !! (n-1) !! (k-1)
a002024_row n = a002024_tabl !! (n-1)
a002024_tabl = iterate (\xs@(x:_) -> map (+ 1) (x : xs)) [1]
a002024_list = concat a002024_tabl
a002024' = round . sqrt . (* 2) . fromIntegral
-- Reinhard Zumkeller, Jul 05 2015, Feb 12 2012, Mar 18 2011

a002024_list = [1..] >>= \n -> replicate n n

a002024 = (!!) $ [1..] >>= \n -> replicate n n
-- Sascha Mücke, May 10 2016

[Floor(Sqrt(2*n) + 1/2): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014

A002024 := n-> ceil((sqrt(1+8*n)-1)/2); seq(A002024(n), n=1..100);

a[1] = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 (* Branko Curgus, May 12 2009 *)
Table[n, {n, 13}, {n}] // Flatten (* Robert G. Wilson v, May 11 2010 *)
Table[PadRight[{},n,n],{n,15}]//Flatten (* Harvey P. Dale, Jan 13 2019 *)

t1(n)=floor(1/2+sqrt(2*n)) /* A002024 = this sequence */

t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1) */

t3(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */

t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1)-1 */

A002024(n)=(sqrtint(n*8)+1)\2 \\ M. F. Hasler, Apr 19 2014

PARI
```
a(n)=(sqrtint(8*n-7)+1)\2
```

a(n)=my(k=1);while(binomial(k+1,2)+1<=n,k++);k \\ R. J. Cano, Mar 17 2014

from math import isqrt
def A002024(n): return (isqrt(8*n)+1)//2 # Chai Wah Wu, Feb 02 2022

[floor(sqrt(2*n) +1/2) for n in (1..80)] # G. C. Greubel, Dec 10 2018

a(n) = floor(1/2 + sqrt(2n)). Also a(n) = ceiling((sqrt(1+8n)-1)/2). [See the Liu link for a large collection of explicit formulas. - N. J. A. Sloane, Oct 30 2019]
a((k-1)*k/2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller, Aug 30 2001
a(n) = a(n - a(n-1)) + 1, with a(1)=1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), Aug 18 2002
a(n) = round(sqrt(2n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
T(n,k) = A003602(A118413(n,k)); = T(n,k) = A001511(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006
G.f.: (x/(1-x))*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, Oct 06 2003
Equals A127899 * A004736. - Gary W. Adamson, Feb 09 2007
Sum_{i=1..n} Sum_{j=i..n+i-1} T(j,i) = A000578(n); Sum_{i=1..n} T(n,i) = A000290(n). - Reinhard Zumkeller, Jun 24 2007
a(n) + n = A014132(n). - Vincenzo Librandi, Jul 08 2010
a(n) = ceiling(-1/2 + sqrt(2n)). - Branko Curgus, May 12 2009
a(A169581(n)) = A038567(n). - Reinhard Zumkeller, Dec 02 2009
a(n) = round(sqrt(2*n)) = round(sqrt(2*n-1)); there exist a and b greater than zero such that 2*n = 2+(a+b)^2 -(a+3*b) and a(n)=(a+b-1). - Fabio Civolani (civox(AT)tiscali.it), Feb 23 2010
A005318(n+1) = 2*A005318(n) - A205744(n), A205744(n) = A005318(A083920(n)), A083920(n) = n - a(n). - N. J. A. Sloane, Feb 11 2012
Expansion of psi(x) * x / (1 - x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Mar 19 2014
G.f.: (x/(1-x)) * Product_{n>=1} (1 + x^n) * (1 - x^(2*n)). - Paul D. Hanna, Feb 27 2016
a(n) = 1 + Sum_{i=1..n/2} ceiling(floor(2(n-1)/(i^2+i))/(2n)). - José de Jesús Camacho Medina, Jan 07 2017
a(n) = floor((sqrt(8*n-7)+1)/2). - Néstor Jofré, Apr 24 2017
a(n) = floor((A000196(8*n)+1)/2). - Pontus von Brömssen, Dec 10 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - Amiram Eldar, Oct 01 2022
G.f. as array: (x^2*(1 - y)^2 + y^2 + x*y*(1 - 2*y))/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Apr 22 2024

A007917 Version 1 of the "previous prime" function: largest prime <= n.

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 71, 71, 73, 73, 73, 73

Offset: 2

R. Muller

Version 2 of the "previous prime" function (see A151799) is "largest prime < n". This produces the same sequence of numerical values, except the offset (or indexing) starts at 3 instead of 2.
Maple's "prevprime" function uses version 2.
Also the largest prime dividing n! or lcm(1,...,n). - Labos Elemer, Jun 22 2000
Also largest prime among terms of (n+1)st row of Pascal's triangle. - Jud McCranie, Jan 17 2000
Also largest integer k such that A000203(k) <= n+1. - Benoit Cloitre, Mar 17 2002. - Corrected by Antti Karttunen, Nov 07 2017
Also largest prime factor of A061355(n) (denominator of Sum_{k=0..n} 1/k!). - Jonathan Sondow, Jan 09 2005
Also prime(pi(x)) where pi(x) is the prime counting function = number of primes <= x. - Cino Hilliard, May 03 2005
Also largest prime factor, occurring to the power p, in denominator of Sum_{k=1..n} 1/k^p, for any positive integer p. - M. F. Hasler, Nov 10 2006
For n > 10, these values are close to the most negative eigenvalues of A191898 (conjecture). - Mats Granvik, Nov 04 2011

K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.

N. J. A. Sloane, Table of n, a(n) for n = 2..10000
Marc Deleglise and Jean-Louis Nicolas, Maximal product of primes whose sum is bounded, arXiv preprint arXiv:1207.0603 [math.NT], 2012. See Fig. 1. - From _N. J. A. Sloane_, Dec 17 2012
Hans Gunter, Puzzle 145. The Inferior Smarandache Prime Part and Superior Smarandache Prime Part functions; Solutions by Jean Marie Charrier, Teresinha DaCosta, Rene Blanch, Richard Kelley and Jim Howell. F. Smarandache, Only Problems, Not Solutions!.
Eric Weisstein's World of Mathematics, Previous Prime

Cf. A000040, A000203, A005145, A007918, A070801, A113523, A151799, A179278, A175851, A000720.

a007917 n = if a010051' n == 1 then n else a007917 (n-1)
-- Reinhard Zumkeller, May 01 2013, Jul 26 2012

[NthPrime(#PrimesUpTo(n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

Maple
```
A007917 := n-> prevprime(n+1);
```

Table[Prime[PrimePi[n]], {n, 2, 70}] (* Stefan Steinerberger, Apr 06 2006 *)
NextPrime[Range[3,80],-1] (* Harvey P. Dale, Jan 23 2011 *)

a=precprime \\ In older versions, use a(n)=precprime(n)
\\ Charles R Greathouse IV, Jun 15 2011

Equals A006530(A000142(n)). - Jonathan Sondow, Jan 09 2005
Equals A006530(A056040(n)). - Peter Luschny, Mar 04 2011
a(n) = A000040(A049084(A007918(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012
From Wesley Ivan Hurt, May 22 2013: (Start)
omega( Product_{i=2..n} a(i) ) = pi(n).
Omega( Product_{i=2..n} a(i) ) = n - 1. (End)
For n >= 2, a(A000203(n)) = A070801(n). - Antti Karttunen, Nov 07 2017
a(n) = n + 1 - Sum_{i=1..n} floor(pi(i)/pi(n)) = n + 1 - A175851(n). - Ridouane Oudra, Jun 24 2024
Conjecture: a(n) = floor(log(Sum_{k=2..n} exp(A000010(k)+1))). - Joseph M. Shunia, Aug 09 2024
a(n) = A000040(A000720(n)). - Ridouane Oudra, Oct 04 2024

Edited by N. J. A. Sloane, Apr 06 2008

A031368 Odd-indexed primes: a(n) = prime(2n-1).

Original entry on oeis.org

2, 5, 11, 17, 23, 31, 41, 47, 59, 67, 73, 83, 97, 103, 109, 127, 137, 149, 157, 167, 179, 191, 197, 211, 227, 233, 241, 257, 269, 277, 283, 307, 313, 331, 347, 353, 367, 379, 389, 401, 419, 431, 439, 449, 461, 467, 487, 499, 509, 523, 547, 563

Offset: 1

Jeff Burch

Appeared as a puzzle in "Stickelers", a nationally distributed feature, by Terry Stickels, Sep 28 2006. - Franklin T. Adams-Watters, Sep 28 2006
Also every second prime starting with 2. - Cino Hilliard, Dec 02 2007
Central terms of the triangle in A005145. - Reinhard Zumkeller, Aug 05 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A031215 (even-indexed primes), A005408.

First differences are A155067.

a031368 = a000040 . ((subtract 1) . (* 2))
a031368_list = map a000040 [1, 3 ..]  -- Reinhard Zumkeller, Nov 25 2012

[ NthPrime(2*n-1): n in [1..1000] ]; // Vincenzo Librandi, Apr 11 2011

A031368 := n->ithprime(2*n-1): seq(A031368(n), n=1..100);

Table[ Prime[ 2n -1], {n, 52}] (* Robert G. Wilson v, Dec 01 2013 *)

a(n) = prime(2*n-1) \\ Jianing Song, Jun 03 2021

a(n) = A219603(n) / A000040(n). - Reinhard Zumkeller, Nov 25 2012

A033286 a(n) = n * prime(n).

Original entry on oeis.org

2, 6, 15, 28, 55, 78, 119, 152, 207, 290, 341, 444, 533, 602, 705, 848, 1003, 1098, 1273, 1420, 1533, 1738, 1909, 2136, 2425, 2626, 2781, 2996, 3161, 3390, 3937, 4192, 4521, 4726, 5215, 5436, 5809, 6194, 6513, 6920, 7339, 7602, 8213, 8492, 8865, 9154, 9917

Offset: 1

N. J. A. Sloane

Does an n exist such that n*prime(n)/(n+prime(n)) is an integer? - Ctibor O. Zizka, Mar 04 2008. The answer to Zizka's question is easily seen to be No: such an integer k would be positive and less than prime(n), but then k*(n + prime(n)) = prime(n)*n would be impossible. - Robert Israel, Apr 20 2015
Sums of rows of the triangle in A005145. - Reinhard Zumkeller, Aug 05 2009
Complement of A171520(n). - Jaroslav Krizek, Dec 13 2009
Partial sums of A090942. - Omar E. Pol, Apr 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 1.
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.

Cf. A000040, A007504, A014689, A090942, A124012, A141042, A152535.

Cf. A005145 (primes repeated), A171520 (complement), A076146 (iterated).

a033286 n = a000040 n * n  -- Reinhard Zumkeller, Jul 24 2013

[ n*NthPrime(n): n in [1..47] ]; // Klaus Brockhaus, Sep 09 2009

A033286 := proc(n) n*ithprime(n) ; end proc:
seq(A033286(n),n=1..20) ; # R. J. Mathar, Mar 21 2011

Table[Prime[n]*n, {n, 38}] (* Alonso del Arte *)

ithprime(i)*i $ i = 1..47 // Zerinvary Lajos, Feb 26 2007

a(n)=n*prime(n) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = n * A000040(n) = n * A008578(n+1) = n * A158611(n+2). - Jaroslav Krizek, Aug 31 2009
a(n) = A007504(n) + A152535(n). - Omar E. Pol, Aug 09 2012
Sum_{n>=1} 1/a(n) = A124012. - Amiram Eldar, Oct 15 2020

Correction for change of offset in A158611 and A008578 in Aug 2009 from Jaroslav Krizek, Jan 27 2010

a(A014284(n)) = a(A175965(n)) = A018252(n). [Reinhard Zumkeller, Mar 18 2011]

This is how A063905 would have been reckoned at the beginning of the 20th century, taking the primes as given in A008578 instead of the way they are given in A000040. [Alonso del Arte, Sep 09 2011]

With this constant f(1) and using the formula f(n+1) = (floor(f(n))*(f(n))) - ((floor(f(n)))^2 - floor(f(n))) it is possible to obtain the prime numbers repeated exactly a number of times corresponding to the position of the prime number. That is, 2 once, 3 twice, 5 thrice, etc.

cp's OEIS Frontend

A002024 k appears k times; a(n) = floor(sqrt(2n) + 1/2).

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A007917 Version 1 of the "previous prime" function: largest prime <= n.

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A031368 Odd-indexed primes: a(n) = prime(2n-1).

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A033286 a(n) = n * prime(n).

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A175944 1 appears once, n-th prime p appears p times.

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