A000792 -id:A000792 - cp's OEIS frontend

A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 0, 3, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1

Offset: 1

Abhimanyu Kumar, Nov 23 2020

The product of the parts of a partition is called its norm.

			For n=6 the partitions and their counts for each norm are given in the table below.
  Relevant partition(s)  | Norm | Count
  1+1+1+1+1+1+1          | 1    | 1
  2+1+1+1+1              | 2    | 1
  3+1+1+1                | 3    | 1
  4+1+1, 2+2+1+1         | 4    | 2
  5+1                    | 5    | 1
  6, 3+2+1               | 6    | 2
  4+2, 2+2+2             | 8    | 2
  3+3                    | 9    | 1
The number of partitions of 6 with norm value 4 are 2, expressed as T(6,4)=2. Similarly, T(6,7)=0 because there is no partition of 6 with norm 7.
So the 6th row is 1, 1, 1, 2, 1, 2, 0, 2, 1.
First few rows of the array are:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 2;
  1, 1, 1, 2, 1, 1;
  1, 1, 1, 2, 1, 2, 0, 2, 1;
  1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2;
  1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1;
  ...

Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).

Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.

Cf. A000041 (row sums), A000792 (row lengths), A001055, A118851, A212721.

row(n) = {my(list = List()); forpart(p=n, listput(list, vecprod(Vec(p)));); my(vlist = Vec(list)); my(v = vector(vecmax(vlist))); for (i=1, #vlist, v[vlist[i]]++); v;} \\ Michel Marcus, Nov 26 2020

Let the number of partitions of n having the norm value k refer to the norm counting function T(n,k). The following properties hold true:
Max_{n=1..oo} T(n,k) = A001055(k).
Sum_{k=1..A000792(n)} T(n,k) = A000041(n).
Sum_{n>=1} T(n+1,k) - T(n,k) = A001055(k) - 1.
G.f.: Sum_{n>=0} Sum_{k>=1} ((q^n)/(k^s))*T(n,k) = Product_{m>=1}(1-((q^m)/(m^s)))^(-1).
n*Sum_{k=1..A000792(n)} T(n,k) = Sum_{m=1..n} (Sum_{k=1..A000792(n-m)} T(n-m,i)*k^(-s))*(Sum_{d|m} (d/m)^(d*s-1))

More terms from Michel Marcus, Nov 26 2020

A062723 Least common multiple (LCM) of the first n+1 terms of A000792.

Original entry on oeis.org

1, 1, 2, 6, 12, 12, 36, 36, 36, 108, 108, 108, 324, 324, 324, 972, 972, 972, 2916, 2916, 2916, 8748, 8748, 8748, 26244, 26244, 26244, 78732, 78732, 78732, 236196, 236196, 236196, 708588, 708588, 708588, 2125764, 2125764, 2125764, 6377292, 6377292

Offset: 0

Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 15 2001

Apparently this sequence (when taken without repeats) is a subsequence of A000792, cf. A202337.

			a(4)=12 because a(4) is the LCM of 1,1,2,3,4 - which is clearly 12.

Harry J. Smith, Table of n, a(n) for n = 0..400
Index entries for sequences related to lcm's
Index entries for linear recurrences with constant coefficients, signature (0, 0, 3).

Cf. A000244, A000792.

a062723 n = a062723_list !! n
a062723_list = scanl1 lcm a000792_list
-- Reinhard Zumkeller, Dec 17 2011

Module[{nn=50,trms},trms=CoefficientList[Series[(1+x+2x^2+x^4)/(1-3x^3),{x,0,nn}],x];Table[LCM@@Take[trms,n],{n,nn}]] (* or *) LinearRecurrence[{0,0,3},{1,1,2,6,12,12,36},50] (* Harvey P. Dale, Oct 04 2024 *)

PARI
```
a(n)=if(n<0,0,if(n<4,n!,4*3^(n\3)))
```

a(n) = 4*3^floor(n/3), n >= 3. - Vladeta Jovovic, Jul 18 2001

G.f.: (1+x+2*x^2+3*x^3+9*x^4+6*x^5+18*x^6)/(1-3*x^3).

Formula and correction from Vladeta Jovovic, Jul 18 2001

More terms from Jason Earls, Jul 21 2001

A349983 a(n) is the largest k such A000792(k) <= n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 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, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Harry Altman, Jan 08 2022

Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217.

Harry Altman, Integer Complexity: The Integer Defect, arXiv:1804.07446 [math.NT], 2018.

Cf. A005245, A000792, A007600.

a(n) = max{ 3*floor(log_3(n)), 3*floor(log_3(n/2))+2, 3*floor(log_3(n/4))+4, 1 }.

a(n) = A007600(n)-1 except when n appears in A000792, in which case a(n) = A007600(n).

A087902 Least m such that A080256(m)=n and has a maximum number A000792(n) of divisors.

Original entry on oeis.org

2, 4, 6, 12, 36, 60, 180, 900, 1260, 6300, 44100, 69300, 485100, 5336100, 6306300, 69369300, 901800900, 1179278100, 15330615300, 260620460100, 291281690700, 4951788741900, 94083986096100, 113891141063700, 2163931680210300

Offset: 2

Lekraj Beedassy, Oct 14 2003

nonn

Omega(m)=A004523(n+1).

			a(6): The numbers m with A080256(m) = 6 are 24, 30, 32, 36, 40..., with number of divisors 8, 8, 6, 9, 8.... None of these have more than A000792(6) = 9 divisors and 36 is the first that has 9, so a(6) = 36.

a(n) = local(x); x = (n + 2)\3; prod(i = 1, x, prime(i)^2)/if (n%3 == 1, prime(x)*prime(x - 1), if (n%3, prime(x), 1)); \\ David Wasserman, Jun 17 2005

More terms from David Wasserman, Jun 17 2005

A001414 Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 12, 18, 16, 67, 21, 26, 14, 71, 12, 73, 39, 13, 23, 18, 18

Offset: 1

N. J. A. Sloane

MacMahon calls this the potency of n.
Downgrades the operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
Consider all ways of writing n as a product of zero, one, or more factors; sequence gives smallest sum of terms. - Amarnath Murthy, Jul 07 2001
a(n) <= n for all n, and a(n) = n iff n is 4 or a prime.
Look at the graph of this sequence. At the lower edge of the logarithmic scatterplot there is a set of fuzzy but unmistakable diagonal fringes, sloping toward the southeast. Their spacing gradually increases, and their slopes gradually decrease; they are more distinct toward the lower edge of the range. Is any explanation known? - Allan C. Wechsler, Oct 11 2015
For n >= 2, the glb and lub are: 3 * log(n) / log(3) <= a(n) <= n, where the lub occurs when n = 3^k, k >= 1. (Jakimczuk 2012) - Daniel Forgues, Oct 12 2015
Except for the initial term, row sums of A027746. - M. F. Hasler, Feb 08 2016
Atanassov proves that a(n) <= A065387(n) - n. - Charles R Greathouse IV, Dec 06 2016
From Robert G. Wilson v, Aug 15 2022: (Start)
Differs from A337310 beginning with n at 64, 192, 256, 320, 448, 512, ..., .
The number of terms which equal k is A000607(k).
The first occurrence of k>1 is A056240(k).
The last occurrence of k>1 is A000792(k).
The Amarnath Murthy comment of Jul 07 2001 is a result of the fundamental theorem of arithmetic.
(End)

			a(24) = 2+2+2+3 = 9.
a(30) = 10: 30 can be written as 30, 15*2, 10*3, 6*5, 5*3*2. The corresponding sums are 30, 17, 13, 11, 10. Among these 10 is the least.

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
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).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Kevin S. Brown, The Sum of the Prime Factors of N.
Es-said En-naoui, Study of the generalized Von Mangoldt function defined by L-additive function, arXiv:2301.09677 [math.GM], 2023.
Hans Havermann, Log plot of 100000 terms
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Rafael Jakimczuk, Sum of Prime Factors in the Prime Factorization of an Integer, International Mathematical Forum, Vol. 7, No. 53 (2012), pp. 2617-2621.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
Eric Weisstein's World of Mathematics, Sum of Prime Factors.
Wikipedia, Table of prime factors.
Steve Witham, Linear-log plot (The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).)
Steve Witham, Log-log plot (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)

Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343, A120007, A036288.
A000607(n) gives the number of values of k for which A001414(k) = n.
Cf. A036349 (indices of even terms), A356163 (their char. function), A335657 (indices of odd terms), A289142 (of multiples of 3), A373371 (their char. function).
For sum of squares of prime factors see A067666, for cubes see A224787.
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A056239 (with a(p)=primepi(p)), A059975 (with a(p)=p-1), A064097 (with a(p)=1+a(p-1)), A113177 (with a(p)=Fib(p)), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
For other completely additive sequences see the cross-references in A104244.

a001414 1 = 0
a001414 n = sum $ a027746_row n
-- Reinhard Zumkeller, Feb 27 2012, Nov 20 2011

[n eq 1 select 0 else (&+[j[1]*j[2]: j in Factorization(n)]): n in [1..100]]; // G. C. Greubel, Jan 10 2019

A001414 := proc(n) add( op(1,i)*op(2,i),i=ifactors(n)[2]) ; end proc:
seq(A001414(n), n=1..100); # Peter Luschny, Jan 17 2011

a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* Ray Chandler, Nov 12 2005 *)

a(n)=local(f); if(n<1,0,f=factor(n); sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))

A001414(n) = (n=factor(n))[,1]~*n[,2] \\ M. F. Hasler, Feb 07 2009

from sympy import factorint
def A001414(n):
    return sum(p*e for p,e in factorint(n).items()) # Chai Wah Wu, Jan 08 2016

[sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0,len(factor(n)))) for n in range(1,79)] # Giuseppe Coppoletta, Jan 19 2015

If n = Product p_j^k_j then a(n) = Sum p_j * k_j.
Dirichlet g.f. f(s)*zeta(s), where f(s) = Sum_{p prime} p/(p^s-1) = Sum_{k>0} primezeta(k*s-1) is the Dirichlet g.f. for A120007. Totally additive with a(p^e) = p*e. - Franklin T. Adams-Watters, Jun 02 2006
For n > 1: a(n) = Sum_{k=1..A001222(n)} A027746(n,k). - Reinhard Zumkeller, Aug 27 2011
Sum_{n>=1} (-1)^a(n)/n^s = ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s), if Re(s)>1 and 0 if s=1 (Alladi and Erdős, 1977). - Amiram Eldar, Nov 02 2020
a(n) >= k*log(n), where k = 3/log(3). This bound is sharp. - Charles R Greathouse IV, Jul 28 2025

A001113 Decimal expansion of e.

Original entry on oeis.org

2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0, 4, 5, 2, 3, 5, 3, 6, 0, 2, 8, 7, 4, 7, 1, 3, 5, 2, 6, 6, 2, 4, 9, 7, 7, 5, 7, 2, 4, 7, 0, 9, 3, 6, 9, 9, 9, 5, 9, 5, 7, 4, 9, 6, 6, 9, 6, 7, 6, 2, 7, 7, 2, 4, 0, 7, 6, 6, 3, 0, 3, 5, 3, 5, 4, 7, 5, 9, 4, 5, 7, 1, 3, 8, 2, 1, 7, 8, 5, 2, 5, 1, 6, 6, 4, 2, 7, 4, 2, 7, 4, 6

Offset: 1

N. J. A. Sloane

e is sometimes called Euler's number or Napier's constant.
Also, decimal expansion of sinh(1)+cosh(1). - Mohammad K. Azarian, Aug 15 2006
If m and n are any integers with n > 1, then |e - m/n| > 1/(S(n)+1)!, where S(n) = A002034(n) is the smallest number such that n divides S(n)!. - Jonathan Sondow, Sep 04 2006
Limit_{n->infinity} A000166(n)*e - A000142(n) = 0. - Seiichi Kirikami, Oct 12 2011
Euler's constant (also known as Euler-Mascheroni constant) is gamma = 0.57721... and Euler's number is e = 2.71828... . - Mohammad K. Azarian, Dec 29 2011
One of the many continued fraction expressions for e is 2+2/(2+3/(3+4/(4+5/(5+6/(6+ ... from Ramanujan (1887-1920). - Robert G. Wilson v, Jul 16 2012
e maximizes the value of x^(c/x) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. This explains why elements of A000792 are composed primarily of factors of 3, and where needed, some factors of 2. These are the two primes closest to e. - Richard R. Forberg, Oct 19 2014
There are two real solutions x to c^x = x^c when c, x > 0 and c != e, one of which is x = c, and only one real solution when c = e, where the solution is x = e. - Richard R. Forberg, Oct 22 2014
This is the expected value of the number of real numbers that are independently and uniformly chosen at random from the interval (0, 1) until their sum exceeds 1 (Bush, 1961). - Amiram Eldar, Jul 21 2020

			2.71828182845904523536028747135266249775724709369995957496696762772407663...

Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 400.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 250-256.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers, p. 85.
E. Maor, e: The Story of a Number, Princeton Univ. Press, 1994.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 52.
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places. Math. Tables and Other Aids to Computation 4, (1950). 11-15.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 1 and 2, equations 1:7:4, 2:5:4 at pages 13, 20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

N. J. A. Sloane, Table of 50000 digits of e labeled from 1 to 50000 [based on the ICON Project link below]
Mohammad K. Azarian, An Expansion of e, Problem # B-765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug. 1995, p. 377.
Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.
L. E. Bush, The William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 18-33, problem 3.
Ed Copeland and Brady Haran, A proof that e is irrational, Numberphile video (2021).
Dave's Math Tables, e
X. Gourdon, Plouffe's Inverter, e to 1.250 billion digits
X. Gourdon and P. Sebah, The constant e and its computation
Brady Haran and James Grime, Incredible Formula, Numberphile YouTube video, 2016.
ICON Project, e to 50000 places
Roger Mansuy, Un intrigant poème... mathématique, Images des Mathématiques, CNRS, 2023. In French.
R. Nemiroff and J. Bonnell, The first 5 million digits of the number e
Remco Niemeijer, Digits Of E, programmingpraxis
J. J. O'Connor & E. F. Robertson, The number e
Michael Penn, e is irrational, YouTube video, 2020.
Simon Plouffe, A million digits
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places, Pi, A Source book, pp 277-281, 2000.
E. Sandifer, How Euler Did It, Who proved e is irrational?, MAA Online (2006)
D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679-680.
Jean-Louis Sigrist, Le premier million de décimales de e.
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 339.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641 (article) and 114 (2007), 659 (addendum).
Jonathan Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
G. Villemin's Almanach of Numbers, Constant "e"
Eric Weisstein's World of Mathematics, e
Eric Weisstein's World of Mathematics, e Digits
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Uniform Sum Distribution
Eric Weisstein's World of Mathematics, e Approximations
Wikipedia, E (mathematical constant)
Index entries for "core" sequences
Index entries for transcendental numbers

Cf. A002034, A003417 (continued fraction), A073229, A122214, A122215, A122216, A122217, A122416, A122417.
Expansion of e in base b: A004593 (b=2), A004594 (b=3), A004595 (b=4), A004596 (b=5), A004597 (b=6), A004598 (b=7), A004599 (b=8), A004600 (b=9), this sequence (b=10), A170873 (b=16). - Jason Kimberley, Dec 05 2012
Powers e^k: A092578 (k = -7), A092577 (k = -6), A092560 (k = -5), A092553 - A092555 (k = -2 to -4), A068985 (k = -1), A072334 (k = 2), A091933 (k = 3), A092426 (k = 4), A092511 - A092513 (k = 5 to 7).

-- See Niemeijer link.
a001113 n = a001113_list !! (n-1)
a001113_list = eStream (1, 0, 1)
   [(n, a * d, d) | (n, d, a) <- map (\k -> (1, k, 1)) [1..]] where
   eStream z xs'@(x:xs)
     | lb /= approx z 2 = eStream (mult z x) xs
     | otherwise = lb : eStream (mult (10, -10 * lb, 1) z) xs'
     where lb = approx z 1
           approx (a, b, c) n = div (a * n + b) c
           mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)
-- Reinhard Zumkeller, Jun 12 2013

Digits := 200: it := evalf((exp(1))/10, 200): for i from 1 to 200 do printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od: # James Sellers, Feb 13 2001

RealDigits[E, 10, 120][[1]] (* Harvey P. Dale, Nov 14 2011 *)

e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x).
e is the unique positive root of the equation Integral_{u = 1..x} du/u = 1.
exp(1) = ((16/31)*(1 + Sum_{n>=1} ((1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!)))^2. Robert Israel confirmed that the above formula is correct, saying: "In fact, Sum_{n=0..oo} n^j*t^n/n! = P_j(t)*exp(t) where P_0(t) = 1 and for j >= 1, P_j(t) = t (P_(j-1)'(t) + P_(j-1)(t)). Your sum is 1/2*P_3(1/2) + 1/2*P_1(1/2) + P_0(1/2)." - Alexander R. Povolotsky, Jan 04 2009
exp(1) = (1 + Sum_{n>=1} ((1+n+n^3)/n!))/7. - Alexander R. Povolotsky, Sep 14 2011
e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2. - Rok Cestnik, Jan 19 2017
From Peter Bala, Nov 13 2019: (Start)
The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,..., where R(n,x) is the n-th row polynomial of A269953.
e = 2 + Sum_{n >= 0} (-1)^n*(n+2)!/(d(n+2)*d(n+3)), where d(n) = A000166(n).
e = Sum_{n >= 0} (x^2 + (n+2)*x + n)/(n!(n + x)*(n + 1 + x)), provided x is not zero or a negative integer. (End)
Equals lim_{n -> oo} (2*3*5*...*prime(n))^(1/prime(n)). - Peter Luschny, May 21 2020
e = 3 - Sum_{n >= 0} 1/((n+1)^2*(n+2)^2*n!). - Peter Bala, Jan 13 2022
e = lim_{n->oo} prime(n)*(1 - 1/n)^prime(n). - Thomas Ordowski, Jan 31 2023
e = 1+(1/1)*(1+(1/2)*(1+(1/3)*(1+(1/4)*(1+(1/5)*(1+(1/6)*(...)))))), equivalent to the first formula. - David Ulgenes, Dec 01 2023
From Michal Paulovic, Dec 12 2023: (Start)
Equals lim_{n->oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)
Equals Product_{k>=1} (1 + 1/k) * (1 - 1/(k + 1)^2)^k. - Antonio Graciá Llorente, May 14 2024
Equals lim_{n->oo} Product_{k=1..n} (n^2 + k)/(n^2 - k) (see Finch). - Stefano Spezia, Oct 19 2024
e ~ (1 + 9^((-4)^(7*6)))^(3^(2^85)), correct to more than 18*10^24 digits (Richard Sabey, 2004); see Haran and Grime link. - Paolo Xausa, Dec 21 2024.

A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 60, 84, 105, 140, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060, 60060, 120120

Offset: 0

N. J. A. Sloane

Also the largest orbit size (cycle length) for the permutation A057511 acting on Catalan objects (e.g., planar rooted trees, parenthesizations). - Antti Karttunen, Sep 07 2000
Grantham mentions that he computed a(n) for n <= 500000.
An easy lower bound is a(n) >= A002110(max{ m | A007504(m) <= n}), with strict inequality if n is not in A007504 (sum of the first m primes). Indeed, if A007504(m) <= n, the partition of n into the first m primes and maybe one additional term will have an LCM greater than or equal to primorial(m). If n > A007504(m) then a(n) >= (3/2)*A002110(m) by replacing the initial 2 by 3. But even for n = A007504(m), one has a(n) > A002110(m) for m > 8, since replacing 2+23 in 2+3+5+7+11+13+17+19+23 by 16+9, one has an LCM of 8*3*primorial(8) > primorial(9) because 24 > 23. - M. F. Hasler, Mar 29 2015
Maximum degree of the splitting field of a polynomial of degree n over a finite field, since over a finite field the degree of the splitting field is the least common multiple of the degrees of the irreducible polynomial factors of the polynomial. - Charles R Greathouse IV, Apr 27 2015
Maximum order of the elements in the symmetric group S_n. - Jianing Song, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 6*x^6 + 12*x^7 + 15*x^8 + ...
From _Joerg Arndt_, Feb 15 2013: (Start)
The 15 partitions of 7 are the following:
[ #]  [ partition ]   lcm( parts )
[ 1]  [ 1 1 1 1 1 1 1 ]   1
[ 2]  [ 1 1 1 1 1 2 ]   2
[ 3]  [ 1 1 1 1 3 ]   3
[ 4]  [ 1 1 1 2 2 ]   2
[ 5]  [ 1 1 1 4 ]   4
[ 6]  [ 1 1 2 3 ]   6
[ 7]  [ 1 1 5 ]   5
[ 8]  [ 1 2 2 2 ]   2
[ 9]  [ 1 2 4 ]   4
[10]  [ 1 3 3 ]   3
[11]  [ 1 6 ]   6
[12]  [ 2 2 3 ]   6
[13]  [ 2 5 ]  10
[14]  [ 3 4 ]  12  (max)
[15]  [ 7 ]   7
The maximum (LCM) value attained is 12, so a(7) = 12.
(End)

J. Haack, "The Mathematics of Steve Reich's Clapping Music," in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 223.
J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.
S. M. Shah, An inequality for the arithmetical function g(x), J. Indian Math. Soc., 3 (1939), 316-318. [See below for a scan of the first page.]
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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..814 from David Wasserman)
Giedrius Alkauskas, Colouring monohedral tilings: defects and grain boundaries, 2024.
Joerg Arndt, Table of n, a(n) for n = 0..65536 (xz compressed).
Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, arXiv:0803.2160 [math.NT], 2008.
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
Marc Deléglise and Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).
John D. Dixon and Daniel Panario, The Degree of the Splitting Field of a Random Polynomial over a Finite Field, The Electronic Journal of Combinatorics 11:1 (2004).
FindStat - Combinatorial Statistic Finder, The order of a permutation
Jon Grantham, The largest prime dividing the maximal order of an element of S_n, Math. Comput. 64, No. 209, 407-410 (1995).
Joel K. Haack, The Mathematics of Steve Reich's Clapping Music,in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Jan Elmar Krauskopf, Andreas Rauh, and Andreas Hein, Discrete simulation of maypole braiding machines to create collision-free braiding programmes, Heliyon (2025), Art. No. e42917.
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, (French) [Explicit upper bound on the maximum order of an element of the symmetric group] Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, 269--281 (1985). MR0799599 (87a:11093).
Jean-Pierre Massias, Jean-Louis Nicolas, and Guy Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53 (1989), no. 188, 665--678. MR0979940 (90e:11139).
W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
Jean-Louis Nicolas, Sur l'ordre maximum d'un élément dans le groupe Sn des permutations, Acta Arith. 14, 315-332 (1968).
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers', Bull. Soc. Math. France 97 (1969), 129-191.
Jean-Louis Nicolas, Calcul de l'ordre maximum d'un élément du groupe symétrique S_n Revue française d'informatique et de recherche opérationnelle, série rouge 3.2 (1969): 43-50.
Jean-Louis Nicolas, Ordre maximal d'un e'le'ment du'un groupe de permutations, C. R. Acad. Sci. Paris, A, 270 (1970), 1-4. [Annotated scanned copy]
Roger D. Nussbaum, Lunel Verduyn, and M. Sjoerd, Asymptotic estimates for the periods of periodic points of non-expansive maps, Ergodic Theory Dynam. Systems 23 (2003), no. 4, pp. 1199-1226. MR1997973 (2004m:37033).
S. M. Shah, An inequality for the arithmetical function g(x) (scan of first page).
A. Wechsler, Re: Question (A000793(A007504(n)) =? A002110(n)), SeqFan mailing list, Mar 29 2015.
Eric Weisstein's World of Mathematics, Landau's Function
Herbert S. Wilf, The asymptotics of e^P(z) and the number of elements of each order in S_n, Bull. Amer. Math. Soc., 15.2 (1986), 225-232.
Index entries for sequences related to lcm's
Index entries for "core" sequences

Cf. A000792, A009490, A034891, A057731, A074859, A128305, A129759, A225655, A225648-A225650, A225651, A225636, A225558.

Row 1 of A225630.

a000793 = maximum . map (foldl lcm 1) . partitions where
   partitions n = ps 1 n where
      ps x 0 = [[]]
      ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Mar 29 2015

A000793 := proc(n)
    l := 1:
    p := combinat[partition](n):
    for i from 1 to combinat[numbpart](n) do
        if ilcm( p[i][j] $ j=1..nops(p[i])) > l then
            l := ilcm( p[i][j] $ j=1..nops(p[i]))
        end if:
    end do:
    l ;
end proc:
seq(A000793(n),n=0..30) ; # James Sellers, Dec 07 2000
seq( max( op( map( x->ilcm(op(x)), combinat[partition](n)))), n=0..30); # David Radcliffe, Feb 28 2006
# third Maple program:
b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013

f[n_] := Max@ Apply[LCM, IntegerPartitions@ n, 1]; Array[f, 47] (* Robert G. Wilson v, Oct 23 2011 *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]]; a[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 07 2014, after Alois P. Heinz *)

{a(n) = my(m, t, j, u); if( n<2, n>=0, m = ceil(n / exp(1)); t = ceil( (n/m)^m ); j=1; for( i=2, t, u = factor(i); u = sum( k=1, matsize(u)[1], u[k,1]^u[k,2]); if( u<=n, j=i)); j)}; /* Michael Somos, Oct 20 2004 */

c=0;A793=apply(t->eval(concat(Vec(t)[#Str(c++) .. -1])),select(t->#t,readstr("/tmp/b000793.txt")));A000793(n)=A793[n+1] \\ Assumes the b-file in the /tmp (or C:\tmp) folder. - M. F. Hasler, Mar 29 2015

A008475(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]^f[i,2]);
a(n)=
{
  if(n<2, return(1));
  forstep(i=ceil(exp(1.05315*sqrt(log(n)*n))), 2, -1,
    if(A008475(i)<=n, return(i))
  );
  1;
} \\ Charles R Greathouse IV, Apr 28 2015

{ \\ translated from code given by Tomas Rokicki
  my( N = 100 );
  my( V = vector(N,j,1) );
   forprime (i=2, N,  \\ primes i
      forstep (j=N, i,  -1,
         my( hi = V[j] );
         my( pp = i );  \\ powers of prime i
         while ( pp<=j,  \\ V[] is 1-based
             hi = max(if(j==pp, pp, V[j-pp]*pp), hi);
             pp *= i;
         );
         V[j] = hi;
      );
   );
   print( V );  \\ all values
\\   print( V[N] );  \\ just a(N)
\\  print("0 1");  for (n=1, N, print(n, " ", V[n]) );  \\ b-file
} \\ Joerg Arndt, Nov 14 2016

{a(n) = my(m=1); if( n<0, 0, forpart(v=n, m = max(m, lcm(Vec(v)))); m)}; /* Michael Somos, Sep 04 2017 */

from sympy import primerange
def aupton(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    for i in primerange(2, N+1):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max((pp if j==pp else V[j-pp]*pp), hi)
                pp *= i
            V[j] = hi
    return V
print(aupton(47)) # Michael S. Branicky, Oct 09 2022 after Joerg Arndt

from sympy import primerange,sqrt,log,Rational
def f(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    if N < 4:
        C = 2
    else:
        C = Rational(166,125)
    for i in primerange(C*sqrt(N*log(N))):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max(V[j-pp]*pp, hi)
                pp *= i
            V[j] = hi
    return V
# Philip Turecek, Mar 31 2023

def a(n):
  return max([lcm(l) for l in Partitions(n)])
# Philip Turecek, Mar 28 2023

;; A naive algorithm searching through all partitions of n:
(define (A000793 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
(define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))
;; From Antti Karttunen, May 17 2013.

Landau: lim_{n->oo} (log a(n)) / sqrt(n log n) = 1.
For bounds, see the Shah and Massias references.
For n >= 2, a(n) = max_{k} A008475(k) <= n. - Joerg Arndt, Nov 13 2016

More terms from David W. Wilson

Removed erroneous comment about a(16) which probably originated from misreading a(15)=105 as a(16) because of offset=0: a(16) = 4*5*7 = 140 is correct as it stands. - M. F. Hasler, Feb 02 2009

A005245 The (Mahler-Popken) complexity of n: minimal number of 1's required to build n using + and *.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 10, 11, 9, 10, 10, 9, 10, 11, 10, 11, 10, 11, 11, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 11, 12, 13, 11, 12, 12, 12, 12, 13, 11, 12, 12, 12, 13, 14, 12, 13, 13, 12, 12, 13, 13, 14, 13, 14, 13, 14, 12, 13, 13, 13, 13, 14, 13, 14

Offset: 1

N. J. A. Sloane

The complexity of an integer n is the least number of 1's needed to represent it using only additions, multiplications and parentheses. This does not allow juxtaposition of 1's to form larger integers, so for example, 2 = 1+1 has complexity 2, but 11 does not ("pasting together" two 1's is not an allowed operation).
The complexity of a number has been defined in several different ways by different authors. See the Index to the OEIS for other definitions.
Guy asks if a(p) = a(p-1) + 1 for prime p. Martin Fuller found the least counterexample p = 353942783 in 2008, see Fuller link. - Charles R Greathouse IV, Oct 04 2012
It appears that this sequence is lower than A348262 {1,+,^} only a finite number of times. - Gordon Hamilton and Brad Ballinger, May 23 2022
The second Altman links proves that {a(n) - 3*log_3(n)} is a well-ordered subset of the reals whose intersection with [0,k) has order type omega^k for each positive integer k, so this set itself has order type omega^omega. - Jianing Song, Apr 13 2024

			From _Lekraj Beedassy_, Jul 04 2009: (Start)
   n.........minimal expression........ a(n) = number of 1's
   1..................1...................1
   2.................1+1..................2
   3................1+1+1.................3
   4.............(1+1)*(1+1)..............4
   5............(1+1)*(1+1)+1.............5
   6............(1+1)*(1+1+1).............5
   7...........(1+1)*(1+1+1)+1............6
   8..........(1+1)*(1+1)*(1+1)...........6
   9...........(1+1+1)*(1+1+1)............6
  10..........(1+1+1)*(1+1+1)+1...........7
  11.........(1+1+1)*(1+1+1)+1+1..........8
  12.........(1+1)*(1+1)*(1+1+1)..........7
  13........(1+1)*(1+1)*(1+1+1)+1.........8
  14.......{(1+1)*(1+1+1)+1}*(1+1)........8
  15.......{(1+1)*(1+1)+1}*(1+1+1)........8
  16.......(1+1)*(1+1)*(1+1)*(1+1)........8
  17......(1+1)*(1+1)*(1+1)*(1+1)+1.......9
  18........(1+1)*(1+1+1)*(1+1+1).........8
  19.......(1+1)*(1+1+1)*(1+1+1)+1........9
  20......{(1+1+1)*(1+1+1)+1}*(1+1).......9
  21......{(1+1)*(1+1+1)+1}*(1+1+1).......9
  22.....{(1+1)*(1+1+1)+1}*(1+1+1)+1.....10
  23....{(1+1)*(1+1+1)+1}*(1+1+1)+1+1....11
  24......(1+1)*(1+1)*(1+1)*(1+1+1).......9
  25.....(1+1)*(1+1)*(1+1)*(1+1+1)+1.....10
  26....{(1+1)*(1+1)*(1+1+1)+1}*(1+1)....10
  27.......(1+1+1)*(1+1+1)*(1+1+1)........9
  28......(1+1+1)*(1+1+1)*(1+1+1)+1......10
  29.....(1+1+1)*(1+1+1)*(1+1+1)+1+1.....11
  30.....{(1+1+1)*(1+1+1)+1}*(1+1+1).....10
  31....{(1+1+1)*(1+1+1)+1}*(1+1+1)+1....11
  32....(1+1)*(1+1)*(1+1)*(1+1)*(1+1)....10
  33...(1+1)*(1+1)*(1+1)*(1+1)*(1+1)+1...11
  34..{(1+1)*(1+1)*(1+1)*(1+1)+1}*(1+1)..11
  .........................................
(End)

M. Criton, "Les uns de Germain", Problem No. 4, pp. 13 ; 68 in '7 x 7 Enigmes Et Défis Mathématiques pour tous', vol. 25, Editions POLE, Paris 2001.
R. K. Guy, Unsolved Problems in Number Theory, Sect. F26.
K. Mahler and J. Popken, Over een Maximumprobleem uit de Rekenkunde (Dutch: On a maximum problem in arithmetic), Nieuw Arch. Wiskunde, (3) 1 (1953), pp. 1-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler, Table of n, a(n) for n = 1..10000
Harry Altman, Highest few sums and products of ones, Preprint, undated but before 2013.
Harry Altman, Integer Complexity and Well-Ordering, arXiv:1310.2894v1 [math.NT] , Oct 10, 2013.
Harry Altman and Joshua Zelinsky, Numbers with integer complexity close to the lower bound, arXiv:1207.4841 [math.NT], 2012.
J. Arias de Reyna, Complejidad de los números naturales, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250.
J. Arias de Reyna, Complejidad de los números naturales, Gaceta R. Soc. Mat. Esp., 3 (2000), 230-250. [Cached copy, with permission]
J. Arias de Reyna, Complexity of natural numbers, arXiv:2111.03345 [math.NT], 2021.
J. Arias de Reyna and J. van de Lune, The question "How many 1's are needed?" revisited, arXiv preprint arXiv:1404.1850 [math.NT], 2014.
J. Arias de Reyna and J. van de Lune, Algorithms for determining integer complexity, arXiv preprint arXiv:1404.2183 [math.NT], 2014.
Juris Cernenoks, Janis Iraids, Martins Opmanis, Rihards Opmanis and Karlis Podnieks, Integer Complexity: Experimental and Analytical Results II, arxiv:1409.0446 [math.NT], 2014.
Katherine Cordwell, Alyssa Epstein, Anand Hemmady, Steven J. Miller, Eyvindur A. Palsson, Aaditya Sharma, Stefan Steinerberger and Yen Nhi Truong Vu, On algorithms to calculate integer complexity, arXiv:1706.08424 [math.NT], 2017.
Martin N. Fuller, C program
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Qizheng He, Improved Algorithms for Integer Complexity, arXiv:2308.10301 [cs.DS], 2023.
Janis Iraids, Online calculator of a(n) for n < 10^12
Jānis Iraids, Kaspars Balodis, Juris Čerņenoks, Mārtiņš Opmanis, Rihards Opmanis, and Kārlis Podnieks, Integer complexity: experimental and analytical results, arXiv:1203.6462 [math.NT], 2012.
Daniel A. Rawsthorne, How many 1's are needed?, Fibonacci Quarterly 27 (1989), pp. 14-17.
Srinivas, Vivek V. and Shankar, B. R., Integer Complexity: Breaking the Theta(n^2) barrier, World Academy of Science 41 (2008), pp. 690-691.
Stefan Steinerberger, A Short Note on Integer Complexity, Contributions to Discrete Mathematics, Vol. 9.1 (2014).
Stefan Steinerberger, A short note on integer complexity, Contributions to Discrete Mathematics, Vol. 9.1 (2014). [Cached copy, with permission.]
Joshua Zelinsky, Upper Bounds on Integer Complexity, arXiv preprint (2022). arXiv:2211.02995 [math.NT]
Venecia Wang, A counterexample to the prime conjecture of expressing numbers using just ones, YouTube video, 2012.
Venecia Wang, A counterexample to the prime conjecture of expressing numbers using just ones, Journal of Number Theory, Volume 133, Issue 2, February 2013, Pages 391-397.
Eric Weisstein's World of Mathematics, Integer Complexity
Dimitri Zucker, The Most Underrated Concept in Number Theory, Combo Class Youtube video that mentions this sequence.
Index to sequences related to the complexity of n

Cf. A025280 (variant using +, *, and ^), A091333 (using +, -, and *), A091334 (using +, -, *, and ^), A348089 (using +, -, *, / and ^), A348262 (using + and ^).

Cf. A000792 (largest integer of given complexity), A003313, A076142, A076091, A061373, A005421, A064097, A005520, A003037, A161906, A161908, A244743.

import Data.List (genericIndex)
a005245 n = a005245_list `genericIndex` (n-1)
a005245_list = 1 : f 2 [1] where
   f x ys = y : f (x + 1) (y : ys) where
     y = minimum $
         (zipWith (+) (take (x `div` 2) ys) (reverse ys)) ++
         (zipWith (+) (map a005245 $ tail $ a161906_row x)
                      (map a005245 $ reverse $ init $ a161908_row x))
-- Reinhard Zumkeller, Mar 08 2013

with(numtheory):
a:= proc(n) option remember;
      `if`(n=1, 1, min(seq(a(i)+a(n-i), i=1..n/2),
       seq(a(d)+a(n/d), d=divisors(n) minus {1, n})))
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 18 2012

a[n_] := a[n] = If[n == 1, 1,
   Min[Table[a[i] + a[n-i], {i, 1, n/2}] ~Join~
   Table[a[d] + a[n/d], {d, Divisors[n] ~Complement~ {1, n}}]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)

A005245(n /* start by calling this with the largest needed n */, lim/* see below */) = { local(d); n<6 && return(n);
if(n<=#A005245, A005245[n]&return(A005245[n]) /* return memoized result if available */,
A005245=vector(n) /* create vector if needed - should better reuse existing data if available */);
for(i=1, n-1, A005245[i] || A005245[i]=A005245(i,lim)); /* compute all previous elements */
A005245[n]=min( vecmin(vector(min(n\2,if(lim>0,lim,n)), k, A005245[k]+A005245[n-k])) /* additive possibilities - if lim>0 is given, consider a(k)+a(n-k) only for k<=lim - we know it is save to use lim=1 up to n=2e7 */, if( isprime(n), n, vecmin(vector((-1+#d=divisors(n))\2, i, A005245[d[i+1]]+A005245[d[ #d-i]]))/* multiplicative possibilities */))}
\\ See also the Python program by Tim Peters at A005421.
\\ M. F. Hasler, Jan 30 2008

from functools import lru_cache
from itertools import takewhile
from sympy import divisors
@lru_cache(maxsize=None)
def A005245(n): return min(min(A005245(a)+A005245(n-a) for a in range(1,(n>>1)+1)),min((A005245(d)+A005245(n//d) for d in takewhile(lambda d:d*d<=n,divisors(n)) if d>1),default=n)) if n>1 else 1 # Chai Wah Wu, Apr 29 2023

It is known that a(n) <= A061373(n) but I think 0 <= A061373(n) - a(n) <= 1 also holds. - Benoit Cloitre, Nov 23 2003 [That's false: the numbers {46, 235, 649, 1081, 7849, 31669, 61993} require {1, 2, 3, 4, 5, 6, 7} fewer 1's in A005245 than in A061373. - Ed Pegg Jr, Apr 13 2004]
It is known from the work of Selfridge and Coppersmith that 3 log_3 n <= a(n) <= 3 log_2 n = 4.754... log_3 n for all n > 1. [Guy, Unsolved Problems in Number Theory, Sect. F26.] - Charles R Greathouse IV, Apr 19 2012 [Comment revised by N. J. A. Sloane, Jul 17 2016]
Zelinsky (2022) improves the upper bound to a(n) <= A*log n where A = 41/log(55296) = 3.754422.... This compares to the constant 2.7307176... of the lower bound. - Charles R Greathouse IV, Dec 11 2022
a(n) >= A007600(n) is a very accurate lower bound. - Mehmet Sarraç, Dec 18 2022

More terms from David W. Wilson, May 15 1997

A034891 Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 81, 98, 117, 140, 166, 196, 231, 271, 317, 369, 429, 496, 573, 660, 758, 869, 993, 1133, 1290, 1465, 1662, 1881, 2125, 2397, 2699, 3035, 3407, 3820, 4276, 4780, 5337, 5951, 6628, 7372, 8191, 9090

Offset: 0

Wouter Meeussen

a(n) = length of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012

Number of partitions of n into noncomposite parts. - Omar E. Pol, Jun 23 2022

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from T. D. Noe)
Index entries for sequences related to partitions

Cf. A000041, A000792, A000793, A009490, A008578, A140436, A353188.

a034891 = length . a212721_row  -- Reinhard Zumkeller, Jun 14 2012

b:= proc(n, i) option remember; `if`(n=0, 1, (p->
      `if`(i<0, 0, b(n, i-1)+ `if`(p>n, 0,
         b(n-p, i))))(`if`(i<1, 1, ithprime(i))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

Table[ Length[ Union[ Apply[ Times, Partitions[ n], 1]]], {n, 30}]
CoefficientList[ Series[ (1/(1 - x)) Product[1/(1 - x^Prime[i]), {i, 100}], {x, 0, 50}], x] (* Robert G. Wilson v, Aug 17 2013 *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<0, 0, b[n, i-1] + If[p>n, 0, b[n-p, i]]]]]; a[n_] := b[n, PrimePi[n] ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

[Partitions(n, parts_in=(prime_range(n+1)+[1])).cardinality() for n in xsrange(1000)] # Giuseppe Coppoletta, Jul 11 2016

G.f.: (1/(1-x))*(1/Product_{k>0} (1-x^prime(k))). a(n) = (1/n)*Sum_{k=1..n} A074372(k)*a(n-k). Partial sums of A000607. - Vladeta Jovovic, Sep 19 2002

a(n) = A000041(n) - A353188(n). - Omar E. Pol, Jun 23 2022

More terms from Vladeta Jovovic

a(0)=1 from Michael Somos, Feb 05 2011

A056240 Smallest number whose prime divisors (taken with multiplicity) add to n.

Original entry on oeis.org

2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13, 33, 26, 39, 17, 65, 19, 51, 38, 57, 23, 95, 46, 69, 92, 115, 29, 161, 31, 87, 62, 93, 124, 155, 37, 217, 74, 111, 41, 185, 43, 123, 86, 129, 47, 215, 94, 141, 188, 235, 53, 329, 106, 159, 212, 265, 59, 371, 61, 177, 122

Offset: 2

Adam Kertesz, Aug 19 2000

a(n) is the index of first occurrence of n in A001414.
From David James Sycamore and Michel Marcus, Jun 16 2017, Jun 28 2017: (Start)
Recursive calculation of a(n):
For prime p, a(p) = p.
For even composite n, let P_n denote the largest prime < n-1 such that n-P_n is prime (except if n = 6).
For odd composite n, let P_n denote the largest prime < n-1 such that n-3-P_n is prime.
Conjecture: a(n) = min { q*a(n-q); q prime, P_n <= q < n-1 }.
Examples:
For n = 9998, P_9998 = 9967 and a(9998) = min { 9973*a(25), 9967*a(31) } = 9967*31 = 308977.
For n = 875, P_875 = 859 and a(875) = min { 863*a(12), 859*a(16) } = 863*35 = 30205.
Note: A000040 and A288313 are both subsequences of this sequence. (End)

			a(8) = 15 = 3*5 because 15 is the smallest number whose prime divisors sum to 8.
a(10000) = 586519: Let pp(n) be the largest prime < n and the candidate being the current value that might be a(10000). Then we see that pp(10000 - 1) = 9973, giving a candidate 9973 * a(10000 - 9973) = 9973 * 92. pp(9973) = 9967, giving a candidate 9967 * a(10000 - 9967) = 9967 * 62. pp(9967) = 9949, giving the candidate 9949 * a(10000 - 9949) = 9962 * 188. This is larger than our candidate so we keep 9967 * 62 as our candidate. pp(9949) = 9941, giving a candidate 9941 * pp(10000 - 9941) = 9941 * 59. We see that (n - p) * a(p) >= (n - p) * p > candidate = 9941 * 59 for p > 59 so we stop iterating to conclude a(10000) = 9941 * 59 = 586519. - _David A. Corneth_, Mar 23 2018, edited by _M. F. Hasler_, Jan 19 2019

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Hans Havermann, Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).

Cf. A000040, A001414, A064502, A288313.
First column of array A064364, n>=2.
See A000792 for the maximal numbers whose prime factors sums up to n.

a056240 = (+ 1) . fromJust . (`elemIndex` a001414_list)
-- Reinhard Zumkeller, Jun 14 2012

A056240 := proc(n)
    local k ;
    for k from 1 do
        if A001414(k) = n then
            return k ;
        end if;
    end do:
end proc:
seq(A056240(n),n=2..80) ; # R. J. Mathar, Apr 15 2024

a = Table[0, {75}]; Do[b = Plus @@ Flatten[ Table[ #1, {#2}] & @@@ FactorInteger[n]]; If[b < 76 && a[[b]] == 0, a[[b]] = n], {n, 2, 1000}]; a (* Robert G. Wilson v, May 04 2002 *)
b[n_] := b[n] = Total[Times @@@ FactorInteger[n]];
a[n_] := For[k = 2, True, k++, If[b[k] == n, Return[k]]];
Table[a[n], {n, 2, 63}] (* Jean-François Alcover, Jul 03 2017 *)

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) == n;
a(n) = my(k=2); while(!isok(k, n), k++); k; \\ Michel Marcus, Jun 21 2017

a(n) = {if(n < 7, return(n + 2*(n==6))); my(p = precprime(n), res); if(p == n, return(p), p = precprime(n - 2); res = p * a(n - p); while(res > (n - p) * p && p > 2, p = precprime(p - 1); res = min(res, a(n - p) * p)); res)} \\ David A. Corneth, Mar 23 2018

A056240(n, p=n-1, m=oo)=if(n<6 || isprime(n), n, n==6, 8, until(p<3 || (n-p=precprime(p-1))*p >= m=min(m,A056240(n-p)*p),); m) \\ M. F. Hasler, Jan 19 2019

Trivial but essential: a(n) >= n. - David A. Corneth, Mar 23 2018

a(n) >= n with equality iff n = 4 or n is prime. - M. F. Hasler, Jan 19 2019

More terms from James Sellers, Aug 25 2000

cp's OEIS Frontend

A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

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A062723 Least common multiple (LCM) of the first n+1 terms of A000792.

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A349983 a(n) is the largest k such A000792(k) <= n.

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A087902 Least m such that A080256(m)=n and has a maximum number A000792(n) of divisors.

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A001414 Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).

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A001113 Decimal expansion of e.

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A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

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