

A005153


Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.
(Formerly M0991)


132



1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252
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OFFSET

1,2


COMMENTS

Equivalently, positive integers m such that every number k <= m is a sum of distinct divisors of m.
2^r is a member for all r as every number < = sigma(2^r) = 2^(r+1)1 is a sum of a distinct subset of divisors {1, 2, 2^2, ..., 2^m}.  Amarnath Murthy, Apr 23 2004
Also, numbers m such that A030057(m) > m. This is a consequence of the following theorem (due to Stewart), found at the McLeman link: An integer m >= 2 with factorization Product_{i=1..k} p_i^e_i with the p_i in ascending order is practical if and only if p_1 = 2 and, for 1 < i <= k, p_i <= sigma(Product_{j < i} p_j^e_j) + 1.  Franklin T. AdamsWatters, Nov 09 2006
Practical numbers first appear in Srinivasan's short paper, which contains terms up to 200. Let m be a practical number. He states that (1) if m>2, m is a multiple of 4 or 6; (2) sigma(m) >= 2*m1 (A103288); and (3) 2^t*m is practical. He also states that highly composite numbers (A002182), perfect numbers (A000396), and primorial numbers (A002110) are practical.  T. D. Noe, Apr 02 2010
Conjecture: The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing to the limit 1.  ZhiWei Sun, Jan 12 2013
Conjecture: For any positive rational number r, there are finitely many pairwise distinct practical numbers q(1)..q(k) such that r = Sum_{j=1..k} 1/q(j). For example, 2 = 1/1 + 1/2 + 1/4 + 1/6 + 1/12 with 1, 2, 4, 6 and 12 all practical, and 10/11 = 1/2 + 1/4 + 1/8 + 1/48 + 1/132 + 1/176 with 2, 4, 8, 48, 132 and 176 all practical.  ZhiWei Sun, Sep 12 2015
Analogous with the {1 union primes} (A008578), practical numbers form a complete sequence. This is because it contains all powers of 2 as a subsequence.  Frank M Jackson, Jun 21 2016
Sun's 2015 conjecture on the existence of Egyptian fractions with practical denominators for any positive rational number is true. See the link "Egyptian fractions with practical denominators".  David Eppstein, Nov 20 2016
Conjecture: if all divisors of m are 1 = d_1 < d_2 < ... < d_k = m, then m is practical if and only if d_(i+1)/d_i <= 2 for 1 <= i <= k1.  Jianing Song, Jul 18 2018
The above conjecture is incorrect. The smallest counterexample is 78 (for which one of these quotients is 13/6; see A174973). m is practical if and only if the divisors of m form a complete subsequence. See Wikipedia links.  Frank M Jackson, Jul 25 2018
Reply to the comment above: Yes, and now I can show the opposite: The largest value of d_(i+1)/d_i is not bounded for practical numbers. Note that sigma(n)/n is not bounded for primorials, and primorials are practical numbers. For any constant c >= 2, let k be a practical number such that sigma(k)/k > 2c. By Bertrand's postulate there exists some prime p such that c*k < p < 2c*k < sigma(k), so k*p is a practical number with consecutive divisors k and p where p/k > c. For example, for k = 78 we have 13/6 > 2, and for 97380 we have 541/180 > 3.  Jianing Song, Jan 05 2019
Erdős (1950) and Erdős and Loxton (1979) proved that the asymptotic density of practical numbers is 0.  Amiram Eldar, Feb 13 2021
Let P(x) denote the number of practical numbers up to x. P(x) has order of magnitude x/log(x) (see Saias 1997). Moreover, we have P(x) = c*x/log(x) + O(x/(log(x))^2), where c = 1.33607... (see Weingartner 2015, 2020 and Remark 1 of Pomerance & Weingartner 2021). As a result, a(n) = k*n*log(n*log(n)) + O(n), where k = 1/c = 0.74846...  Andreas Weingartner, Jun 26 2021
Every number of least prime signature (A025487) is practical, thereby including two classes of number mentioned in Noe's comment. This follows from Stewart's characterization of practical numbers, mentioned in AdamsWatters's comment, combined with Bertrand's postulate (there is a prime between every natural number and its double, inclusive).
Also, the first condition in Stewart's characterization (p_1 = 2) is equivalent to the second condition with index i = 1, given that an empty product is equal to 1. (End)
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number. Note that this conjecture occupies the space between the unproven Goldbach conjecture and the theorem that every even number, beginning with 2, is the sum of two practical numbers (Melfi's 1996 proof of Margenstern's conjecture).  Hal M. Switkay, Jan 28 2023


REFERENCES

H. Heller, Mathematical Buds, Vol. 1, Chap. 2, pp. 1022, Mu Alpha Theta OK, 1978.
Malcolm R. Heyworth, More on Panarithmic Numbers, New Zealand Math. Mag., Vol. 17 (1980), pp. 2834 [ ISSN 05490510 ].
Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179180.


LINKS

Paul Erdős, On a Diophantine equation (in Hungarian, with Russian and English summaries), Mat. Lapok, Vol. 1 (1950), pp. 192210.
Harvey J. Hindin, Quasipractical numbers, IEEE Communications Magazine, Vol. 18, No. 2 (March 1980), pp. 4145.
Maurice Margenstern, Sur les nombres pratiques, (in French), Groupe d'étude en théorie analytique des nombres, 1 (19841985), Exposé No. 21, 13 p.
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided.  N. J. A. Sloane, May 20 2023]


FORMULA

Weingartner proves that a(n) ~ k*n log n, strengthening an earlier result of Saias. In particular, a(n) = k*n log n + O(n log log n).  Charles R Greathouse IV, May 10 2013
More precisely, a(n) = k*n*log(n*log(n)) + O(n), where k = 0.74846... (see comments).  Andreas Weingartner, Jun 26 2021


MAPLE

with(numtheory): with(combinat): P:=proc(n) local a, b, k, j, ok; a:=choose([op(divisors(n))]); b:=[]; for k from 1 to nops(a) do b:=[op(b), add(a[k][j], j=1..nops(a[k]))]; od; for j from 0 to n1 do ok:=0; for k in b do if k=j then ok:=1; break; fi; od;
if ok=0 then break; fi; od; if ok=1 then n; fi; end:


MATHEMATICA

PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1  (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[Range[200], PracticalQ] (* T. D. Noe, Apr 02 2010 *)


PROG

(Haskell)
a005153 n = a005153_list !! (n1)
a005153_list = filter (\x > all (p $ a027750_row x) [1..x]) [1..]
where p _ 0 = True
p [] _ = False
p ds'@(d:ds) m = d <= m && (p ds (m  d)  p ds m)
(PARI) is_A005153(n)=bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i1]^n[2, i1])) && return) \\ M. F. Hasler, Jan 13 2013
(Python)
from sympy import factorint
if n & 1: return n == 1
f = factorint(n) ; P = (2 << f.pop(2))  1
for p in f: # factorint must have prime factors in increasing order
if p > 1 + P: return
P *= p**(f[p]+1)//(p1)
(Python)
from sympy import divisors; from more_itertools import powerset
[i for i in range(1, 253) if (lambda x:len(set(map(sum, powerset(x))))>sum(x))(divisors(i))] # Nicholas Stefan Georgescu, May 20 2023


CROSSREFS



KEYWORD

nonn,nice,easy,changed


AUTHOR



EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
Erroneous comment removed by T. D. Noe, Nov 14 2010
Definition changed to exclude n = 0 explicitly by M. F. Hasler, Jan 19 2013


STATUS

approved



