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 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) 124
 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 (list; graph; refs; listen; history; text; internal format)
 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. Adams-Watters, 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*m-1 (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. - Zhi-Wei 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. - Zhi-Wei 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 <= k-1. - 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 REFERENCES H. Heller, Mathematical Buds, Vol. 1, Chap. 2, pp. 10-22, Mu Alpha Theta OK, 1978. Malcolm R. Heyworth, More on Panarithmic Numbers, New Zealand Math. Mag., Vol. 17 (1980), pp. 28-34 [ ISSN 0549-0510 ]. 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), 179-180. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) Wayne Dymacek, Letter to N. J. A. Sloane, Jun 15 1978. David Eppstein, Egyptian fractions with practical denominators, Nov 20, 2016 David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018. Paul Erdős, On a Diophantine equation (in Hungarian, with Russian and English summaries), Mat. Lapok, Vol. 1 (1950), pp. 192-210. Paul Erdős and J. H. Loxton, Some problems in partitio numerorum, Journal of the Australian Mathematical Society, Vol. 27, No. 3 (1979), pp. 319-331. Harvey J. Hindin, Quasipractical numbers, IEEE Communications Magazine, Vol. 18, No. 2 (March 1980), pp. 41-45. Paolo Leonetti and Carlo Sanna, Practical numbers among the binomial coefficients, Journal of Number Theory, Vol. 207 (2020), pp. 145-155; arXiv preprint, arXiv:1905.12023 [math.NT], 2019. Maurice Margenstern, Sur les nombres pratiques, (in French), Groupe d'étude en théorie analytique des nombres, 1 (1984-1985), Exposé No. 21, 13 p. Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory, Volume 37, Issue 1 (January 1991), pp. 1-36. C. McLeman, Practical number, PlanetMath.org. Giuseppe Melfi, On two conjectures about practical numbers, J. Number Theory, Vol. 56, No. 1 (1996), pp. 205-210 [MR96i:11106]. Giuseppe Melfi, On certain positive integer sequences, arXiv:0404555 [math.NT], 2004. Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Politec. Torino, Vol. 53, No. 4 (1995), pp. 347-359. Giuseppe Melfi, Practical Numbers (old link). Paul Pollack and Lola Thompson, Practical pretenders, arXiv:1201.3168v1 [math.NT], Jan 16, 2012 Carl Pomerance, Lola Thompson and Andreas Weingartner, On integers n for which X^n-1 has a divisor of every degree, arXiv:1511.03357 [math.NT], 2015. Carl Pomerance and Andreas Weingartner, On primes and practical numbers, Ramanujan J. (2021); arXiv preprint, arXiv:2007.11062 [math.NT], 2020. Eric Saias, Entiers à diviseurs denses 1, J. Number Theory, Vol. 62, No. 1 (1997), pp. 163-191; uses this definition. Carlo Sanna, Practical central binomial coefficients, arXiv:2004.05376 [math.NT], 2020. A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180. B. M. Stewart, Sums of distinct divisors, Amer. J. Math., Vol. 76, No. 4 (1954), pp. 779-785 [MR64800] Zhi-Wei Sun, A conjecture on unit fractions involving primes, preprint, 2015. Peter Taylor, Table of n, a(n) for n = 1..1000000. Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015. Andreas Weingartner, The constant factor in the asymptotic for practical numbers, Int. J. Number Theory, 16 (2020), no. 3, 629-638;  arXiv preprint, arXiv:1906.07819 [math.NT], 2019. Eric Weisstein's World of Mathematics, Practical Number. Wikipedia, Complete sequence. Wikipedia, Practical number. Robert G. Wilson v, Letter to N. J. A. Sloane, date unknown. 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 n-1 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: seq(P(i), i=1..252); # Paolo P. Lava, Jun 13 2017 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, PracticalQ] (* T. D. Noe, Apr 02 2010 *) PROG (Haskell) a005153 n = a005153_list !! (n-1) 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) -- Reinhard Zumkeller, Feb 23 2014, Oct 27 2011 (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, i-1]^n[2, i-1])) && return) \\ M. F. Hasler, Jan 13 2013 CROSSREFS Cf. A002093, A007620 (second definition), A030057, A033630, A119348, A174533, A174973. Cf. A027750. Sequence in context: A103288 A125225 A092903 * A174973 A238443 A325795 Adjacent sequences:  A005150 A005151 A005152 * A005154 A005155 A005156 KEYWORD nonn,nice,easy 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

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Last modified October 19 22:05 EDT 2021. Contains 348095 sequences. (Running on oeis4.)