A119348 -id:A119348 - cp's OEIS frontend

A005153 Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.

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

Offset: 1

N. J. A. Sloane

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
From Hal M. Switkay, Dec 22 2022: (Start)
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 Adams-Watters'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

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.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 146-147.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Thomas Bloom, Problem 18, Erdős Problems.
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.
James Grimes and Brady Haran, Practical Numbers, Numberphile video (2023).
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:1201.3168 [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.
Sai Teja Somu, Ting Hon Stanford Li, and Andrzej Kukla, On Some Results on Practical Numbers, INTEGERS, Volume 23, A68, 2023 [MR4643065].
Sai Teja Somu and Duc Van Khanh Tran, On Sums of Practical Numbers and Polygonal Numbers, arXiv:2403.13533 [math.NT], 2024.
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.
Terence Tao, Erdős problem database, see entry no. 18.
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.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Practical Number.
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]
Wikipedia, Practical number.
Robert G. Wilson v, Letter to N. J. A. Sloane, date unknown.

Subsequence of A103288.
Cf. A002093, A007620 (second definition), A030057, A033630, A119348, A174533, A174973.
Cf. A027750.

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

isA005153 := proc(n)
    local ifs,pprod,p,i ;
    if n = 1 then
        return true;
    elif type(n,'odd') then
        return false ;
    end if;
    # not using ifactors here directly because no guarantee primes are sorted...
    ifs := ifactors(n)[2] ;
    pprod := 1;
    for p in sort(numtheory[factorset](n) ) do
        for i in ifs do
            if op(1,i) = p then
                if p > 2 and p > 1+numtheory[sigma](pprod) then
                    return false ;
                end if;
                pprod := pprod*p^op(2,i) ;
            end if;
        end do:
    end do:
    return true ;
end proc:
for n from 1 to 300 do
    if isA005153(n)  then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2023

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 *)

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

from sympy import factorint
def is_A005153(n):
    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)//(p-1)
    return True # M. F. Hasler, Jan 02 2023

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

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

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

A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

Original entry on oeis.org

1, 3, 3, 7, 3, 12, 3, 15, 7, 15, 3, 28, 3, 15, 15, 31, 3, 39, 3, 42, 15, 15, 3, 60, 7, 15, 15, 56, 3, 72, 3, 63, 15, 15, 15, 91, 3, 15, 15, 90, 3, 96, 3, 63, 55, 15, 3, 124, 7, 63, 15, 63, 3, 120, 15, 120, 15, 15, 3, 168, 3, 15, 59, 127, 15, 144, 3, 63, 15, 142, 3, 195, 3, 15, 63, 63

Offset: 1

Emeric Deutsch, May 15 2006

nonn

If a(n)=sigma(n) (=sum of the divisors of n =A000203(n); i.e. all numbers from 1 to sigma(n) are sums of distinct divisors of n), then n is called a practical number (A005153). The actual sums obtained from the divisors of n are given in row n of the triangle A119348.
The records appear to occur at the highly abundant numbers, A002093, excluding 3 and 10. For n in A174533, a(n) = sigma(n)-2. - T. D. Noe, Mar 29 2010
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020
Zumkeller numbers A083207 give the positions of even terms in this sequence (likewise, the positions of odd terms in A308605). - Antti Karttunen and Ilya Gutkovskiy, Nov 29 2024

			a(5)=3 because the divisors of 5 are 1 and 5 and all the possible sums: are 1,5 and 6; a(6)=12 because we can form all sums 1,2,...,12 by adding up the terms of a nonempty subset of the divisors 1,2,3,6 of 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Jon Maiga, Computer-generated formulas for A119347, Sequence Machine.
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics 76 (1954), pp. 779-785.

One less than A308605.
Cf. A000203, A002093, A005153, A027750, A030057, A033630, A093890, A100587, A119348, A193279, A225561, A237290, A378447, A378450, A378451.
Cf. A083207 (positions of even terms).

import Data.List (subsequences, nub)
a119347 = length . nub . map sum . tail . subsequences . a027750_row'
-- Reinhard Zumkeller, Jun 27 2015

with(numtheory): with(linalg): a:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: seq(a(n),n=1..90);

a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Jan 27 2018 *)

A119347(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=1,poldegree(p),(0Antti Karttunen, Nov 28 2024

A119347(n) = { my(c=[0]); fordiv(n, d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A119347(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    return len(c)-1 # Chai Wah Wu, Jul 05 2023

For n > 1, 3 <= a(n) <= sigma(n). - Charles R Greathouse IV, Feb 11 2019
For p prime, a(p) = 3. For k >= 0, a(2^k) = 2^(k + 1) - 1. - Ctibor O. Zizka, Oct 19 2023
From Antti Karttunen, Nov 29 2024: (Start)
a(n) = A308605(n)-1.
a(n) = 2*(A237290(n)/A000203(n)) - 1. [Found by Sequence Machine. See A237290.]
a(n) <= A100587(n).
(End)

Definition clarified by Antti Karttunen, Nov 29 2024

A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 78, 16, 120, 52, 144, 24, 406, 28, 192, 192, 496, 36, 780, 40, 903, 256, 288, 48, 1830, 124, 336, 320, 1596, 60, 2628, 64, 2016, 384, 432, 384, 4186, 76, 480, 448, 4095, 84, 4656, 88, 2688, 2184, 576, 96, 7750, 228, 2976, 576, 3136, 108, 7260

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 200 terms from Vincenzo Librandi)
Jon Maiga, Computer-generated formulas for A237290, Sequence Machine.

Cf. A000203, A119348, A005153, A119347 (count of the same numbers), A184387, A229335, A237287, A237289.

isSumDist := proc(n,k)
    local dvs,s ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237290 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
end proc:
seq(A237290(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* Giovanni Resta, Mar 13 2014 *)

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek));); sum(i=1, #vks, vks[i]);} \\ Michel Marcus, Mar 09 2014

A237290(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); vecsum(c); }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A237290(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237289(n).
a(p) = 2(p+2) for odd primes p.
a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).
a(n) = A000203(n) * (A119347(n)+1) / 2. [Found by Sequence Machine and easily seen to be true. Compare for example to the formulas of A229335.] - Antti Karttunen, Nov 29 2024

A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 9, 0, 20, 0, 39, 27, 54, 0, 77, 108, 108, 0, 135, 0, 170, 0, 272, 378, 252, 0, 372, 567, 500, 0, 405, 0, 464, 0, 792, 1053, 792, 0, 665, 1350, 1148, 0, 819, 0, 902, 882, 897, 2052, 1080, 0, 1425, 1395, 2052, 1715, 1377, 0, 2052, 0, 2600, 3375, 1710

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 2 + 3 + 4 = 9 (numbers 2, 3 and 4 are not a sum of any subset of distinct divisors of 5).
Numbers n = 14 and 15 are an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237290(14) = A237290(15).
a(14) = 4+5+6+11+12+13+18+19+20 = a(15) = 2+7+10+11+12+13+14+17+22 = 108.
a(6) = 0 as 6 is practical; the sums into distinct divisors from 1 through 12 are 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 2 + 3, 7 through 12 are (1 through 6) + 6. So none are not a sum distinct divisors of 6. - _David A. Corneth_, Jul 22 2025

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A119348, A005153, A184387, A237287, A237290.

isSumDist := proc(n,k)
    local dvs ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237289 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if not isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
    a ;
end proc:
seq(A237289(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Block[{d = Divisors@n, s}, s = Plus @@ d; s*(s + 1)/2 - Plus @@ Union[Plus @@@ Subsets@d]]; m = Array[a, 59] (* Giovanni Resta, Mar 13 2014 *)

from sympy import divisors
def A237289(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return (s*(s+1)>>1)-sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237290(n).
a(p) = p(p - 1) / 2 - 1 for p = prime > 2.
a(n) = 0 for practical numbers (A005153), a(n) > 0 for numbers that are not practical (A237287).
a(n) = A184387(n) - A229335(n) for numbers n such that A119347(n) = A100587(n).

a(55) and a(57)-a(59) corrected by Giovanni Resta, Mar 13 2014

A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 29 2019

T(n, k) > 0 for all values of k iff n is practical (A005153).

			Table begins as:
  1
  1,1,1
  1,0,1,1
  1,1,1,1,1,1,1
  1,0,0,0,1,1
  1,1,2,1,1,2,1,1,2,1,1,1
  1,0,0,0,0,0,1,1
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  1,0,1,1,0,0,0,0,1,1,0,1,1
  1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1

Amiram Eldar, Table of n, a(n) for n = 1..8299 (rows 1..100 flattened)

Cf. A000203 (row lengths), A307224 (row products).

Cf. A005153, A027750, A030057, A033630, A119348, A225561, A237287, A322860.

T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; Table[T[n, k], {n,1,10}, {k, 1, DivisorSigma[1,n]}] // Flatten

T(n, n) = A033630(n).

T(n, A030057(n)) = 0 if there is a 0 in the n-th row, i.e. A030057(n) <= sigma(n) or n is not practical.

A119357 Numbers k such that the number of distinct nonzero sums of distinct divisors of k is less than 2^tau(k) - 1 (the largest number of possible distinct sums, tau(k) being the number of divisors of k (A000005)).

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 144, 150, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 186, 189, 192, 195

Offset: 1

Emeric Deutsch, May 18 2006

nonn

Equivalently, numbers k for which there exist two distinct subsets of the set of divisors of k having the same sum.
The sequence is closed with respect to multiplication by positive integers (i.e. any multiple of any term in the sequence is in the sequence). The primitive entries of the sequence, i.e. those that are not multiples of other terms of the sequence, are given in A119425 (the first five are 6,20,28,45 and 63).
The number of distinct sums of distinct divisors of n are given in A119347 and the actual sums are given in row n of the triangle A119348.
Subsequence of A051774 (Max Alekseyev).

			6 is in the sequence because from the divisors of 6, namely 1,2,3,6, we can form by addition 12 sums (1,2,3,...,12) and 12 < 2^tau(6)-1=2^4-1=15.
Sequence contains, for example, all multiples of 6 (1+2=3), all multiples of 20 (1+4=5), all multiples of 28 (1+2+4=7), all multiples of 63 (1+9=3+7).

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

Cf. A000005, A051774, A119347, A119348, A119425.

with(numtheory): with(linalg): s:=proc(n) local dl,t:dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: a:=proc(n) if s(n)<2^tau(n)-1 then n else fi end: seq(a(n),n=1..230);

q[n_] := Module[{d = Divisors[n], x}, Max[CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, Total[d]}], x]] > 1]; Select[Range[200], q] (* Amiram Eldar, Jan 02 2022 *)

A193280 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 3, 4, 1, 2, 3, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 1, 2, 3, 7, 8, 9, 10, 1, 3, 4, 5, 6, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 1

Michael Engling, Jul 20 2011

Row n > 1 contains A193279(n) terms. In row n the first term is 1 and the last term is sigma(n) - n (= A000203(n) - n). Row 1 contains 0 because 1 has no proper divisors.

			Row 10 is 1,2,3,5,6,7,8 the possible sums obtained from the proper divisors 1, 2, and 5 of 10.
Triangle starts:
  0;
  1;
  1;
  1,2,3;
  1;
  1,2,3,4,5,6;
  1;
  1,2,3,4,5,6,7;
  1,3,4;
  1,2,3,5,6,7,8;
  1;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16;

Nathaniel Johnston, Rows 1..150, flattened

Cf. A119347, A119348, A193279.

with(linalg): print(0); for n from 2 to 12 do dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): print(op({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)})): od: # Nathaniel Johnston, Jul 23 2011

A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 48, 53, 54, 56, 59, 60, 61, 64, 66, 67, 71, 72, 73, 78, 79, 80, 83, 84, 88, 89, 90, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 120, 126, 127, 128

Offset: 1

Andrzej Kukla, Nov 11 2024

Equivalently, positive integers m such that every number k <= d is a sum of distinct proper divisors of m, where d is the largest proper divisor of m (follows from Corollary 2.11 in the Kukla and Miska paper).

Rao and Peng (2013) proved that a number is quasi practical if and only if it is prime or practical (also Theorem 2.9 in Kukla/Miska paper).

Andrzej Kukla, Table of n, a(n) for n = 1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Union of A000040 and A005153.

Cf. A002093, A103288, A033630, A119348, A174533, A174973, A083207.

QuasiPracticalQ[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]] || PrimeQ[n]]; Select[Range[200], QuasiPracticalQ] (* Created based on code by T. D. Noe, Apr 02 2010 *)

A385904 a(n) is the number of nonempty subsets of the divisors of n that sum to a perfect square.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 3, 3, 2, 1, 11, 1, 3, 4, 5, 1, 9, 1, 9, 3, 3, 1, 27, 2, 2, 4, 8, 1, 27, 1, 7, 3, 2, 2, 49, 1, 1, 3, 22, 1, 21, 1, 7, 8, 3, 1, 77, 2, 5, 2, 4, 1, 22, 2, 21, 2, 1, 1, 248, 1, 2, 7, 11, 1, 21, 1, 4, 2, 17, 1, 235, 1, 1, 9, 7, 1, 20, 1, 64, 6, 1

Offset: 1

Felix Huber, Jul 21 2025

nonn

			a(6) = 4 because exactly the 4 nonempty subsets {1}, {1, 3}, {1, 2, 6} and {3, 6} of the divisors of 6 sum to a perfect square: 1 = 1^2, 1 + 3 = 2^2, 1 + 2 + 6 = 3^2.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000290, A005835, A027750, A119347, A119348, A237290, A385645, A385646.

with(NumberTheory):
A385904:=proc(n)
    local b,l,j;
    l:=[(Divisors(n))[]]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
	end;
    add(b(j^2,nops(l)),j=1..floor(sqrt(sigma(n))));
end:
seq(A385904(n),n=1..82);

a[n_]:=Module[{nb = 0, d = Divisors[n]},Length[Select[Subsets[d],IntegerQ[Sqrt[Total[#]]]&]]]-1;Array[a,82] (* James C. McMahon, Jul 27 2025 *)

a(n) = my(nb=0, d=divisors(n)); forsubset(#d, s, nb+=issquare(sum(i=1, #s, d[s[i]]))); nb-1; \\ Michel Marcus, Jul 22 2025

a(p) = 1 for primes p != 3.

cp's OEIS Frontend

A005153 Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.

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A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

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A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

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A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

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A119357 Numbers k such that the number of distinct nonzero sums of distinct divisors of k is less than 2^tau(k) - 1 (the largest number of possible distinct sums, tau(k) being the number of divisors of k (A000005)).

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