A237289 -id:A237289 - cp's OEIS frontend

A237287 Numbers that are not practical: positive integers n such that there exists at least one number k <= sigma(n) that is not a sum of distinct divisors of n.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

Complement of A005153 (practical numbers).
Numbers n such that A030057(n) < n.
First differs from A237046 at a(48).
First differs from A238524 at a(55). - Omar E. Pol, Mar 09 2014
First differs from A378471 at a(72). - Hartmut F. W. Hoft, Nov 27 2024

			5 is in the sequence because there are 3 numbers <= sigma(5) = 6 that are not a sum of any subset of distinct divisors of 5: 2, 3 and 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A000203, A005153, A237046, A237289, A237290, A378471.

from itertools import count, islice
from sympy import factorint
def A237287_gen(startvalue=1): # generator of terms
    for m in count(max(startvalue,1)):
        if m > 1:
            l = (~m & m-1).bit_length()
            if l>0:
                P = (1<>l).items():
                    if p > 1+P:
                        yield m
                        break
                    P *= (p**(e+1)-1)//(p-1)
            else:
                yield m
A237387_list = list(islice(A237287_gen(),30)) # Chai Wah Wu, Jul 05 2023

More terms added by Hartmut F. W. Hoft, Nov 27 2024, in order to show the difference from A378471.

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

A378450 a(n) is the number 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, 1, 0, 3, 0, 5, 0, 6, 3, 9, 0, 11, 9, 9, 0, 15, 0, 17, 0, 17, 21, 21, 0, 24, 27, 25, 0, 27, 0, 29, 0, 33, 39, 33, 0, 35, 45, 41, 0, 39, 0, 41, 21, 23, 57, 45, 0, 50, 30, 57, 35, 51, 0, 57, 0, 65, 75, 57, 0, 59, 81, 45, 0, 69, 0, 65, 63, 81, 2, 69, 0, 71, 99, 61, 77, 81, 0, 77, 0, 90, 111, 81, 0, 93, 117, 105

Offset: 1

Antti Karttunen, Nov 29 2024

nonn

Difference between the sum of divisors n and the number of distinct sums of distinct divisors of n.

			For n = 3, with divisors [1, 3] and sigma(3)=4, only 2 in range 1..4 cannot be represented as a sum of a subset of [1, 3], therefore a(3) = 1.
For n = 15, with divisors [1, 3, 5, 15] and sigma(15) = 24, the subset sums are 1, 3, 1+3, 5, 1+5, 3+5, 1+3+5, 15, 1+15, 3+15, 1+3+15, 5+15, 1+5+15, 3+5+15, 1+3+5+15 i.e., [1, 3, 4, 5, 6, 8, 9, 15, 16, 18, 19, 20, 21, 23, 24], which leaves 2, 7, 10, 11, 12, 13, 14, 17, 22 as unrepresented numbers, therefore a(15) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A119347, A237289 (gives the sums of unrepresented numbers), A322860.

Cf. A005153 (positions of 0's), A237287 (of nonzeros), A030057.

A119347(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; };
A378450(n) = (sigma(n)-A119347(n));

a(n) = A000203(n) - A119347(n).

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A237287 Numbers that are not practical: positive integers n such that there exists at least one number k <= sigma(n) that is not a sum of distinct divisors of n.

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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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A378450 a(n) is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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