A294266 -id:A294266 - cp's OEIS frontend

A294265 Number of partitions of n into squares that do not divide n.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 3, 3, 1, 0, 3, 3, 1, 0, 5, 3, 3, 0, 2, 3, 3, 0, 7, 3, 3, 3, 8, 1, 3, 0, 10, 9, 6, 0, 14, 9, 0, 0, 15, 12, 9, 4, 15, 16, 9, 0, 18, 18, 7, 1, 23, 18, 17, 0, 9, 22, 19, 5, 30, 28, 19, 5, 34, 6, 24, 2, 40, 36, 30

Offset: 0

Ilya Gutkovskiy, Oct 26 2017

nonn

			a(25) = 2 because we have [16, 9] and [9, 4, 4, 4, 4].

Cf. A001156, A098743, A284345, A294266.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] > 0 && OddQ[DivisorSigma[0, k]]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 95}]

A300715 Number of compositions (ordered partitions) of n into squares that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 3, 0, 0, 7, 6, 0, 0, 14, 10, 4, 0, 22, 20, 10, 0, 32, 39, 20, 0, 49, 70, 42, 0, 12, 116, 88, 0, 128, 156, 174, 11, 207, 3, 320, 0, 333, 551, 575, 0, 555, 914, 0, 0, 959, 1502, 1829, 44, 1691, 2486, 3192, 0, 3000, 4172, 4005

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

			a(21) = 4 because we have [9, 4, 4, 4], [4, 9, 4, 4], [4, 4, 9, 4] and [4, 4, 4, 9].

Cf. A000290, A006456, A294105, A294265, A294266, A300702, A300703, A300704, A300706.

a:= proc(m) option remember; local b; b:= proc(n) option
      remember; `if`(n=0, 1, add((s->`if`(s>n or irem(m, s)
       =0, 0, b(n-s)))(j^2), j=2..isqrt(n))) end; b(m)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 11 2018

Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && IntegerQ[k^(1/2)]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 75}]

A371968 Numbers k that are not the sum of distinct squares that do not divide k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 54, 55, 56, 57, 59, 60, 63, 64, 66, 67, 68, 71, 72, 75, 76, 79, 80, 82, 84, 88, 91, 92, 95, 96, 99, 104, 107, 108, 111, 112, 120, 124, 127, 128, 132, 135, 140, 144, 147, 148, 156, 160, 168, 172, 176, 184, 188, 192, 200, 216, 224, 252, 256, 288, 300, 432

Offset: 1

Robert Israel, Apr 15 2024

nonn

Numbers k such that A294266(k) = 0.

No other terms <= 100000.

			a(20) = 21 is a term because the only way to write 21 as the sum of distinct squares is 1^2 + 2^2 + 4^2, but 21 is divisible by 1^2.

Cf. A294266.

filter:= proc(n) local P,k,x;
P:= 1;
for k from 2 to floor(sqrt(n)) do
  if n mod k^2 = 0 then next fi;
  P:= series(P*(1+x^(k^2)),x,n+1);
  if coeff(P,x,n) > 0 then return false fi;
od;
true
end proc:
select(filter, [$1..500]);

More terms than usual in the DATA section, because these are probably all the terms.

cp's OEIS Frontend

A294265 Number of partitions of n into squares that do not divide n.

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A300715 Number of compositions (ordered partitions) of n into squares that do not divide n.

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Mathematica

A371968 Numbers k that are not the sum of distinct squares that do not divide k.

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