A111900 -id:A111900 - cp's OEIS frontend

A090677 Number of ways to partition n into sums of squares of primes.

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 3, 4, 4, 2, 3, 4, 5, 3, 3, 5, 5, 4, 3, 5, 5, 5, 4, 5, 6, 5, 5, 5, 7, 6, 6, 6, 7, 7, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 9, 8, 9, 9, 10, 9, 9, 10, 11, 11, 10, 11

Offset: 0

N. J. A. Sloane, Dec 19 2003

nonn

From Hieronymus Fischer, Nov 11 2007: (Start)
First statement of monotony: a(n+p^2)>=a(n) for all primes p. Proof: we restrict ourselves on a(n)>0 (the case a(n)=0 is trivial). Let T(i), 1<=i<=a(n), be the a(n) different sums of squares of primes representing n. Then, adding p^2 to those expressions, we get a(n) sums of squares of primes T(i)+p^2, obviously representing n+p^2, thus a(n+p^2) cannot be less than a(n).
Second statement of monotony: a(n+m)>=max(a(n),a(m)) for all m with a(m)>1. Proof: let T(i), 1<=i<=a(n), be the a(n) different sums of squares of primes representing n; let S(i), 1<=i<=a(m), be the a(m) different sums of squares of primes representing m. Then, adding these expressions, we get a(n) sums of squares of primes T(i)+S(1), representing n+m, further we get a(m) sums T(1)+S(i), also representing n+m. Thus a(n+m) cannot be less than the maximum of a(n) and a(m).
The minimum b(k):=min( n | a(j)>k for all j>n) exists for all k>=0. See A134755 for that sequence representing b(k). (End)

			a(25)=2 because 25 = 5^2 = 4*(2^2)+3^2.
a(83)=8 because 83 = 3^2+5^2+7^2 = 4*(2^2)+2*(3^2)+7^2
                   = 2*(2^2)+3*(5^2) = 6*(2^2)+3^2+2*(5^2)
                   = 2^2+6*(3^2)+5^2 = 10*(2^2)+2*(3^2)+5^2
                   = 5*(2^2)+7*(3^2) = 14*(2^2)+3*(3^2).

R. F. Churchouse, Representation of integers as sums of squares of primes. Caribbean J. Math. 5 (1986), no. 2, 59-65.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Christian N. K. Anderson, Table of coefficients b_i solving n=sum(b_i * prime(i)^2), for n=0..500
Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.

Cf. A111900, A001248, A001156, A023893, A111901.

Cf. A078134, A078135, A078136, A078139, A134600, A078137, A134754, A134755.

CoefficientList[ Series[ Product[1/(1 - x^Prime[i]^2), {i, 111}], {x, 0, 101}], x] (* Robert G. Wilson v, Sep 20 2004 *)

G.f.: 1/((1-x^4)*(1-x^9)*(1-x^25)*(1-x^49)*(1-x^121)*(1-x^169)*(1-x^289)...).

G.f.: 1 + Sum_{i>=1} x^(prime(i)^2) / Product_{j=1..i} (1 - x^(prime(j)^2)). - Ilya Gutkovskiy, May 07 2017

A167700 Number of partitions of n into distinct odd squares.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Nov 09 2009

A167701 and A167702 give record values and where they occur: A167701(n)=a(A167702(n)) and a(m) < A167701(n) for m < A167702(n);

a(A167703(n)) = 0.

			a(50) = #{49+1} = 1;
a(130) = #{121+9, 81+49} = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio
Index entries for sequences related to sums of squares.

Cf. A033461, A016754, A167661, A000700, A111900.

a167700 = p a016754_list where
   p _  0 = 1
   p (q:qs) m = if m < q then 0 else p qs (m - q) + p qs m
-- Reinhard Zumkeller, Mar 15 2014

nmax = 100; CoefficientList[Series[Product[1 + x^((2*k-1)^2), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)

a(n) = f(n,1,8) with f(x,y,z) = if x

G.f.: Product_{k>=0} (1 + x^((2*k+1)^2)). - Ilya Gutkovskiy, Jan 11 2017

a(n) ~ exp(3 * 2^(-7/3) * Pi^(1/3) * (sqrt(2)-1)^(2/3) * Zeta(3/2)^(2/3) * n^(1/3)) * (sqrt(2)-1)^(1/3) * Zeta(3/2)^(1/3) / (2^(7/6) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017

A111902 Number of partitions of n into distinct parts that are primes or squares of primes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 4, 4, 4, 6, 4, 8, 5, 9, 7, 10, 9, 11, 12, 12, 15, 14, 17, 17, 20, 20, 23, 24, 26, 28, 30, 32, 35, 36, 40, 41, 46, 47, 52, 54, 58, 62, 65, 71, 73, 80, 82, 90, 93, 101, 104, 113, 117, 125, 132, 139, 148, 154, 165, 171, 183, 191

Offset: 1

Reinhard Zumkeller, Aug 20 2005

nonn

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 3*x^7 + x^8 + 4*x^9 + 2*x^10 + ...
a(12) = #{3^2+3, 7+5, 7+3+2, 5+2^2+3} = 4.

Cf. A000586, A111900, A106244, A111901.

{a(n) = if(n < 0, 0, polcoeff( prod(k=1, primepi(n), (1 + x^prime(k)^2 + x*O(x^n)) * (1 + x^prime(k))), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=1} (1 + x^prime(k))*(1 + x^(prime(k)^2)). - Ilya Gutkovskiy, Dec 26 2016

A287965 Smallest number which can be represented as the sum of distinct squares of primes in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

4, 410, 1014, 1494, 1685, 2188, 2335, 2573, 2717, 2863, 3054, 3389, 3224, 3654, 3534, 4014, 4232, 4183, 4254, 4064, 4589, 4618, 4544, 4593, 4903, 5193, 5503, 5215, 5579, 5433, 5455, 5673, 5962, 5983, 6158, 6178, 5744, 5864, 5984, 5913, 6223, 6273, 6678, 6393, 6442, 6513, 6870, 6535, 7038, 7015

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

It appears that 1275 is the first k for which a(k) = 0. - Robert Israel, Oct 14 2024

			a(2) = 410 because 410 = 7^2 + 19^2 = 11^2 + 17^2 and this is the smallest number that can be written as the sum of distinct squares of primes in 2 different ways.

Robert Israel, Table of n, a(n) for n = 1..1274
Index entries for sequences related to sums of squares

Cf. A001248, A048261, A097563, A111900, A121518.

N:= 100: # to try with primes up to N
P:= select(isprime, [2,seq(i,i=3..N,2)]):
nP:= nops(P):
S:= mul(1+x^(P[i]^2), i=1..nP):
M:= 100: # for a(1) .. a(M)
V:= Vector(M): count:= 0:
for i from 4 to N^2 while count < M do
  r:= coeff(S,x,i);
  if r >= 1 and r <= M and V[r] = 0 then count:= count+1; V[r]:= i; fi
od:
convert(V,list); # Robert Israel, Oct 14 2024

A111900(a(n)) = n.

A280126 Expansion of Product_{k>=1} (1 + x^(prime(k)^2))*(1 + x^(prime(k)^3)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 26 2016

nonn

Number of partitions of n into distinct parts that are squares of primes (A001248) or cubes of primes (A030078).

			a(61) = 2 because we have [49, 8, 4] and [25, 27, 9].

Index entries for related partition-counting sequences

Cf. A000586, A001248, A030078, A106244, A111900, A111902, A131799, A168363, A280125.

nmax = 120; CoefficientList[Series[Product[(1 + x^Prime[k]^2) (1 + x^Prime[k]^3), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(prime(k)^2))*(1 + x^(prime(k)^3)).

A281477 Expansion of Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

Total number of parts in all partitions of n into distinct squares of primes (A001248).

			a(38) = 3 because we have [25, 9, 4].

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example
Index entries for related partition-counting sequences

Cf. A001248, A024938, A048261, A111900, A121518, A281449, A281542, A281668.

Primes:= select(isprime, [$1..20]):
g:= add(x^(p^2)/(1+x^(p^2)),p=Primes)*mul(1+x^(p^2),p=Primes):
S:= series(g, x, 20^2+1):
seq(coeff(S,x,n),n=1..20^2); # Robert Israel, Feb 08 2017

nmax = 125; Rest[CoefficientList[Series[Sum[x^Prime[k]^2/(1 + x^Prime[k]^2), {k, 1, nmax}] Product[1 + x^Prime[k]^2, {k, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

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A090677 Number of ways to partition n into sums of squares of primes.

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A167700 Number of partitions of n into distinct odd squares.

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A111902 Number of partitions of n into distinct parts that are primes or squares of primes.

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A287965 Smallest number which can be represented as the sum of distinct squares of primes in exactly n ways, or 0 if no such integer exists.

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A280126 Expansion of Product_{k>=1} (1 + x^(prime(k)^2))*(1 + x^(prime(k)^3)).

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A281477 Expansion of Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

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