cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane, Dec 19 2003

Keywords

Comments

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)

Examples

			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).

References

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

Links

Crossrefs

Cf. A111900, A001248, A001156, A023893, A111901.
Cf. A078134, A078135, A078136, A078139, A134600, A078137, A134754, A134755.

Programs

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

Formula

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

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

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Author

Reinhard Zumkeller, Aug 20 2005

Keywords

Examples

			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.

Crossrefs

Cf. A000586, A111900, A106244, A111901.

Programs

  • PARI
    {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 */

Formula

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

A280125 Expansion of Product_{k>=1} 1/((1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, 0, 4, 4, 2, 2, 4, 4, 2, 2, 5, 6, 4, 4, 7, 6, 4, 4, 8, 8, 6, 7, 10, 10, 6, 7, 11, 13, 9, 10, 15, 15, 12, 10, 16, 18, 16, 14, 20, 22, 19, 17, 21, 25, 23, 22, 26, 29, 28, 25, 30, 32, 33, 31, 37, 38, 38, 37
Offset: 0

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Author

Ilya Gutkovskiy, Dec 26 2016

Keywords

Comments

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

Examples

			a(16) = 3 because we have [8, 8], [8, 4, 4] and [4, 4, 4, 4].

Links

Crossrefs

Cf. A000607, A001248, A023893, A030078, A090677, A111901, A131799, A168363, A279760, A280126.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/((1 - x^Prime[k]^2) (1 - x^Prime[k]^3)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A280715 Expansion of Product_{k>=1} 1/((1 - x^prime(k))*(1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 36, 44, 53, 65, 78, 94, 112, 134, 159, 189, 222, 263, 307, 361, 420, 491, 569, 661, 764, 883, 1017, 1170, 1343, 1539, 1761, 2011, 2293, 2611, 2968, 3369, 3819, 4323, 4887, 5518, 6222, 7007, 7883, 8857, 9942, 11144, 12483, 13964, 15609, 17426, 19440, 21664
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 07 2017

Keywords

Comments

Number of partitions of n into parts that are primes (A000040), squares of primes (A001248) or cubes of primes (A030078).

Examples

			a(8) = 6 because we have [8], [5, 3], [4, 4], [4, 2, 2], [3, 3, 2], [2, 2, 2, 2].

Links

Crossrefs

Cf. A000040, A000607, A001248, A023893, A030078, A087797, A090677, A111901, A279760, A280125.

Programs

  • Mathematica
    nmax = 61; CoefficientList[Series[Product[1/((1 - x^Prime[k]) (1 - x^Prime[k]^2) (1 - x^Prime[k]^3)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/((1 - x^prime(k))*(1 - x^(prime(k)^2))*(1 - x^(prime(k)^3))).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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