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.

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A046676 Expansion of 1 + Sum_{k>=1} x^(p_1+p_2+...+p_k)/((1-x)*(1-x^2)*(1-x^3)*...*(1-x^k)) (where p_i are the primes).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 60, 69, 78, 89, 99, 113, 126, 143, 159, 179, 199, 224, 248, 277, 307, 343, 378, 421, 464, 515, 567, 628, 690, 763, 837, 923, 1012, 1115, 1219, 1340, 1465, 1607
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan considered that this could equal the prime parts partition numbers A000607, but they differ from the 20th term on, cf. A192541. See A238804 for a correct variant, where the coefficient and power of x^{...} are adjusted to match A000607. - M. F. Hasler, Mar 06 2014

References

  • B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.

Links

Crossrefs

Differs from A000607 at the 20th term. Cf. A192541.

Programs

  • Maple
    t3:=1+add(q^sum(ithprime(i),i=1..j)/mul(1-q^i,i=1..j), j=1..51);
t4:=series(t3,q,50);
t5:=seriestolist(%);
  • PARI
    Vec(sum(i=0,25,x^sum(k=1,i,prime(k))/prod(k=1,i,1-x^k),O(x^99))) \\ M. F. Hasler, Mar 05 2014
  • PARI
    A046676(n,S=1,P=1+O(x^(n+1)))={for(k=1,n, nM. F. Hasler, Mar 05 2014

A192541 A046676(n) - A000607(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 1, 2, 1, 2, 2, 3, 2, 4, 2, 5, 4, 5, 5, 7, 6, 8, 8, 11, 10, 14, 13, 19, 18, 24, 25, 32, 33, 42, 45, 55, 60, 72, 77, 94, 102, 120, 132, 155, 169, 196, 218, 249, 275, 315, 346, 395, 435, 492, 542, 613, 673, 756, 833, 931, 1024, 1143, 1253, 1397, 1532, 1699, 1864, 2063, 2258, 2496
Offset: 0

Views

Author

N. J. A. Sloane, Jul 03 2011

Keywords

Comments

See A046676 for explanations and further references. - M. F. Hasler, Mar 05 2014

Links

Crossrefs

Cf. A238804, A238882.

Programs

A238882 Coefficients in a variant of Ramanujan's wrong identity for prime number partitions.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -2, 2, -1, 3, 1, 2, 1, -1, -4, 1, -4, -4, -10, -2, -8, -4, -5, -4, -1, 1, 2, 5, 6, 13, 12, 16, 18, 21, 25, 23, 30, 22, 23, 21, 21, 18, 14, 8, -1, -9, -20, -36, -36, -51, -61, -75, -80, -96, -103
Offset: 0

Views

Author

M. F. Hasler, Mar 06 2014

Keywords

Comments

Consider the g.f. of the prime parts partition numbers, GF = 1/Product_{k>=1} (1-x^prime(k)), cf. A000607. Then consecutively subtract a(n)*x^b(n)/Product_{k=1..n} (1-x^k), n=0,1,2,3,... where a(n)*x^b(n) is the leading term of the remaining expression, GF - previously subtracted terms. Sequence A238804 lists the exponents b(n), here we list the coefficients a(n).
The identity Ramanujan considered, GF = Sum_{n>=0} x^Sum_{k=1..n} prime(k)/Product_{k=1..n} (1-x^k), or A000607 = A046676, is wrong: In the way they are defined above, the pattern of b(n) = (sum of first n primes) breaks after b(4)=17; the pattern a(n)=1 breaks also after n=4 (which yields this sequence), and the nontrivial cancellations stop after the power b(5)=21, followed by 22, 24, 25, 26, 27, ...

Examples

			GF = 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)...) = 1+x^2+x^3+x^4+2*x^5+... (cf. A000607)
=> a(0)=1, b(0)=0, GF - 1 = x^2 + ....
=> a(1)=1, b(1)=2, GF - 1 - x^2/(1-x) = x^5 + ...
=> a(2)=1, b(2)=5, GF - 1 - x^2/(1-x) - x^5/(1-x)(1-x^2) = x^10 + ...
=> a(3)=1, b(3)=10, GF - ... - x^10/(1-x)(1-x^2)(1-x^3) = x^17 + ...
=> a(4)=1, b(4)=17, GF - ... - x^17/(1-x)(1-x^2)(1-x^3)(1-x^4) = -x^21+...
=> a(5)=-1, b(5)=21, GF - ... + x^21/... etc.

Crossrefs

Cf. A000607, A046676, A192541, A238804.

Programs

  • PARI
    p=1/prod(k=1,25,1-x^prime(k),1+O(x^999))/* Note: p1+...+p25 > 1000 */; for(k=0,99, print1(polcoeff(p,c=valuation(p,x)),",");p-=polcoeff(p,c)*x^c/prod(j=1,k,1-x^j,O(x^199)+1))

Extensions

Example section corrected by Vaclav Kotesovec, Sep 12 2019
Showing 1-3 of 3 results.

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

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