A053253 -id:A053253 - cp's OEIS frontend

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Gus Wiseman, Mar 18 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Intersection of A066208 and A346635.

A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 8, 10, 10, 12, 14, 15, 18, 20, 22, 26, 29, 32, 36, 40, 44, 50, 56, 60, 68, 76, 82, 92, 101, 110, 122, 134, 146, 160, 176, 191, 210, 230, 248, 272, 296, 320, 350, 380, 410, 446, 484, 522, 566, 612, 660, 715, 772, 830, 896, 966, 1038, 1120

Offset: 0

Naohiro Nomoto, Feb 24 2002

Also number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k (Dean Hickerson). Absolute values of the terms of A053254. - Emeric Deutsch, Feb 10 2006
The number of self-conjugate partitions of n into odd parts is nonzero if and only if n = 4*k + 1 for some nonnegative integer k. - Michael Somos, Jul 25 2015
Also number of C3v plane partitions of n = 3*k + 1 with rank 1 ; equivalently number of self-conjugate integer partitions with (weight-length) = n. - Wouter Meeussen, May 23 2025

			a(5)=3 because we have [11,1,1,1,1,1,1,1,1,1,1], [9,3,3,1,1,1,1,1,1] and [5,5,5,3,3].
G.f. = 1 + x + 2*x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + ...

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 260, Article 512.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. S. Andersen, A Bijection Between Two Different Classes of Partitions Enumerated by p_nu(n), arXiv:2007.09794 [math.CO], 2020.
George E. Andrews, The Bhargava-Adiga Summation and Partitions, Journal of the Indian Mathematical Society, Volume 84, Issue 3-4, 2017.
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.

Cf. A000700, A000701.

Cf. A053253, A053254, A085140, A047993.

g:=sum(q^(k*(k+1))/product(1-q^(2*j+1),j=0..k),k=0..8): gser:=series(g,q=0,80): seq(coeff(gser,q,n),n=0..75); # Emeric Deutsch, Feb 10 2006

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / Product[ 1 - x^i, {i, 1, 2 k + 1, 2}], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jul 25 2015 *)
Table[Length[Flatten[Table[Select[IntegerPartitions[w], (w-Length[#])== r && TransposePartition[#] == # &],{w,r,1+2r}],1]],{r,1,17}] (* Wouter Meeussen, May 24 2025 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 4*n+1) -1) \ 2, x^(k^2 + k) / prod(j=0, k, 1 - x^(2*j+1), 1 + x * O(x^(n - k^2 - k)))), n))}; /* Michael Somos, Jan 27 2008 */

/* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + (-x)^(n-k+1)*(1 - (-x)^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013

G.f.: Sum_{k>=0} q^(k*(k+1)) / ((1-q) * (1-q^3) ... (1-q^(2*k+1))). - Emeric Deutsch and Dean Hickerson
G.f.: Sum_{k>=0} q^k * (1+q) * (1+q^3) ... (1+q^(2*k-1)). - Dean Hickerson and Vladeta Jovovic
G.f.: 1/(1 - x*(1+x)/(1 + x^2*(1-x^2)/(1 - x^3*(1+x^3)/(1 + x^4*(1-x^4)/(1 - x^5*(1+x^5)/(1 - ...)))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
From Michael Somos, Jul 25 2015: (Start)
Expansion of nu(-x) in powers of x where nu() is a 3rd-order mock theta function.
a(n) = (-1)^n * A053254(n).
a(2*n) = A085140(n).
a(2*n + 1) = A053253(n). (End)
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019

More terms from Emeric Deutsch, Feb 10 2006

A097196 Expansion of psi(x^3)^2 / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 4, 4, 6, 8, 9, 12, 16, 18, 22, 28, 33, 40, 50, 58, 70, 84, 98, 116, 138, 160, 188, 222, 256, 298, 348, 400, 463, 536, 614, 706, 812, 926, 1060, 1212, 1378, 1568, 1785, 2022, 2292, 2598, 2932, 3312, 3740, 4208, 4736, 5328, 5978, 6708, 7522, 8416, 9416

Offset: 0

N. J. A. Sloane, Sep 17 2004

nonn

On page 63 of Watson 1936 is an equation with left side 2*rho(q) + omega(q) and the right side is 3 times the g.f. of this sequence. - Michael Somos, Jul 14 2015

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + ...
G.f. = q^2 + q^8 + 2*q^11 + 2*q^14 + 2*q^17 + 4*q^20 + 4*q^23 + 6*q^26 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 50, Eq. (25.4).
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053253, A053255.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)]^2 / (4 x^(3/4) QPochhammer[ x^2]), {x, 0, n}]; (* Michael Somos, Jul 14 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^4 * (1-x^(3*k))^2 / (1-x^(2*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^4 / (eta(x^2 + A) * eta(x^3 + A)^2), n))}; /* Michael Somos, Jul 14 2015 */

Expansion of q^(-2/3) * eta(x^6)^4 / (eta(x^2) * eta(x^3)^2) in powers of q. - Michael Somos, Jul 14 2015
G.f.: Product_{n >= 1} (1+q^(3*n))^4*(1-q^(3*n))^2/(1-q^(2*n)).
3 * a(n) = A053253(n) + 2 * A053255(n). - Michael Somos, Jul 29 2015
a(n) ~ exp(Pi*sqrt(n/3)) / (12*sqrt(n)). - Vaclav Kotesovec, Oct 14 2015

A095913 Number of plasma partitions of 2n-1.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, 36, 44, 56, 68, 82, 101, 122, 146, 176, 210, 248, 296, 350, 410, 484, 566, 660, 772, 896, 1038, 1204, 1391, 1602, 1846, 2120, 2428, 2784, 3182, 3628, 4138, 4708, 5347, 6072, 6880, 7784, 8804, 9940, 11208, 12630

Offset: 1

Jon Perry, Jul 13 2004

nonn

			A plasma partition is a partition of n into 1 distinct odd part and an even number of odd parts and at least 2 parts of 1, so looking like plasma.
E.g. a(7) counts the plasma partitions of 13, has 11+1+1 = 9+1+1 = 7+1+1+1+1 = 5+1+1+1+1+1+1 = 5+3+3+1+1 = 3+1+1+1+1+1+1+1+1, so a(7)=6.
Graphically, these are;
.....*..........*........*......*.....*....*
***********.....*........*......*....***...*
.....*......*********....*......*...*****..*
................*.....*******...*....***...*
................*........*....*****...*....*
.........................*......*.........***
.........................*......*..........*
................................*..........*
................................*..........*
...........................................*
...........................................*

Seiichi Manyama, Table of n, a(n) for n = 1..1000

a(n)=A053253(n-3).

{a(n)=local(A); if(n<3, 0, n-=2; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* Michael Somos, Aug 18 2006 */

G.f.: sum(i>=1, x^(i+2)/prod(j=1..i, 1-x^(2*j-1))) . - Michael Somos, Aug 18 2006
G.f.: x^2*(1 - G(0) )/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 10 2019

A292445 Expansion of a q-series used by Ramanujan in his Lost Notebook.

Original entry on oeis.org

1, 4, 10, 22, 44, 82, 145, 248, 410, 658, 1036, 1598, 2420, 3614, 5322, 7738, 11132, 15850, 22353, 31260, 43366, 59708, 81650, 110932, 149788, 201112, 268562, 356790, 471732, 620834, 813480, 1061496, 1379626, 1786282, 2304440, 2962566, 3795921, 4848160

Offset: 0

Michael Somos, Sep 16 2017

nonn

Similar to A292420 but with a=1.

			G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 44*x^4 + 82*x^5 + 145*x^6 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A022567, A053253, A256209, A279715, A292420.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 / QPochhammer[ x]^2 Sum[ x^k / Product[ 1 - x^(2 i + 1), {i, 0, k}], {k, 0, n}], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2 * sum(k=0, n, x^k / prod(i=0, k, 1 - x^(2*i+1), 1 + A/x^k)), n))};

a(n) = 2 * A256209(n) - A279715(n).

G.f. is the product of the g.f. of A022567 and A053253.

A294599 Expansion of 1/(Sum_{i>=0} q^(2i(i+1))/Product_{j=0..i} (1 - q^(2*j+1))^2).

Original entry on oeis.org

1, -2, 1, 0, -1, 2, -1, -2, 5, -4, -2, 10, -13, 4, 16, -32, 24, 14, -62, 76, -17, -100, 185, -126, -108, 382, -426, 36, 655, -1098, 650, 798, -2352, 2402, 115, -4186, 6441, -3234, -5612, 14296, -13307, -2750, 26556, -37524, 15220, 38448, -86366, 72836, 28545, -166734, 216788, -65702, -257380

Offset: 0

Ilya Gutkovskiy, Nov 03 2017

sign

Convolution inverse of the 3rd-order mock theta function omega(q) (A053253).

Eric Weisstein's World of Mathematics, Mock Theta Function

Cf. A053253, A294407, A294408, A294598, A294600.

nmax = 52; CoefficientList[Series[1/Sum[q^(2 i (i + 1))/Product[(1 - q^(2 j + 1))^2, {j, 0, i}], {i, 0, nmax}], {q, 0, nmax}], q]
nmax = 52; CoefficientList[Series[1/Sum[q^i/Product[1 - q^(2 j + 1), {j, 0, i}], {i, 0, nmax}], {q, 0, nmax}], q]

G.f.: 1/(Sum_{i>=0} q^(2*i*(i+1))/Product_{j=0..i} (1 - q^(2*j+1))^2).

G.f.: 1/(Sum_{i>=0} q^i/Product_{j=0..i} (1 - q^(2*j+1))).

A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 6, 11, 17, 27, 38, 55, 76, 103, 136, 182, 235, 303, 385, 489, 612, 766, 945, 1166, 1428, 1742, 2111, 2557, 3072, 3686, 4401, 5246, 6223, 7371, 8692, 10236, 12014, 14074, 16435, 19171, 22292, 25884, 29981, 34677, 40017, 46122, 53038, 60920

Offset: 0

Jonathan Vos Post, Aug 28 2005

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000039, A000019, A053250, A053251, A053252, A053253, A053254, A053255, A064053.

nmax = 200; f[q_, s_] := Sum[q^(n^2)/Product[1 + q^k, {k, n}]^2, {n, 0, s}]; A000039:= CoefficientList[Series[f[q, nmax], {q, 0, nmax}], q][[1 ;; -1 ;; 2]]; Table[Sum[Abs[A000039[[k]]], {k,1,n}], {n,1,51}] (* G. C. Greubel, Feb 18 2018 *)

a(n) = Sum_{k=0..n} abs(A000039(k)). [corrected by Joerg Arndt, Feb 25 2018]

a(n) ~ sqrt(3/2) * exp(sqrt(n/3)*Pi) / Pi. - Vaclav Kotesovec, Jun 12 2019

cp's OEIS Frontend

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

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A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

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A097196 Expansion of psi(x^3)^2 / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions.

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A095913 Number of plasma partitions of 2n-1.

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A292445 Expansion of a q-series used by Ramanujan in his Lost Notebook.

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A294599 Expansion of 1/(Sum_{i>=0} q^(2*i*(i+1))/Product_{j=0..i} (1 - q^(2*j+1))^2).

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A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

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A294599 Expansion of 1/(Sum_{i>=0} q^(2i(i+1))/Product_{j=0..i} (1 - q^(2*j+1))^2).