A035359 -id:A035359 - cp's OEIS frontend

A051044 Odd values of the PartitionsQ function A000009.

1, 1, 1, 3, 5, 15, 27, 89, 165, 585, 1113, 4097, 7917, 29927, 58499, 225585, 444793, 1741521, 3457027, 13699699, 27342421, 109420549, 219358315, 884987529, 1780751883, 7233519619, 14600965705, 59656252987, 120742510607, 495811828759, 1005862035461

Offset: 0

Eric W. Weisstein

nonn

A000009(n) is odd iff n is of the form k*(3*k - 1)/2 or k*(3*k + 1)/2. - Jonathan Vos Post, Jun 18 2005
Eric W. Weisstein comments: "The values of n for which A000009(n) is prime are 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, ... (A035359). These values correspond to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (A051005). It is not known if a(n) is infinitely often prime, but Gordon and Ono (1997) proved that it is 'almost always' divisible by any given power of 2 (1997)."
Semiprime values begin: a(5) = 15 = 3 * 5, a(11) = 4097 = 17 * 241, a(20) = 27342421 = 389 * 70289, a(24) = 1780751883 = 3 * 593583961, a(28) = 120742510607 = 31 * 3894919697. - Jonathan Vos Post, Jun 18 2005

Alois P. Heinz, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Partition Function Q Congruences

Cf. A000009, A001318, A035359, A051005, A118303.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> b((m->m*(3*m-1)/2)(ceil(-n*(-1)^n/2))):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2021

PartitionsQ /@ Table[n*((n + 1)/6), {n, Select[Range[50], Mod[#, 3] != 1 & ]}] (* Jean-François Alcover, Oct 31 2012, after Reinhard Zumkeller *)

a(n) = A000009(A001318(n)). - Reinhard Zumkeller, Apr 22 2006

Missing initial 1 inserted by Sean A. Irvine, Aug 23 2021

A330996 Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 44, 46, 48, 51, 53, 56, 59, 61, 64, 68, 71, 74, 78, 81, 85, 89, 93, 97, 101, 106, 111, 115, 120, 126

Offset: 0

Gus Wiseman, Jan 08 2020

nonn

Conjecture: This sequence is nondecreasing. More generally, the rational sequence A000041(n)/A000009(n) is nondecreasing for n > 5.

The numerators are A330994.
The denominators are A330995.
The same for factorizations is A331048.
Cf. A000009, A000041, A001055, A003238, A035359, A045778, A046063, A331022.

Table[Round[PartitionsP[n]/PartitionsQ[n]],{n,0,100}]

A285216 Indices of primes in A000219.

Original entry on oeis.org

2, 4, 11, 30, 32, 40, 50, 85, 100, 237, 381, 733, 805, 882, 1015, 1650, 2439, 3163, 3335, 3506, 3675, 4152, 4446, 4576, 5010, 5101, 6045, 6760, 7412, 8178, 8562, 10026, 10527, 10888, 12406, 12693, 13479, 16109, 16978, 17962, 20696, 22483, 25383, 31458, 38956

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			11 is in the sequence because A000219(11) = 859 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..82

Cf. A000219, A035359, A046063.

A000219 = Cases[Import["https://oeis.org/A000219/b000219.txt", "Table"], {, }][[All, 2]];
Flatten[Position[A000219, ?PrimeQ]] - 1 (* _Jean-François Alcover, Sep 17 2020, as of todate, this only gives 27 terms *)

A285217 Indices of primes in A000712.

Original entry on oeis.org

1, 2, 70, 106, 330, 366, 370, 546, 836, 1370, 1870, 2126, 2616, 4240, 4836, 4956, 9520, 10896, 11446, 14250, 15836, 16170, 18040, 18566, 26516, 28676, 37060, 40546, 40760, 46850, 52060, 57176, 67726, 74776, 78460, 90810, 98216, 108870, 115400, 115990, 123930

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			70 is in the sequence because A000712(70) = 7592053897 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..69

Cf. A000712, A035359, A046063, A265835.

A285222 Indices of primes in A022629.

Original entry on oeis.org

2, 3, 4, 7, 12, 39, 49, 56, 76, 91, 216, 221, 235, 272, 277, 331, 345, 362, 385, 655, 686, 749, 1023, 1591, 1742, 1825, 1836, 1911, 2179, 2511, 2638, 2951, 3004, 3327, 3692, 4743, 4868, 5546, 5658, 5997, 6032, 7867, 8261, 8620, 8811, 13762, 13950, 14150

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			12 is in the sequence because A022629(12) = 463 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..245

Cf. A022629, A035359.

A285223 Indices of primes in A026007.

Original entry on oeis.org

2, 3, 8, 15, 18, 34, 106, 169, 404, 411, 421, 501, 504, 558, 586, 616, 822, 1051, 1573, 1791, 1894, 1935, 2178, 2536, 2823, 2908, 5002, 5037, 5383, 5826, 6418, 8778, 8968, 9816, 9928, 11512, 12011, 12052, 12524, 13117, 13268, 13640, 15614, 16657, 17938, 19869

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			8 is in the sequence because A026007(8) = 83 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..109

Cf. A026007, A035359.

A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319237 in lacking 300.

The version for strict integer partitions is A001318.
The version for integer partitions is A052002.
The version for set partitions appears to be A032766.
The non-strict version is A331050.
The version for primes (instead of odds) is A331201.
The even version is A331231.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A035359, A045780, A318286, A328966, A330991, A330997, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],OddQ[Length[strfacs[#]]]&]

A285219 Indices of primes in A003105.

Original entry on oeis.org

5, 6, 7, 8, 9, 11, 13, 22, 28, 29, 31, 38, 47, 53, 56, 59, 63, 64, 76, 85, 88, 91, 110, 111, 124, 135, 165, 202, 210, 214, 234, 243, 256, 262, 280, 322, 335, 346, 438, 443, 458, 463, 508, 580, 590, 696, 790, 865, 903, 951, 993, 996, 1004, 1163, 1338, 1396

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			28 is in the sequence because A003105(28) = 47 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..373

Cf. A003105, A035359, A046063, A265835.

A295291 Indices of primes in sequence A000700.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 24, 29, 32, 35, 38, 39, 51, 56, 61, 77, 82, 88, 90, 91, 92, 107, 118, 119, 123, 139, 148, 161, 162, 166, 185, 189, 190, 194, 208, 214, 333, 346, 355, 373, 401, 402, 493, 543, 567, 578, 603, 629, 653, 665, 666, 678

Offset: 1

Vaclav Kotesovec, Nov 19 2017

nonn

			51 is in the sequence because A000700(51) = 107 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1755

Cf. A000700, A035359, A046063, A265835, A285217.

A254269 Largest prime factor of the strict partition numbers Q(n) (partitions into distinct parts, A000009).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 3, 2, 5, 3, 5, 3, 11, 3, 2, 19, 23, 3, 2, 19, 89, 13, 61, 71, 11, 3, 37, 2, 37, 17, 13, 7, 2, 13, 167, 19, 3, 491, 53, 7, 31, 23, 227, 2, 3, 37, 97, 17, 59, 241, 79, 5, 953, 1063, 1777, 29, 367, 17, 17, 3019, 181, 29, 4111

Offset: 0

Jean-François Alcover, Jan 27 2015

nonn

A035359 is the sequence of indices n such that a(n) = A000009(n).

Jean-François Alcover and Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 1001 terms from Jean-François Alcover)
Eric Weisstein's MathWorld, Partition Function Q

Cf. A000009, A035359, A071963.

Table[FactorInteger[PartitionsQ[n]][[-1, 1]], {n, 0, 100}]

cp's OEIS Frontend

A051044 Odd values of the PartitionsQ function A000009.

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A330996 Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).

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A285216 Indices of primes in A000219.

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A285217 Indices of primes in A000712.

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A285222 Indices of primes in A022629.

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A285223 Indices of primes in A026007.

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A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

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A285219 Indices of primes in A003105.

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A295291 Indices of primes in sequence A000700.

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A254269 Largest prime factor of the strict partition numbers Q(n) (partitions into distinct parts, A000009).

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