A067567 -id:A067567 - cp's OEIS frontend

A066897 Total number of odd parts in all partitions of n.

1, 2, 5, 8, 15, 24, 39, 58, 90, 130, 190, 268, 379, 522, 722, 974, 1317, 1754, 2330, 3058, 4010, 5200, 6731, 8642, 11068, 14076, 17864, 22528, 28347, 35490, 44320, 55100, 68355, 84450, 104111, 127898, 156779, 191574, 233625, 284070, 344745, 417292, 504151

Offset: 1

Naohiro Nomoto, Jan 24 2002

Also sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n (Cf. A206563). - Omar E. Pol, Feb 12 2012
Column 1 of A206563. - Omar E. Pol, Feb 15 2012
Suppose that p=[p(1),p(2),p(3),...] is a partition of n with parts in nonincreasing order.  Let f(p) = p(1) - p(2) + p(3) - ... be the alternating sum of parts of p and let F(n) = sum of alternating sums of all partitions of n.  Conjecture: F(n) = A066897(n) for n >= 1. - Clark Kimberling, May 17 2019
From Omar E. Pol, Apr 02 2023: (Start)
Convolution of A000041 and A001227.
Convolution of A002865 and A060831.
a(n) is also the total number of odd divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned odd divisors are also all odd parts of all partitions of n. (End)
a(n) is odd iff n is a term of A067567 (proof: n*p(n) = the sum of the parts in all the partitions of n == the number of odd parts in all partitions of n (mod 2). Hence the number of odd parts in all partitions of n is odd iff n*p(n) is odd, equivalently, iff both n and p(n) are odd). - Peter Bala, Jan 11 2025

			a(4) = 8 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], we have a total of 0+2+0+2+4=8 odd parts.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A000041, A001227, A001620, A002865, A006128, A060831, A066898, A066966, A066967, A103919, A206563, A207381, A207382, A209423, A338156.

a066897 = p 0 1 where
   p o _             0 = o
   p o k m | m < k     = 0
           | otherwise = p (o + mod k 2) k (m - k) + p o (k + 1) m
-- Reinhard Zumkeller, Mar 09 2012

a066897 = length . filter odd . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

g:=sum(x^(2*j-1)/(1-x^(2*j-1)),j=1..70)/product(1-x^j,j=1..70): gser:=series(g,x=0,45): seq(coeff(gser,x^n),n=1..44);
# Emeric Deutsch, Mar 13 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
o[n_] := Sum[f[n, i], {i, 1, n, 2}]
e[n_] := Sum[f[n, i], {i, 2, n, 2}]
Table[o[n], {n, 1, 45}]  (* A066897 *)
Table[e[n], {n, 1, 45}]  (* A066898 *)
%% - %                   (* A209423 *)
(* Clark Kimberling, Mar 08 2012 *)
b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 26 2015, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A001227(k)=number of odd divisors of k and numbpart() is A000041. - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k=0..n} k*A103919(n,k). - Emeric Deutsch, Mar 13 2006
G.f.: Sum_{j>=1}(x^(2j-1)/(1-x^(2j-1)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Mar 13 2006
a(n) = A066898(n) + A209423(n) = A006128(n) - A066898(n). [Reinhard Zumkeller, Mar 09 2012]
a(n) = A207381(n) - A207382(n). - Omar E. Pol, Mar 11 2012
a(n) = (A006128(n) + A209423(n))/2. - Vaclav Kotesovec, May 25 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(24*n/Pi^2)) / (8*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

More terms from Vladeta Jovovic, Jan 26 2002

A067589 Numbers k such that A067588(k) is an odd number.

Original entry on oeis.org

1, 5, 7, 15, 35, 51, 57, 77, 117, 145, 155, 187, 247, 287, 301, 345, 425, 477, 495, 551, 651, 715, 737, 805, 925, 1001, 1027, 1107, 1247, 1335, 1365, 1457, 1617, 1717, 1751, 1855, 2035, 2147, 2185, 2301, 2501, 2625, 2667, 2795, 3015, 3151, 3197, 3337

Offset: 1

Naohiro Nomoto, Jan 31 2002

The terms are exactly the odd pentagonal numbers; that is, they are all the odd numbers of the form k*(3*k-1)/2 where k is an integer. - James Sellers, Jun 09 2007
Apparently groups of two odd pentagonal numbers (A000326, A014632) followed by two odd 2nd pentagonal numbers (A005449), which leads to the conjectured generating function x*(x^2+4*x+1)*(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(1-x)^3). - R. J. Mathar, Jul 26 2009
Odd generalized pentagonal numbers. - Omar E. Pol, Aug 19 2011
From Peter Bala, Jan 10 2025: (Start)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} x^(2*n+1)/(Product_{k = 1..2*n+1} 1 + x^(2*k+1)) = x + x^5 - x^7 - x^15 + x^35 + x^51 - x^57 - x^77 + + - - ... (follows from Berndt et al., Theorem 3.3). Cf. A193828.
For positive integer m, define b_m(n) = Sum_{k = 1..n} k^(2*m+1)*A000009(k)*A000009(n-k). We conjecture that
i) for odd n, b(n)/ n is an integer
ii) b(2*n)/n is an integer, which is odd iff n is a member of this sequence.
Cf. A067567. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

Cf. A067567, A067588, A000009.
Cf. A001318, A193828.
Cf. A014493, A128880, A154293.

With[{nn=50},Sort[Select[Table[(n(3n-1))/2,{n,-nn,nn}],OddQ]]] (* Harvey P. Dale, Feb 16 2014 *)

Sum_{n>=1} 1/a(n) = Pi/2. - Amiram Eldar, Aug 18 2022

Corrected by T. D. Noe, Oct 25 2006

A194798 Numbers n having the same parity as the number of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 13, 17, 22, 23, 26, 28, 29, 30, 33, 34, 35, 37, 39, 40, 41, 42, 43, 46, 49, 50, 51, 53, 58, 61, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 77, 78, 80, 81, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 96, 98, 99, 100, 105, 106, 107, 108, 110, 111

Offset: 1

Omar E. Pol, Jan 29 2012

Odd positive integers with an odd number of partitions and even positive integers with an even number of partitions. - Omar E. Pol, Mar 17 2012

Union of A067567 and A127219. Note that the union of A163096 and A163097 gives A209920 and the union of A209920 and this sequence gives A001477. - Omar E. Pol, Mar 22 2012

			10 is in the sequence because the number of partitions of 10 is equal to 42 and both 10 and 42 have the same parity.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659, A209920.

with(combinat):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while irem(k+numbpart(k), 2)=1 do od; k
    end:
seq(a(n), n=1..80); # Alois P. Heinz, Mar 16 2012

Select[Range[200], Mod[PartitionsP[#] - #, 2] == 0 &] (* T. D. Noe, Mar 16 2012 *)

More terms from Alois P. Heinz, Mar 16 2012

A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 1, 0, 3, 2, 7, 6, 15, 16, 32, 36, 62, 74, 117, 142, 214, 264, 377, 468, 648, 806, 1090, 1354, 1791, 2224, 2894, 3580, 4598, 5670, 7193, 8838, 11102, 13588, 16925, 20632, 25501, 30972, 38021, 46000, 56135, 67668, 82119, 98642, 119115, 142592, 171412, 204520

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Conjecture: a(n) is odd iff n is a term of A067567. - Peter Bala, Jan 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A001227, A006128, A067567, A209423, A066898, A305121, A066897.

Cf. A116676, A305124.

nmax = 60; CoefficientList[Series[Sum[x^(2*k-1)/(1+x^(2*k-1)), {k, 1, nmax}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

For n > 0, a(n) = A209423(n) - A305121(n).

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)).

A306903 Sum over all partitions of n of the bitwise XOR of the parts.

Original entry on oeis.org

0, 1, 2, 7, 8, 19, 26, 61, 70, 126, 146, 270, 308, 519, 604, 1054, 1222, 1929, 2208, 3454, 3930, 5862, 6576, 9833, 11102, 16052, 17904, 25752, 28764, 40479, 44830, 62988, 70188, 97151, 107662, 148141, 164710, 223783, 247380, 334035, 370406, 495313, 547000

Offset: 0

Alois P. Heinz, Mar 15 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A006906, A066186, A067567, A306884, A306895, A306901, A306902, A306925.

b:= proc(n, i, r) option remember; `if`(i<1, 0, (t->
      `if`(i b(n$2, 0):
seq(a(n), n=0..45);

a(n) is odd <=> n in { A067567 }.

A209920 Numbers n having distinct parity as the number of partitions of n.

Original entry on oeis.org

0, 4, 6, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 24, 25, 27, 31, 32, 36, 38, 44, 45, 47, 48, 52, 54, 55, 56, 57, 59, 60, 65, 68, 72, 75, 76, 79, 82, 88, 90, 92, 97, 101, 102, 103, 104, 109, 113, 114, 117, 118, 125, 129, 131, 132, 133, 134, 135, 137, 138, 140

Offset: 1

Omar E. Pol, Mar 16 2012

nonn

Odd positive integers with an even number of partitions and nonnegative even integers with an odd number of partitions. Union of A163097 and A163096. Note that the union of A067567 and A127219 gives A194798 and the union of A194798 and this sequence gives A001477.

			4 is in the sequence because the number of partitions of 4 is equal to 5 and the parity of 4 is distinct to the parity of 5 because 4 is even and 5 is odd.
9 is in the sequence because the number of partitions of 9 is equal to 30 and the parity of 9 is distinct to the parity of 30 because 9 is odd and 30 is even.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Complement of A194798.

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659.

A209658 Partition numbers p(n) having the same parity as n.

Original entry on oeis.org

1, 2, 3, 7, 15, 22, 42, 101, 297, 1002, 1255, 2436, 3718, 4565, 5604, 10143, 12310, 14883, 21637, 31185, 37338, 44583, 53174, 63261, 105558, 173525, 204226, 239943, 329931, 715220, 1121505, 1300156, 1505499, 1741630, 2323520, 2679689, 3554345

Offset: 1

Omar E. Pol, Mar 22 2012

nonn

Union of A154795 and A154798. The union of A209659 and this sequence gives A000041.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A194798, A209659, A209920.

A209659 Partition numbers p(n) having opposite parity of n.

Original entry on oeis.org

1, 5, 11, 30, 56, 77, 135, 176, 231, 385, 490, 627, 792, 1575, 1958, 3010, 6842, 8349, 17977, 26015, 75175, 89134, 124754, 147273, 281589, 386155, 451276, 526823, 614154, 831820, 966467, 2012558, 3087735, 5392783, 8118264, 9289091, 13848650

Offset: 1

Omar E. Pol, Mar 22 2012

nonn

Union of A154797 and A154796. The union of this sequence and A209658 gives A000041.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A194798, A209658, A209920.

A111354 Numbers n such that the numerator of Sum_{i=1..n} (1/i^2), in reduced form, is prime.

Original entry on oeis.org

2, 7, 13, 19, 121, 188, 252, 368, 605, 745, 1085, 1127, 1406, 1743, 1774, 2042, 2087, 2936, 3196, 3207, 3457, 4045, 7584, 10307, 12603, 12632, 14438, 14526, 14641, 15662, 15950, 16261, 18084, 18937, 19676, 40984, 45531, 46009, 48292, 48590

Offset: 1

Ryan Propper, Nov 05 2005

nonn

Numbers n such that A007406(n) is prime.
Some of the larger entries may only correspond to probable primes.
A007406(n) are the Wolstenholme numbers: numerator of Sum 1/k^2, k = 1..n. Primes in A007406(n) are listed in A123751(n) = A007406(a(n)) = {5,266681,40799043101,86364397717734821,...}.
For prime p>3, Wolstenholme's theorem says that p divides A007406(p-1). Hence n+1 cannot be prime for any n>2 in this sequence. - 12 more terms from T. D. Noe, Nov 11 2005
No other n<50000. All n<=1406 yield provable primes. - T. D. Noe, Mar 08 2006

			A007406(n) begins {1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141,...}.
Thus a(1) = 2 because A007406(2) = 5 is prime but A007406(1) = 1 is not prime.
a(2) = 7 because A007406(7) = 266681 is prime but all A007406(k) are composite for 2 < k < 7.

Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Generalized cosecant numbers and trigonometric inverse power sums, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1 (2018), 70-109.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme Number

Cf. A007406 (numerator of Sum_{i=1..n} (1/i^2)).

Cf. A123751, A001008, A007407, A067567, A056903.

s = 0; Do[s += 1/n^2; If[PrimeQ[Numerator[s]], Print[n]], {n, 1, 10^4}]
Module[{nn=10400,t},t=Accumulate[1/Range[nn]^2];Select[Thread[{Range[nn],Numerator[t]}],PrimeQ[#[[2]]]&]][[;;,1]] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, May 18 2025 *)

12 more terms from T. D. Noe, Nov 11 2005
More terms from T. D. Noe, Mar 08 2006
Additional comments from Alexander Adamchuk, Oct 11 2006
Edited by N. J. A. Sloane, Nov 11 2006

A123751 Primes in A007406.

Original entry on oeis.org

5, 266681, 40799043101, 86364397717734821, 36190908596780862323291147613117849902036356128879432564211412588793094572280300268379995976006474252029, 334279880945246012373031736295774418479420559664800307123320901500922509788908032831003901108510816091067151027837158805812525361841612048446489305085140033

Offset: 1

Alexander Adamchuk, Oct 11 2006

nonn

A007406 lists the Wolstenholme numbers.

Numbers k such that A007406(k) is prime are listed in A111354.

			A007406 begins {1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, ...}.
Thus a(1) = 5 because A007406(2) = 5 is prime but A007406(1) = 1 is not prime.
a(2) = 266681 because A007406(7) = 266681 is prime but all A007406(k) are composite for 2 < k < 7.

Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Generalized cosecant numbers and trigonometric inverse power sums, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1 (2018), 70-109.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
Eric Weisstein's World of Mathematics, Wolstenholme Number

Cf. A111354, A007406, A001008, A007407, A067567, A056903.

Do[f=Numerator[Sum[1/i^2,{i,1,n}]]; If[PrimeQ[f],Print[{n,f}]],{n,1,250}]

a(n) = A007406(A111354(n)).

cp's OEIS Frontend

A066897 Total number of odd parts in all partitions of n.

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A067589 Numbers k such that A067588(k) is an odd number.

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A194798 Numbers n having the same parity as the number of partitions of n.

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A305123 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} 1/(1-x^k).

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A306903 Sum over all partitions of n of the bitwise XOR of the parts.

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A209920 Numbers n having distinct parity as the number of partitions of n.

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A209658 Partition numbers p(n) having the same parity as n.

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A209659 Partition numbers p(n) having opposite parity of n.

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A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).