A259825 -id:A259825 - cp's OEIS frontend

A321440 Number of partitions of n into consecutive parts, all singletons except the largest.

1, 1, 2, 3, 3, 4, 5, 4, 5, 7, 5, 6, 8, 5, 8, 10, 5, 8, 10, 7, 10, 11, 7, 8, 13, 9, 9, 14, 7, 12, 15, 6, 12, 13, 11, 15, 14, 8, 10, 19, 10, 12, 18, 8, 16, 19, 9, 12, 17, 14, 16, 16, 10, 15, 21, 15, 14, 20, 7, 16, 25, 7, 20, 21, 14, 18, 18, 14, 12, 26, 16, 17

Offset: 0

Allan C. Wechsler, Nov 09 2018

nonn

Number of representations of n as the difference of two distinct triangular numbers, plus any multiple of the order of the larger triangular number.
From Jeremy Lovejoy, Nov 10 2022: (Start)
For n > 0, a(n) is also equal to the Hurwitz class number H(8n-1).
a(n) is also equal to the number of partitions y of n having no repeated even parts and smallest part odd, counted according to the weight w(y) = (-1)^(the number of even parts)*(the number of occurrences of the smallest part). For example, the partitions of 6 having no repeated even parts and smallest part odd are [5,1], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,1,1,1,1], and [1,1,1,1,1,1], which are counted with weights 1,-2,2,-1,3,-4, and 6, giving a(6) = 1-2+2-1+3-4+6 = 5. (End)

			Here are the derivations of the terms given. Partitions are listed as strings of digits.
n = 0: (empty partition)
n = 1: 1
n = 2: 11, 2
n = 3: 111, 12, 3
n = 4: 1111, 22, 4
n = 5: 11111, 122, 23, 5
n = 6: 111111, 123, 222, 33, 6
n = 7: 1111111, 1222, 34, 7
n = 8: 11111111, 2222, 233, 44, 8
n = 9: 111111111, 12222, 1233, 234, 333, 45, 9
n = 10: 1111111111, 1234, 22222, 55, (10)

Chai Wah Wu, Table of n, a(n) for n = 0..10000
C. Alfes, K. Bringmann, and J. Lovejoy, Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 3, 385-406.
Dandan Chen and Rong Chen, Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions, arXiv:2107.04809 [math.NT], 2021.
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

See comment by Emeric Deutsch at A001227 (partitions into consecutive parts, all singletons); the partitions considered in the present sequence are a superset of those described by Deutsch.

Cf. A259825, A321441, A321443.

from sympy.utilities.iterables import partitions
def A321440(n):
    return 1 if n == 0 else sum(1 for s,p in partitions(n,size=True) if len(p)-1 == max(p)-min(p) == s-p[max(p)]) # Chai Wah Wu, Nov 09 2018

from _future_ import division
def A321440(n): # a faster program based on the characterization in the comments
    if n == 0:
        return 1
    c = 0
    for i in range(n):
        mi = i*(i+1)//2 + n
        for j in range(i+1,n+1):
            k = mi - j*(j+1)//2
            if k < 0:
                break
            if not k % j:
                c += 1
    return c # Chai Wah Wu, Nov 09 2018

From Jeremy Lovejoy, Nov 10 2022: (Start)
G.f.: 1 + Sum_{n>=0} x^(n+1)*Product_{k=1..n} (1-x^(2*k))/Product_{k=1..n+1} (1-x^(2*k-1)).
G.f.: 1 + Sum_{n>=1} (-1)^(n+1)*x^(n^2)/((1-x^(2*n-1))*Product_{k=1..n} (1-x^(2*k-1))). (End)

More terms from Chai Wah Wu, Nov 09 2018

A117003 a(n) = sigma(n) + A079667(n).

Original entry on oeis.org

1, 4, 6, 10, 10, 18, 14, 24, 21, 30, 22, 44, 26, 42, 40, 52, 34, 66, 38, 70, 56, 66, 46, 100, 55, 78, 72, 98, 58, 122, 62, 112, 88, 102, 84, 156, 74, 114, 104, 156, 82, 168, 86, 154, 138, 138, 94, 216, 105, 170, 136, 182, 106, 216, 132, 212, 152, 174, 118, 294, 122, 186, 186

Offset: 1

N. J. A. Sloane, Apr 15 2006

H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 322.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A013661, A079667, A117004, A259825.

a[n_] := DivisorSum[n, Max[#, n/#] &]; Array[a, 100] (* Amiram Eldar, Jan 12 2025 *)

{a(n) = sumdiv(n, d, max(d, n/d))} \\ Seiichi Manyama, Dec 27 2017

a(n) = Sum_{d|n} max(d, n/d). - Seiichi Manyama, Dec 27 2017
a(n) = Sum_{k in Z} H(4*n-k^2) where H() is the Hurwitz class number. - Seiichi Manyama, Jan 06 2018
G.f.: Sum_{n >= 1} x^(n^2)*(n + 2*x^n - n*x^(2*n))/(1 - x^n)^2 = x + 4*x^2 + 6*x^3 + 10*x^4 + 10*x^5 + .... Cf. A117004. - Peter Bala, Jan 19 2021
Sum_{k=1..n} a(k) ~ zeta(2) * n^2. - Amiram Eldar, Jan 12 2025

A297793 a(n) = Sum_{d|n} min(d, n/d)^3.

Original entry on oeis.org

1, 2, 2, 10, 2, 18, 2, 18, 29, 18, 2, 72, 2, 18, 56, 82, 2, 72, 2, 146, 56, 18, 2, 200, 127, 18, 56, 146, 2, 322, 2, 146, 56, 18, 252, 416, 2, 18, 56, 396, 2, 504, 2, 146, 306, 18, 2, 632, 345, 268, 56, 146, 2, 504, 252, 832, 56, 18, 2, 882, 2, 18, 742, 658, 252

Offset: 1

Seiichi Manyama, Jan 06 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Henri Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080).

Sum_{d|n} min(d, n/d)^k: A117004 (k=1), A297792 (k=2), this sequence (k=3), A297794 (k=4), A297795 (k=5).

Cf. A259825.

a[n_] := DivisorSum[n, Min[#, n/#]^3 &]; Array[a, 65] (* Amiram Eldar, Oct 04 2023*)

PARI
```
{a(n) = sumdiv(n, d, min(d, n/d)^3)}
```

a(n) = - Sum_{k in Z} (k^2-n)*H(4*n-k^2) where H() is the Hurwitz class number.

A297795 a(n) = Sum_{d|n} min(d, n/d)^5.

Original entry on oeis.org

1, 2, 2, 34, 2, 66, 2, 66, 245, 66, 2, 552, 2, 66, 488, 1090, 2, 552, 2, 2114, 488, 66, 2, 2600, 3127, 66, 488, 2114, 2, 6802, 2, 2114, 488, 66, 6252, 10376, 2, 66, 488, 8364, 2, 16104, 2, 2114, 6738, 66, 2, 18152, 16809, 6316, 488, 2114, 2, 16104, 6252, 35728, 488

Offset: 1

Seiichi Manyama, Jan 06 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285. MR0382192 (52 #3080)

Sum_{d|n} min(d, n/d)^k: A117004 (k=1), A297792 (k=2), A297793 (k=3), A297794 (k=4), this sequence (k=5).

Cf. A259825.

f[n_] := Block[{d = Divisors@n}, Plus @@ (Min[#, n/#]^5 & /@ d)]; Array[f, 57] (* Robert G. Wilson v, Jan 06 2018 *)

PARI
```
{a(n) = sumdiv(n, d, min(d, n/d)^5)}
```

a(n) = - Sum_{k in Z} (k^4-3*n*k^2+n^2)*H(4*n-k^2) where H() is the Hurwitz class number.

A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

9, 44, 234, 664, 2628, 4354, 9774, 13660, 24264, 48690, 59488, 101194, 137718, 158884, 207504, 297594, 410580, 453778, 601324, 715608, 777814, 985840, 1143324, 1409670, 1825054, 2060298, 2185144, 2449764, 2589730, 2885454, 4096384, 4495788, 5142294

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), this sequence (m=4), A297492 (m=6), A297493 (m=8), A297494 (m=10).

Cf. A259825.

Let b(n) = 2*n^3 - 3*n - 1.

a(n) = b(prime(n)).

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

Original entry on oeis.org

33, 308, 2874, 11528, 72060, 141218, 414918, 648260, 1394328, 3528690, 4608800, 9358298, 14113470, 17077148, 24378288, 39426858, 60555180, 69195410, 100714868, 127012680, 141942878, 194693840, 237229188, 313639470, 442561238, 520209690, 562658408, 655294428

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), this sequence (m=6), A297493 (m=8), A297494 (m=10).

Cf. A259825.

b(n) = 5*n^4 - 9*n^2 - 5*n - 1;
a(n) = b(prime(n)); \\ Michel Marcus, Jan 01 2018

Let b(n) = 5*n^4 - 9*n^2 - 5*n - 1.

a(n) = b(prime(n)).

A297493 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^8H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

129, 2444, 39714, 224664, 2214948, 5133114, 19734534, 34465980, 89757384, 286456170, 399954528, 969369474, 1620023118, 2055854724, 3207878544, 5850511794, 10003119540, 11817917898, 18893239884, 25249088088, 29012002734, 43064859120, 55130420604

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), this sequence (m=8), A297494 (m=10).

Cf. A259825.

lista(nn) = forprime(p=2, nn, print1(14*p^5-28*p^3-20*p^2-7*p-1, ", ")); \\ Altug Alkan, Jan 01 2018

Let b(n) = 14*n^5 - 28*n^3 - 20*n^2 - 7*n - 1.

a(n) = b(prime(n)).

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

513, 20708, 584874, 4714408, 72449100, 200562418, 1012788198, 1953009460, 6172747128, 24788658690, 37242612640, 107770200778, 198936710910, 265200653548, 449592659568, 931777815258, 1775665528380, 2155635964450, 3812897562148, 5368106367720, 6351988507678

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), A297493 (m=8), this sequence (m=10).

Cf. A000594, A259825, A295645, A297127.

Let b(n) = 42*n^6 - 90*n^4 - 75*n^3 - 35*n^2 - 9*n - 1.
a(n) = b(prime(n)) - tau(prime(n)) where tau(n)=A000594(n) is Ramanujan's tau function.
So tau(prime(n)) + 1 == -a(n) (mod prime(n)).

A306934 Coefficients of q-expansion of Eisenstein series G_{5/2}(tau) multiplied by 120.

Original entry on oeis.org

1, -10, 0, 0, -70, -48, 0, 0, -120, -250, 0, 0, -240, -240, 0, 0, -550, -480, 0, 0, -528, -480, 0, 0, -720, -1210, 0, 0, -960, -720, 0, 0, -1080, -1440, 0, 0, -1750, -1200, 0, 0, -1680, -1920, 0, 0, -1680, -1488, 0, 0, -2160, -3370, 0, 0, -2640, -1680, 0, 0, -2400, -3360, 0, 0, -2880, -2640

Offset: 0

N. J. A. Sloane, Mar 16 2019

sign

H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285.
X. Wang and D. Pei, Modular Forms with Integral and Half-Integral Weights, Science Press Beijing, Springer Berlin Heidelberg, 2012. x+432 pp.
D. Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50 (erroneously gives a(5) = -72).

Cf. A259825, A306935, A306936, A306937.

def a(n):
    if n==0: return 1
    if (n%4) not in [0,1]: return 0
    D = Integer(n).squarefree_part()
    f = Integer(sqrt(n/D))
    if (D%4) not in [0,1]: D, f = 4*D, f//2
    X = kronecker_character(D)
    s = sum([moebius(d)*X(d)*d*sigma(f/d, 3) for d in f.divisors()])
    return round((120*X.lfunction(100)(-1)*s).real()) # Robin Visser, Feb 24 2024

Corrected and more terms from Robin Visser, Feb 24 2024

A306937 Coefficients of q-expansion of Eisenstein series G_{11/2}(tau) multiplied by 132.

Original entry on oeis.org

-1, 0, 0, 44, 330, 0, 0, 4224, 7524, 0, 0, 30600, 23276, 0, 0, 130944

Offset: 0

N. J. A. Sloane, Mar 16 2019

D. Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Cf. A259825, A306934-A306936.

cp's OEIS Frontend

A321440 Number of partitions of n into consecutive parts, all singletons except the largest.

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A117003 a(n) = sigma(n) + A079667(n).

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A297793 a(n) = Sum_{d|n} min(d, n/d)^3.

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A297795 a(n) = Sum_{d|n} min(d, n/d)^5.

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A297491 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^4*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

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A297492 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^6*H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

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A297493 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^8*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

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A297494 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^10*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

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A306934 Coefficients of q-expansion of Eisenstein series G_{5/2}(tau) multiplied by 120.

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Author

A297491 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^4H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

A297492 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^6H(4*p-k^2) where H() is the Hurwitz class number and p is the n-th prime.

A297493 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^8H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.