A048272 -id:A048272 - cp's OEIS frontend

A015723 Number of parts in all partitions of n into distinct parts.

1, 1, 3, 3, 5, 8, 10, 13, 18, 25, 30, 40, 49, 63, 80, 98, 119, 149, 179, 218, 266, 318, 380, 455, 541, 640, 760, 895, 1050, 1234, 1442, 1679, 1960, 2272, 2635, 3052, 3520, 4054, 4669, 5359, 6142, 7035, 8037, 9170, 10460, 11896, 13517, 15349, 17394, 19691

Offset: 1

Clark Kimberling

nonn

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with a total of 1 + 2 + 2 + 3 = 8 parts, so a(6) = 8. - _Gus Wiseman_, May 09 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Arnold Knopfmacher, and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See s(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A006128, A008289, A079499, A067619, A186545.
Column k=1 of: A210485, A213177, A327622.
Row lengths of A325537.
Cf. A022629, A066186, A066189, A325504, A325505, A325506, A325513, A325515.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013

nn=50; Rest[CoefficientList[Series[D[Product[1+y x^i,{i,1,nn}],y]/.y->1,{x,0,nn}],x]]  (* Geoffrey Critzer, Oct 29 2012; fixed by Vaclav Kotesovec, Apr 16 2016 *)
q[n_, k_] := q[n, k] = If[nVaclav Kotesovec, Apr 16 2016 *)
Table[Length[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,50}] (* Gus Wiseman, May 09 2019 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},
   Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

N=66;  q='q+O('q^N); gf=sum(n=0,N, n*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf) /* Joerg Arndt, Oct 20 2012 */

G.f.: sum(k>=1, x^k/(1+x^k) ) * prod(m>=1, 1+x^m ). Convolution of A048272 and A000009. - Vladeta Jovovic, Nov 26 2002
G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005
a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - Mircea Merca, Feb 26 2014
a(n) = Sum_{k>=1} k*A008289(n,k). - Vaclav Kotesovec, Apr 16 2016
G.f.: -(-1; x)inf * (log(1-x) + psi_x(1 - log(-1)/log(x)))/(2*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - _Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, May 19 2018
For n > 0, a(n) = A116676(n) + A116680(n). - Vaclav Kotesovec, May 26 2018

Extended and corrected by Naohiro Nomoto, Feb 24 2002

A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

Original entry on oeis.org

1, -3, 10, -19, 26, -30, 50, -83, 91, -78, 122, -190, 170, -150, 260, -339, 290, -273, 362, -494, 500, -366, 530, -830, 651, -510, 820, -950, 842, -780, 962, -1363, 1220, -870, 1300, -1729, 1370, -1086, 1700, -2158, 1682, -1500, 1850, -2318, 2366, -1590, 2210, -3390, 2451, -1953, 2900, -3230, 2810, -2460, 3172

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Apart from signs, same as A064027.
Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A042968 Numbers not divisible by 4.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

N. J. A. Sloane, Dec 11 1999

Equivalently, numbers whose square part is odd. Cf. A028982. - Peter Munn, Jul 14 2020
More generally the sequence of numbers not divisible by some fixed integer m >= 2 is given by a(n,m) = 1 + n + floor(n/(m-1)). - Benoit Cloitre, Jul 11 2009
Also a(n,m) = floor((m*n-1)/(m-1)) [with offset 1]. - Gary Detlefs, May 14 2011
Numbers not having more even than odd divisors: A048272(a(n)) >= 0. - Reinhard Zumkeller, Jan 21 2012
Extending the comments of Benoit Cloitre (Jul 11 2009) and Gary Detlefs (May 14 2011), the g.f. is A(m,x) = (1-x^m) / ((1-x^(m-1))*(1-x)^2) where m >= 2 is fixed. - Werner Schulte, Apr 26 2018

			G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 6*x^4 + 7*x^5 + 9*x^6 + 10*x^7 + 11*x^8 + ... - _Michael Somos_, Jun 17 2018

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001651, A001935, A028982, A070048, A042965, A064680.
Cf. A071619 (partial sums); A008586 (complement).
Numbers that are congruent to {k0,k1,k2} mod 4: A004772, A004773, A042965, a(n).

a042968 = (`div` 3) . (subtract 1) . (* 4)
a042968_list = filter ((/= 0) . (`mod` 4)) [1..]
-- Reinhard Zumkeller, Sep 02 2012

[n+1+Floor(n/3): n in [0..80]]; // Vincenzo Librandi, Aug 03 2015

seq(n+floor((n-1)/3), n=1..80); # Muniru A Asiru, Feb 17 2019

Select[Table[n,{n,200}], Mod[#,4] != 0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
LinearRecurrence[{1,0,1,-1},{1,2,3,5},80]  (* or *) Drop[Range[110],{4,-1,4}] (* Harvey P. Dale, Jan 07 2023 *)

PARI
```
{a(n) = 1 + n + n\3};
```

def A042968(n): return n+(n-1)//3 # Chai Wah Wu, Apr 13 2025

[1+n+floor(n/3) for n in (0..80)] # G. C. Greubel, Feb 17 2019

a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = a(n-3) + 4, with a(1) = 1.
G.f.: x * (1+x) * (1+x^2) / ( (1+x+x^2)*(1-x)^2 ). - Michael Somos, Jan 12 2000
A064680(A064680(a(n))) = a(n). - Reinhard Zumkeller, Oct 19 2001
Nearest integer to (Sum_{k>n} 1/k^4)/(Sum_{k>n} 1/k^5). - Benoit Cloitre, Jun 12 2003
a(n) = n + 1 + floor(n/3). - Benoit Cloitre, Jul 11 2009
a(n) = floor((4*n+3)/3). - Gary Detlefs, May 14 2011
A214546(a(n)) >= 0 for n > 0. - Reinhard Zumkeller, Jul 20 2012
a(n) = 2*n - ceiling(2*n/3) + 1. - Arkadiusz Wesolowski, Sep 21 2012
Sum_{k=0..n} a(n) = A071619(n+1). - L. Edson Jeffery, Jul 30 2014
The g.f. A(x) satisfies x*A(x)^2 = (B(x)/x)^2 + (B(x)/x), where B(x) is the o.g.f. of A042965. - Peter Bala, Apr 12 2017
a(n) = (12*n + 6 + 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
Euler transform of length 4 sequence [2, 0, 1, -1]. - Michael Somos, Jun 17 2018
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jun 17 2018
E.g.f.: (2/3)*exp(x)*(1 + 2*x) + (1/9)*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)). - Stefano Spezia, Nov 16 2019
a(n) = (12*n + 6 + w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2. - Guenther Schrack, Jun 07 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/8. - Amiram Eldar, Dec 05 2021

Edited by Peter Munn, Nov 16 2019

I restored my original (1999) definition and offset, which in the intervening 21 years had been lost. - N. J. A. Sloane, Jun 12 2021

A030513 Numbers with 4 divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

Jeff Burch

Essentially the same as A007422.
Numbers which are either the product of two distinct primes (A006881) or the cube of a prime (A030078).
4*a(n) are the solutions to A048272(x) = Sum_{d|x} (-1)^d = 4. - Benoit Cloitre, Apr 14 2002
Since A119479(4)=3, there are never more than 3 consecutive integers in the sequence. Triples of consecutive integers start at 33, 85, 93, 141, 201, ... (A039833). No such triple contains a term of the form p^3. - Ivan Neretin, Feb 08 2016
Numbers that are equal to the product of their proper divisors (A007956) (proof in Sierpiński). - Bernard Schott, Apr 04 2022

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Cf. A000005, A007422, A007956, A030515, A035533, A048272, A119479.

Equals the disjoint union of A006881 and A030078.

[n: n in [1..200] | DivisorSigma(0, n) eq 4]; // Vincenzo Librandi, Jul 16 2015

Select[Range[200], DivisorSigma[0,#]==4&] (* Harvey P. Dale, Apr 06 2011 *)

is(n)=numdiv(n)==4 \\ Charles R Greathouse IV, May 18 2015

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A030513(n):
    def f(x): return int(n+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

{n : A000005(n) = 4}. - Juri-Stepan Gerasimov, Oct 10 2009

Incorrect comments removed by Charles R Greathouse IV, Mar 18 2010

A059851 a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).

Original entry on oeis.org

0, 1, 1, 3, 2, 4, 4, 6, 4, 7, 7, 9, 7, 9, 9, 13, 10, 12, 12, 14, 12, 16, 16, 18, 14, 17, 17, 21, 19, 21, 21, 23, 19, 23, 23, 27, 24, 26, 26, 30, 26, 28, 28, 30, 28, 34, 34, 36, 30, 33, 33, 37, 35, 37, 37, 41, 37, 41, 41, 43, 39, 41, 41, 47, 42, 46, 46, 48, 46, 50, 50, 52, 46, 48, 48

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

As n goes to infinity we have the asymptotic formula: a(n) ~ n * log(2).

More precisely, a(n) = n * log(2) + O(n^(131/416) * (log n)^(26947/8320)). - V Sai Prabhav, Jun 02 2025

			a(5) = 4 because floor(5) - floor(5/2) + floor(5/3) - floor(5/4) + floor(5/5) - floor(5/6) + ... = 5 - 2 + 1 - 1 + 1 - 0 + 0 - 0 + ... = 4.

T. D. Noe, Table of n, a(n) for n = 0..10000
V Sai Prabhav, Asymptotic Expansion of a(n)

Cf. A000005, A075997, A271860, A263086.
Partial sums of A048272.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), this sequence (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A059851:= func< n | (&+[Floor(n/j)*(-1)^(j-1): j in [1..n]]) >;
[A059851(n): n in [1..80]]; // G. C. Greubel, Jun 27 2024

for n from 0 to 200 do printf(`%d,`, sum((-1)^(i+1)*floor(n/i), i=1..n)) od:

f[list_, i_] := list[[i]]; nn = 200; a = Table[1, {n, 1, nn}]; b =
Table[If[OddQ[n], 1, -1], {n, 1, nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] // Accumulate (* Geoffrey Critzer, Mar 29 2015 *)
Table[Sum[Floor[n/k] - 2*Floor[n/(2*k)], {k, 1, n}], {n, 0, 100}] (* Vaclav Kotesovec, Dec 23 2020 *)

{ for (n=0, 10000, s=1; d=2; a=n; while ((f=floor(n/d)) > 0, a-=s*f; s=-s; d++); write("b059851.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 29 2009

from math import isqrt
def A059851(n): return ((t:=isqrt(m:=n>>1))**2<<1)-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

def A059851(n): return sum((n//j)*(-1)^(j-1) for j in range(1,n+1))
[A059851(n) for n in range(81)] # G. C. Greubel, Jun 27 2024

From Vladeta Jovovic, Oct 15 2002: (Start)
a(n) = A006218(n) - 2*A006218(floor(n/2)).
G.f.: 1/(1-x)*Sum_{n>=1} x^n/(1+x^n). (End)
a(n) = Sum_{n/2 < k < =n} d(k) - Sum_{1 < =k <= n/2} d(k), where d(k) = A000005(k). Also, a(n) = number of terms among {floor(n/k)}, 1<=k<=n, that are odd. - Leroy Quet, Jan 19 2006
From Ridouane Oudra, Aug 15 2019: (Start)
a(n) = Sum_{k=1..n} (floor(n/k) mod 2).
a(n) = (1/2)*(n + A271860(n)).
a(n) =  Sum_{k=1..n} round(n/(2*k)) - floor(n/(2*k)), where round(1/2) = 1. (End)
a(n) = 2*A263086(n) - 3*A006218(n). - Ridouane Oudra, Aug 17 2024

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Feb 27 2001

A001749 Primes multiplied by 4.

Original entry on oeis.org

8, 12, 20, 28, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964, 1004, 1028, 1052, 1076, 1084, 1108, 1124, 1132, 1172, 1228, 1244, 1252

Offset: 1

N. J. A. Sloane

Numbers k such that Sum_{d|k} (-1)^d = A048272(k) = 2. - Benoit Cloitre, Apr 14 2002

Paolo P. Lava, Table of n, a(n) for n = 1..20000

Cf. A000040, A003415, A048272.

a001749 = (* 4) . a000040  -- Reinhard Zumkeller, Feb 20 2012

[4*p: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 26 2014

4 * Prime[Range[66]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=4*prime(n) \\ Charles R Greathouse IV, Mar 25 2014

4 * primes(66) \\ Joerg Arndt, Mar 26 2014

A048298 a(n) = n if n=2^i for i >= 0, otherwise a(n) = 0.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Adam Kertesz

Expand x/(x-1) = Sum_{n >= 0} 1/x^n as Sum a(n) / (1+x^n).

Nim-binomial transform of the natural numbers. If {t(n)} is the Nim-binomial transform of {a(n)}, then t(n)=(S^n)a(0), where Sf(n) denotes the Nim-sum of f(n) and f(n+1); and S^n=S(S^(n-1)). - John W. Layman, Mar 06 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche and Jeffrey Shallit, The Ring of k-regular Sequences, II.
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.

A kind of inverse to A048272. Cf. A060147.
This is Guy Steele's sequence GS(5, 1) (see A135416).
Cf. A209229 (characteristic function of powers of 2).

a048298 n = a209229 n * n  -- Reinhard Zumkeller, Oct 17 2015

[n eq 2^Valuation(n,2) select n else 0: n in [0..120]]; // Vincenzo Librandi, improved by Bruno Berselli, Mar 27 2015

0, seq(op([2^n,0$(2^n-1)]), n=0..10); # Robert Israel, Mar 25 2015
a := n -> if n = 2^ilog2(n) then n else 0 fi: # Peter Luschny, Oct 03 2022

Table[n*Boole[Or[n == 1, First /@ FactorInteger@ n == {2}]], {n, 0, 120}] (* Michael De Vlieger, Mar 25 2015 *)
a[n_] := If[n == 2^IntegerExponent[n, 2], n, 0]; Array[a, 100, 0] (* Amiram Eldar, Oct 10 2023 *)

a(n)=direuler(p=1,n,if(p==2,1/(1-2*X),1))[n] /* Ralf Stephan, Mar 27 2015 */

a(n) = if(n == 0, 0, if(n == 1 << valuation(n, 2), n, 0)); \\ Amiram Eldar, Oct 10 2023

def A048298(n): return n if n and not(n&-n)^n else 0 # Chai Wah Wu, Dec 01 2022

Multiplicative with a(2^e)=2^e and a(p^e)=0 for p > 2. - Vladeta Jovovic, Jan 27 2002
Inverse mod 2 binomial transform of n. a(n) = sum{k=0..n, (-1)^A010060(n-k)*mod(C(n, k), 2)*k}. - Paul Barry, Jan 03 2005
If n=1 we have a(n)=1; if n=p is prime, then (-1)^(p+1)+a(p)=1, thus a(2)=2, and a(p)=0, if p>2. - Vladimir Shevelev, Jun 09 2009
Dirichlet g.f.: 2^s/(2^s-2). - Ralf Stephan, Jun 17 2007
Dirichlet g.f.: zeta(s)/eta(s). - Ralf Stephan, Mar 25 2015
For n>=1, we have a recursion Sum_{d|n}(-1)^(1+(n/d))a(d)=1. - Vladimir Shevelev, Jun 09 2009
For n>=1, there is the recurrence n=Sum_{k=1..n} a(k)*g(n/k) where g(x) = floor(x) - 2*floor(x/2). - Benoit Cloitre, Nov 11 2010
a(n) = A209229(n)*n. - Reinhard Zumkeller, Oct 17 2015
a(n) = n if 2^n mod n == 0 and a(n) = 0 otherwise. - Chai Wah Wu, Dec 01 2022

More terms from Keiko L. Noble (s1180624(AT)cedarville.edu)

A152649 Decimal expansion of Pi^4/72.

Original entry on oeis.org

1, 3, 5, 2, 9, 0, 4, 0, 4, 2, 1, 3, 8, 9, 2, 2, 7, 3, 9, 3, 9, 5, 0, 0, 4, 6, 2, 0, 6, 7, 6, 4, 5, 9, 8, 7, 8, 4, 6, 8, 4, 3, 8, 6, 8, 9, 8, 9, 8, 4, 0, 8, 6, 3, 4, 6, 0, 3, 7, 2, 0, 2, 6, 9, 3, 0, 5, 1, 5, 0, 7, 7, 0, 2, 3, 3, 7, 1, 1, 0, 5, 8, 1, 9, 6, 1, 3, 7, 0, 4, 4, 9, 2, 7, 1, 2, 4, 8, 9, 6, 5, 4, 1, 2, 3

Offset: 1

R. J. Mathar, Dec 10 2008

A division by 2 is missing in Mezo's penultimate formula on page 4.

			Equals 1.352904042138922739395004620676459878468438689898408634603...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
David Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995), 1191-1198.
I. Gradsteyn and I. Ryzhik, Table of integrals, series and products, Academic Press, 1980, page 7 (formulas from 0.233.3 to 0.233.5).
Istvan Mezo, Summation of Hyperharmonic Numbers, arXiv:0811.0042 [math.CO], 2008.
Index entries for transcendental numbers.

Cf. A001008, A002805, A048272, A092425, A098198.

evalf(Pi^4/72,120); # Muniru A Asiru, Sep 18 2018

RealDigits[Pi^4/72,10,120][[1]] (* Harvey P. Dale, Feb 10 2013 *)

PARI
```
Pi^4/72 \\ Michel Marcus, Jul 07 2015
```

Equals A098198/2 = A092425/72.
Equals Sum_{j >= 1} H(j)/j^3 where H(j) = A001008(j)/A002805(j).
Equals 20*Sum_{j >= 1} (2*j)^(-4) (see Gradsteyn and Ryzhik in Links section). - A.H.M. Smeets, Sep 18 2018
Equals Sum_{k>=1} A048272(k)/k^2. - Amiram Eldar, Jan 25 2024

A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 225, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815

Offset: 1

Stefano Lanfranco (lastefano(AT)yahoo.it), Mar 21 2000

Also numbers k such that the number of partitions of k into consecutive integers is a record. For example, 45 = 22+23 = 14+15+16 = 7+8+9+10+11 = 5+6+7+8+9+10 = 1+2+3+4+5+6+7+8+9, six such partitions, but all smaller terms have fewer such partitions (15 has four). See A000005 comments and A038547 formula. - Rick L. Shepherd, Apr 20 2008
From Hartmut F. W. Hoft, Mar 29 2022: (Start)
Also the odd parts of the numbers in A340506, see also comments in A250071.
A140864 is a subsequence. (End)
Positions of records in A001227, i.e., integers whose number of odd divisors sets a new record. - Bernard Schott, Jul 18 2022
Conjecture: all terms after the first three terms are congruent to 5 mod 10. - Harvey P. Dale, Jul 05 2023
From Keith F. Lynch, Jan 12 2024: (Start)
Dale's conjecture is correct.  a(n) can't be even, since then a(n)/2 would be a smaller number with the same number of odd divisors.  The respective powers of the successive odd primes can't increase, since if they did, swapping them would give a smaller number with the same number of divisors, e.g., 3^2 * 5^4 has the same number of divisors as 3^4 * 5^2, and the latter is smaller.  As such, every a(n) must be an odd multiple of 5, hence congruent to 5 mod 10, unless it's simply a power of 3.  But multiplying a power of 3 by 3 gives just one more divisor while multiplying a power of 3 by 5 doubles the number of divisors, so after a(n) = 9 all a(n) must be congruent to 5 mod 10, i.e., have a rightmost decimal digit of 5.
This has three equivalent definitions:
* Odd numbers with more divisors than any smaller odd number.
* Numbers with more odd divisors than any smaller number, i.e., record high values of A001227.
* Numbers with a greater excess of odd divisors over even divisors than any smaller number, i.e., record high values of A048272. (End)

			9 is in the sequence because 9 has 3 divisors {1, 3, 9}, which is more than any previous odd number.

Amiram Eldar, Table of n, a(n) for n = 1..363 (terms 1..170 from Ray Chandler)
David Ryan, Mathematical Harmony Analysis, arXiv preprint arXiv:1603.08904 [cs.SD], 2016.

Cf. A002182, A053640, A000005. Subsequence of A147516.
Cf. A140864, A237593, A250071, A279387, A340506.
Cf. A005408, A001227, A038547.

nn = 10^6; maxd = 0;
Reap[For[n = 1, n <= nn, n += 2, If[(nd = DivisorSigma[0, n]) > maxd, Print[n]; Sow[n]; maxd = nd]]][[2, 1]] (* Jean-François Alcover, Sep 20 2018, from PARI *)
next[n_] := Module[{k=n, r=DivisorSigma[0, n]}, While[DivisorSigma[0, k]<=r, k+=2]; k]
a053624[n_] := NestList[next, 1, n-1]/; n>=1 (* returns n numbers *)
a053624[31] (* Hartmut F. W. Hoft, Mar 29 2022 *)
DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,1,131*10^6,2}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 05 2023 *)

lista(nn) = {maxd = 0; forstep (n=1, nn, 2, if ((nd = numdiv(n)) > maxd, print1(n, ", "); maxd = nd;););} \\ Michel Marcus, Apr 21 2014

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

Offset: 0

Vladeta Jovovic, Feb 12 2004

Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.

Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

			a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
  (11)  (22)   (221)  (33)   (322)   (44)    (441)   (55)    (443)
        (211)  (311)  (411)  (331)   (332)   (522)   (433)   (533)
                             (511)   (422)   (711)   (442)   (551)
                             (3211)  (611)   (3321)  (622)   (722)
                                     (3221)  (4221)  (811)   (911)
                                     (4211)  (4311)  (5221)  (4322)
                                             (5211)  (5311)  (4331)
                                                     (6211)  (4421)
                                                             (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
  (2)  (22)  (32)   (222)   (322)    (332)     (432)      (3322)
       (31)  (311)  (3111)  (331)    (431)     (3222)     (3331)
                            (421)    (2222)    (4221)     (22222)
                            (31111)  (3311)    (4311)     (42211)
                                     (4211)    (33111)    (43111)
                                     (311111)  (42111)    (331111)
                                               (3111111)  (421111)
                                                          (31111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A047967, A265251.
Column k=2 of A266477.
Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0,  b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)

alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015

G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

a(0) added by Franklin T. Adams-Watters, Nov 02 2015

cp's OEIS Frontend

A015723 Number of parts in all partitions of n into distinct parts.

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A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

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A042968 Numbers not divisible by 4.

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A030513 Numbers with 4 divisors.

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A059851 a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).

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A001749 Primes multiplied by 4.

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