A022831 -id:A022831 - cp's OEIS frontend

A096765 Number of partitions of n into distinct parts, the least being 1.

0, 1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 17, 21, 25, 29, 35, 41, 48, 56, 66, 76, 89, 103, 119, 137, 159, 181, 209, 239, 273, 312, 356, 404, 460, 522, 591, 669, 757, 853, 963, 1085, 1219, 1371, 1539, 1725, 1933, 2164, 2418, 2702, 3016, 3362, 3746, 4171, 4637

Offset: 0

N. J. A. Sloane, Sep 28 2008

nonn

The old entry with this sequence number was a duplicate of A071569.
a(n) is also the total number of 1's in all partitions of n into distinct parts. For n=6 there are partitions [6], [5,1], [4,2], [3,2,1] and only two contain a 1, hence a(6) = 2. - T. Amdeberhan, May 13 2012
a(n), n > 1 is the Euler transform of [0,1,1] joined with period [0,1]. - Georg Fischer, Aug 15 2020

			G.f. = x + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 5*x^10 + 5*x^11 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A025147, A071569.

Cf. A096749 (least=2), A022824 (3), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-1)*(i+2)/2 `if`(n<1, 0, b(n-1$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014
# Using the function EULER from Transforms (see link at the bottom of the page).
[0,1,op(EULER([0,1,seq(irem(n,2),n=1..56)]))]; # Peter Luschny, Aug 19 2020

p[, 0] = 1; p[k, n_] := p[k, n] = If[n < k, 0, p[k+1, n-k] + p[k+1, n]]; a[n_] := p[2, n-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Apr 17 2014, after Reinhard Zumkeller *)
a[ n_] := SeriesCoefficient[ x / ((1 + x) Product[ 1 - x^j, {j, 1, n, 2}]), {x, 0, n}]; (* Michael Somos, Sep 10 2016 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[  Sum[ x^(k (k + 1)/2) / Product[ 1 - x^j, {j, 1, k - 1}], {k, 1, Quotient[-1 + Sqrt[8 n + 1], 2]}], {x, 0, n}]]; (* Michael Somos, Sep 10 2016 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 1], {n, 66}]] (* Robert Price, Jun 13 2020 *)

{a(n) = if( n<1, 0, polcoeff( x / ((1 + x) * prod(k=1, (n+1)\2, 1 - x^(2*k-1), 1 + O(x^n))), n))}; /* Michael Somos, Sep 10 2016 */

a(n) = A025147(n-1), n>1. - R. J. Mathar, Jul 31 2008
G.f.: x*Product_{j=2..infinity} (1+x^j). - R. J. Mathar, Jul 31 2008
G.f.: x / ((1 + x) * Product_{k>0} (1 - x^(2*k-1))). - Michael Somos, Sep 10 2016
G.f.: Sum_{k>0} x^(k*(k+1)/2) / Product_{j=1..k-1} (1 - x^j). - Michael Somos, Sep 10 2016

A008348 a(0)=0; thereafter a(n) = a(n-1) + prime(n) if a(n-1) < prime(n), otherwise a(n) = a(n-1) - prime(n).

Original entry on oeis.org

0, 2, 5, 0, 7, 18, 5, 22, 3, 26, 55, 24, 61, 20, 63, 16, 69, 10, 71, 4, 75, 2, 81, 164, 75, 172, 71, 174, 67, 176, 63, 190, 59, 196, 57, 206, 55, 212, 49, 216, 43, 222, 41, 232, 39, 236, 37, 248, 25, 252, 23, 256, 17, 258, 7, 264, 1, 270, 541, 264, 545, 262, 555

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) < 2*prime(n). Conjecture: a(n) > 0 for n > 3. - Thomas Ordowski, Dec 03 2016 [This conjecture is false, because a(369019)=0. The next counterexample occurs at n = 22877145. - Dmitry Kamenetsky, Feb 14 2017. (Cf. A309225.)]

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A008344, A022831, A022837, A309225.

A008348 := proc(n) option remember; if n = 0 then 0 elif A008348(n-1)>=ithprime(n) then A008348(n-1)-ithprime(n); else A008348(n-1)+ithprime(n); fi; end;
# Maple from N. J. A. Sloane, Aug 31 2019 (Start)
# Riecaman transform
Riecaman := proc(a,s,M)
# Start with s, add or subtract a[n], get M terms. If a has w terms, can get M=w+1 terms.
local b,M2,n,t;
if whattype(a) <> list then ERROR("First argument should be a list"); fi;
if a[1]=0 then ERROR("a[1] should not be zero"); fi;
M2 := min(nops(a),M-1);
b:=[s]; t:=s;
for n from 1 to M2 do
   if a[n]>t then t:=t+a[n] else t:=t-a[n]; fi; b:=[op(b),t]; od:
b; end;
# Riecaman transform of primes, starting at s=0
p1:=[seq(ithprime(i),i=1..100)];
q0:=Riecaman(p1,0,99);
# End

a := {0}; For[n = 2, n < 100, n++, If[a[[n - 1]] >= Prime[n - 1], b := a[[n - 1]] - Prime[n - 1], b := a[[n - 1]] + Prime[n - 1];]; a = Append[a, b]]; a (* Stefan Steinerberger, May 02 2006 *)
nxt[{n_,a_}]:={n+1,If[aHarvey P. Dale, Sep 13 2024 *)

lista(nn) = {print1(a=0, ", "); for (n=1, nn, if (a < (p=prime(n)), a += p, a -= p); print1(a, ", "););} \\ Michel Marcus, Dec 04 2016

a(n) = c(1)p(1) + ... + c(n)p(n), where c(i) = 1 if a(i-1) > p(i) and c(i) = -1 if a(i-1) <= p(i) (p(i) = primes). - Clark Kimberling

More terms from Clark Kimberling

Name edited by Dmitry Kamenetsky, Feb 14 2017

A096749 Number of partitions of n into distinct parts, the least being 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 59, 68, 78, 91, 103, 118, 136, 155, 176, 201, 228, 259, 294, 332, 375, 425, 478, 538, 607, 681, 764, 858, 961, 1075, 1203, 1343, 1499, 1673, 1863, 2073, 2308, 2564, 2847, 3161, 3504

Offset: 0

N. J. A. Sloane, Sep 28 2008

nonn

The old entry with this sequence number was a duplicate of A071569.

a(n), n>2 is the Euler transform of [0,0,1,1,1] joined with period [0,1]. - Georg Fischer, Aug 15 2020

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A025147, A071569.

Cf. A096765 (least=1), A022824 (3), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-2)*(i+3)/2 `if`(n<2, 0, b(n-2$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 07 2014
# Using the function EULER from Transforms (see link at the bottom of the page).
[0,0,1,op(EULER([0,0,1,1,seq(irem(n,2),n=1..57)]))]; # Peter Luschny, Aug 19 2020

b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-2)*(i+3)/2Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 2], {n, 66}]] (* Robert Price, Jun 13 2020 *)

G.f.: x^2*Product_{j>=3} (1+x^j). - R. J. Mathar, Jul 31 2008
a(n) = A025148(n-2), n>1. - R. J. Mathar, Sep 30 2008
G.f.: Sum_{k>=1} x^(k*(k + 3)/2) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020

A064365 a(0) = 0; thereafter a(n) = a(n-1)-prime(n) if positive and new, otherwise a(n) = a(n-1)+prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

0, 2, 5, 10, 3, 14, 1, 18, 37, 60, 31, 62, 25, 66, 23, 70, 17, 76, 15, 82, 11, 84, 163, 80, 169, 72, 173, 276, 383, 274, 161, 34, 165, 28, 167, 316, 467, 310, 147, 314, 141, 320, 139, 330, 137, 334, 135, 346, 123, 350, 121, 354, 115, 356, 105, 362, 99, 368, 97, 374, 93

Offset: 0

Neil Fernandez, Sep 25 2001

'Recamán transform' (see A005132) of the prime sequence. Note that the definition permits repeated terms [though only by addition] (and there are many repeated terms, just as there are in A005132).
Does every positive integer appear in the sequence? This seems unlikely, since 4 has not appeared in 70000 terms.
Note: this is similar to Clark Kimberling's A022831, except in the latter sequence the words 'and new' have been omitted.
The smallest numbers not occurring in the first million terms: 4, 6, 7, 12, 13, 16, 19, 20, 21, 22, 24, 26, 27, 29, 30, 32, 36, 39, 41, 42. - Reinhard Zumkeller, Apr 26 2012

			To find a(9) we try subtracting the 9th prime, which is 23, from a(8), which is 37. 37 - 23 = 14, but 14 is already in the sequence (it is a(5)), so we must add. a(9) = 37 + 23 = 60.

N. J. A. Sloane, First 70000 terms
Index entries for sequences related to Recamán's sequence

Cf. A005132, A022831, A117128.

import Data.Set (singleton, notMember, insert)
a064365 n = a064365_list !! n
a064365_list = 0 : f 0 a000040_list (singleton 0) where
   f x (p:ps) s | x' > 0 && x' `notMember` s = x' : f x' ps (insert x' s)
                | otherwise                  = xp : f xp ps (insert xp s)
                where x' = x - p; xp = x + p
-- Reinhard Zumkeller, Apr 26 2012

a = {0}; Do[ If[ a[ [ -1 ] ] - Prime[ n ] > 0 && Position[ a, a[ [ -1 ] ] - Prime[ n ] ] == {}, a = Append[ a, a[ [ -1 ] ] - Prime[ n ] ], a = Append[ a, a[ [ -1 ] ] + Prime[ n ] ] ], {n, 1, 70} ]; a (* Modified by Ivan N. Ianakiev, Aug 05 2019, to accommodate the new initial term of a(0). *)

A064365(N,s/*=1 to print all terms*/)={ my(a=0,u=0); N & forprime(p=1,prime(N), s & print1(a","); u=bitor(u,2^a+=if(a<=p || bittest(u,a-p),p,-p)));a}  \\ M. F. Hasler, Mar 07 2012

from sympy import primerange, prime
def aupton(terms):
  alst = [0]
  for n, pn in enumerate(primerange(1, prime(terms)+1), start=1):
    x = alst[-1] - pn
    alst += [x if x > 0 and x not in alst else alst[-1] + pn]
  return alst
print(aupton(60)) # Michael S. Branicky, May 30 2021

a(n) = A117128(n) - 1. - Thomas Ordowski, Dec 05 2016

More terms from Robert G. Wilson v, Sep 26 2001
Further terms from N. J. A. Sloane, Feb 10 2002
Added initial term a(0)=0, in analogy with A128204, A005132, A053461, A117073/A078783. - M. F. Hasler, Mar 07 2012

A026824 Number of partitions of n into distinct parts, the least being 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 17, 20, 23, 27, 31, 36, 41, 47, 55, 62, 71, 81, 93, 105, 120, 135, 154, 174, 197, 221, 251, 281, 317, 356, 400, 447, 502, 561, 628, 701, 782, 871, 972, 1081, 1202, 1336, 1483, 1645, 1825, 2021, 2237, 2476

Offset: 0

Clark Kimberling

nonn

Also, number of partitions of n such that if k is the largest part, then k occurs exactly 3 times and each of the numbers 1,2,...,k-1 occur at least once (these are the conjugates of the partitions described in the definition). Example: a(14)=3 because we have [3,3,3,2,2,1],[3,3,3,2,1,1,1] and [2,2,2,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

For n > 3, a(n) is the Euler transform of [0,0,0,1,1,1,1] joined with the period 2 sequence [0,1, ...]. - Georg Fischer, Aug 18 2020

			a(14) = 3 because we have [11,3], [7,4,3] and [6,5,3].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A025147, A025149.

Cf. A096765 (least=1), A096749 (2), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

g:=x^3*product(1+x^j,j=4..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..59); # Emeric Deutsch, Apr 17 2006
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-3)*(i+4)/2 `if`(n<3, 0, b(n-3$2)):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 07 2014

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[(i-3)(i+4)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<3, 0, b[n-3, n-3]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[x^3/((1+x)*(1+x^2)*(1+x^3)) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 3], {n, 1, 66}]] (* Robert Price, Jun 13 2020 *)

From Emeric Deutsch, Apr 17 2006: (Start)
G.f.: (x^3)*Product_{j=4..infinity} (1+x^j).
G.f.: Sum_{k=1..infinity} x^(k*(k+5)/2)/(Product_{j=1..k-1} (1-x^j)). (End)
a(n) = A025149(n-3), n>3. - R. J. Mathar, Jul 31 2008
a(n) ~ exp(Pi*sqrt(n/3)) / (32*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

More terms from Emeric Deutsch, Apr 17 2006

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A096765 Number of partitions of n into distinct parts, the least being 1.

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A008348 a(0)=0; thereafter a(n) = a(n-1) + prime(n) if a(n-1) < prime(n), otherwise a(n) = a(n-1) - prime(n).

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A096749 Number of partitions of n into distinct parts, the least being 2.

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A064365 a(0) = 0; thereafter a(n) = a(n-1)-prime(n) if positive and new, otherwise a(n) = a(n-1)+prime(n), where prime(n) is the n-th prime.

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A026824 Number of partitions of n into distinct parts, the least being 3.

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