A181522 -id:A181522 - cp's OEIS frontend

A126024 Number of subsets of {1,2,3,...,n} whose sum is a square integer (including the empty subset).

1, 2, 2, 3, 5, 7, 12, 20, 34, 60, 106, 190, 346, 639, 1183, 2204, 4129, 7758, 14642, 27728, 52648, 100236, 191294, 365827, 700975, 1345561, 2587057, 4981567, 9605777, 18546389, 35851756, 69382558, 134414736, 260658770, 505941852, 982896850

Offset: 0

John W. Layman, Feb 27 2007

nonn

			The subsets of {1,2,3,4,5} that sum to a square are {}, {1}, {1,3}, {4}, {2,3,4}, {1,3,5} and {4,5}. Thus a(5)=7.

Alois P. Heinz, Table of n, a(n) for n = 0..990 (terms n=1..100 from T. D. Noe)

Cf. A053632, A127542.
Cf. A181522. - Reinhard Zumkeller, Oct 27 2010
Cf. A010052, A284250.
Row sums of A281871.

import Data.List (subsequences)
a126024 = length . filter ((== 1) . a010052 . sum) .
                          subsequences . enumFromTo 1
-- Reinhard Zumkeller, Feb 22 2012, Oct 27 2010

b:= proc(n, i) option remember; (m->
      `if`(n=0 or n=m, 1, `if`(n<0 or n>m, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+
      add(b(j^2-n, n-1), j=isqrt(n)..isqrt(n*(n+1)/2)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 02 2017

g[n_] := Block[{p = Product[1 + z^i, {i, n}]},Sum[Boole[IntegerQ[Sqrt[k]]]*Coefficient[p, z, k], {k, 0, n*(n + 1)/2}]];Array[g, 35] (* Ray Chandler, Mar 05 2007 *)

Extended by Ray Chandler, Mar 05 2007

a(0)=1 prepended by Alois P. Heinz, Jan 30 2017

A127542 Number of subsets of {1,2,3,...,n} whose sum is prime.

Original entry on oeis.org

0, 2, 4, 7, 12, 22, 42, 76, 139, 267, 516, 999, 1951, 3824, 7486, 14681, 28797, 56191, 108921, 210746, 410016, 804971, 1591352, 3153835, 6249154, 12380967, 24553237, 48731373, 96622022, 191012244, 376293782, 739671592, 1454332766, 2867413428, 5678310305

Offset: 1

Emeric Deutsch, Mar 03 2007

nonn

			The subsets of {1,2,3} that sum to a prime are {1,2}, {2}, {3}, {2,3}. Thus a(3)=4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000 (first 100 terms from T. D. Noe)

Cf. A053632, A126024, A181522, A010051.

Row sums of A282516.

import Data.List (subsequences)
a127542 = length . filter ((== 1) . a010051 . sum) .
                          subsequences . enumFromTo 1
-- Reinhard Zumkeller, Feb 22 2012, Oct 27 2010

with(combinat): a:=proc(n) local ct, pn, j:ct:=0: pn:=powerset(n): for j from 1 to 2^n do if isprime(add(pn[j][i],i=1..nops(pn[j]))) =true then ct:=ct+1 else ct:=ct fi: od: end: seq(a(n),n=1..18);
# second Maple program:
b:= proc(n, s) option remember; `if`(n=0,
     `if`(isprime(s), 1, 0), b(n-1, s)+b(n-1, s+n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..44);  # Alois P. Heinz, Oct 22 2023

g[n_] := Block[{p = Product[1 + z^i, {i, n}]},Sum[Boole[PrimeQ[k]]*Coefficient[p, z, k], {k, 0, n*(n + 1)/2}]];Array[g, 34] (* Ray Chandler, Mar 05 2007 *)

a(n)=my(v=Vec(prod(i=1,n,x^i+1)),s);forprime(p=2,#v,s+=v[p]);s \\ Charles R Greathouse IV, Dec 19 2014

first(n)=my(v=vector(n),P=1,s); for(k=1,n, P*=1+'x^n; s=0; forprime(p=2,k*(k+1)/2,s+=polcoeff(P,p)); v[k]=s); v \\ Charles R Greathouse IV, Dec 19 2014

Extended by Ray Chandler, Mar 05 2007

cp's OEIS Frontend

A126024 Number of subsets of {1,2,3,...,n} whose sum is a square integer (including the empty subset).

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