A002828 -id:A002828 - cp's OEIS frontend

A278498 a(n) = A065338(A276573(n)).

3, 6, 8, 3, 3, 16, 18, 9, 24, 27, 6, 32, 3, 6, 8, 3, 9, 48, 3, 1, 24, 3, 27, 64, 3, 6, 72, 3, 6, 16, 3, 1, 24, 18, 9, 96, 27, 6, 9, 108, 48, 3, 9, 24, 3, 54, 128, 3, 6, 8, 3, 3, 144, 27, 1, 24, 3, 6, 32, 162, 9, 72, 27, 1, 48, 3, 3, 18, 81, 192, 3, 54, 8, 3, 27, 18, 9, 216, 3, 1, 96, 3, 27, 1, 12, 48, 243, 9, 24, 3, 3, 256, 3, 9, 72, 3, 54, 16, 3, 1, 24, 3, 6

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10028

Cf. A002828, A065338, A276573, A278497, A278499.

(define (A278498 n) (A065338 (A276573 n)))

a(n) = A065338(A276573(n)).

A278499 a(n) = A276573(n) modulo 4.

Original entry on oeis.org

0, 3, 2, 0, 3, 3, 0, 2, 1, 0, 3, 2, 0, 3, 2, 0, 3, 1, 0, 3, 1, 0, 3, 3, 0, 3, 2, 0, 3, 2, 0, 3, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 3, 1, 0, 3, 2, 0, 3, 2, 0, 3, 3, 0, 3, 1, 0, 3, 2, 0, 2, 1, 0, 3, 1, 0, 3, 3, 2, 1, 0, 3, 2, 0, 3, 3, 2, 1, 0, 3, 1, 0, 3, 3, 1, 0, 0, 3, 1, 0, 3, 3, 0, 3, 1, 0, 3, 2, 0, 3, 1, 0, 3, 2, 0, 3, 2, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 3, 2, 0, 3

Offset: 0

Antti Karttunen, Nov 25 2016

nonn

No subsequence 2, 3 will ever occur.

Antti Karttunen, Table of n, a(n) for n = 0..10028

Cf. A002828, A010873, A276573, A276575, A278497, A278498.

(define (A278499 n) (modulo (A276573 n) 4))

a(n) = A010873(A276573(n)) = A276573(n) modulo 4.

A096436 a(n) = the number of squared primes and 1's needed to sum to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 2, 1, 2, 3, 3, 2, 3, 4, 4, 3, 2, 3, 4, 4, 3, 4, 5, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 4, 3, 4, 5, 5, 4, 3, 4, 5, 5, 4, 5, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 4, 3, 4, 5, 5, 4, 3, 4, 5, 5, 4, 5, 6, 2, 3, 4, 5, 3, 4, 5, 6, 4, 3, 4, 5, 5, 4, 5, 6, 6, 5, 4, 5, 6, 6, 5, 6, 2, 3, 4, 5, 3, 4, 5, 6

Offset: 1

Tom Raes (tommy1729(AT)hotmail.com), Aug 10 2004

a(n) has a new maximum at n=1,2,3,7,24,73,266,795.

I suspect that a(n) <= 9 for all n. - Robert G. Wilson v, Sep 18 2004

			a(5) = 2 because 5=4+1.
a(17) = 3 because 17=9+4+4.
A number may have many such sums: 27=25+1+1=9+9+9, 50=25+25=49+1.

Nicholas Matteo, Table of n, a(n) for n = 1..10000

Cf. A001248, A002828, A045698, A051034, A063274.

f[n_] := Block[{d = n, k = PrimePi[ Sqrt[n]], sp = {}}, While[d > 3, While[p = Prime[k]; d >= p^2, AppendTo[sp, p]; d = d - p^2]; k-- ]; While[d != 0, AppendTo[sp, 1]; d = d - 1]; If[Position[sp, 3] != {} && sp[[ -3]] == 1, sp = Delete[Drop[sp, -3], Position[sp, 3][[1]]]; AppendTo[sp, {2, 2, 2}]]; Reverse[ Sort[ Flatten[ sp]]]]; Table[ Length[ f[n]], {n, 105}] (* Robert G. Wilson v, Sep 20 2004 *)

Edited and extended by Robert G. Wilson v, Sep 18 2004

Edited by Don Reble, Apr 23 2006

A188462 Least number of 5th powers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Offset: 1

Jean-François Alcover, Apr 01 2011

nonn

Vaughan & Wooley (1995) prove that a(n) <= 17 for large enough n; in fact it is conjectured that a(n) <= 6 for large enough n. The maximum value is a(223) = 37. - Charles R Greathouse IV, Jul 05 2017

			33 = 2^5 + 1^5 (least decomposition) hence a(33) = 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, Number theory for the millennium, III (Urbana, IL, 2000), 301-340, A K Peters, Natick, MA, 2002.
Robert C. Vaughan and Trevor D. Wooley, Further improvements in Waring's problem, Acta Mathematica 174:2 (1995), pp. 147-240.

Cf. A002828 (squares), A002376 (cubes), A002377 (4th powers), A374012 (6th powers).

Cnt5[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 5]] == 0, k++]; k]; Array[Cnt5, 105] (* T. D. Noe, Apr 01 2011 *)

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A188462(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,5,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

A193101 Minimal number of numbers of the form (m^3+5m)/6 (see A004006) needed to sum to n.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 2, 3, 1, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 4, 3, 4, 2, 3, 3, 3, 4, 4, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 2, 3, 4, 3, 4, 2, 3, 3, 3, 4, 4, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 2, 3, 4, 3

Offset: 1

N. J. A. Sloane, Jul 15 2011

nonn

Watson showed that a(n) <= 8 for all n.

It is conjectured that a(n) <= 5 for all n.

Lars Blomberg, Table of n, a(n) for n = 1..10000
H. E. Salzer and N. Levine, Proof that every integer <= 452,479,659 is a sum of five numbers of the form Q_x = (x^3+5x)/6, x>= 0, Math. Comp., (1968), 191-192.
N. J. A. Sloane, Transforms
G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc., 27 (1952), 217-224.

Cf. A004006. A002828, A104246, A193105.

# LAGRANGE transform of a sequence {a(n)}
# Suggested by Lagrange's theorem that at most 4 squares are needed to sum to n.
# Returns b(n) = minimal number of terms of {a} needed to sum to n for 1 <= n <= M.
# C = maximal number of terms of {a} to try to build n
# M = upper limit on n
# Internally, the initial terms of both a and b are taken to be 0, but since this is a number-theoretic function, the output starts at n=1
LAGRANGE:=proc(a,C,M)
local t1,ip,i,j,a1,a2,b,c,N1,N2,Nc;
if whattype(a) <> list then RETURN([]); fi:
# sort a, remove duplicates, include 0
t1:=sort(a);
a1:=sort(convert(convert(a,set),list));
if not member(0,a1) then a1:=[0,op(a1)]; fi;
N1:=nops(a1);
b:=Array(1..M+1,-1);
for i from 1 to N1 while a1[i]<=M do b[a1[i]+1]:=1; od;
a2:=a1; N2:=N1;
for ip from 2 to C do
c:={}:
   for i from 1 to N1 while a1[i] <= M do
      for j from 1 to N2 while a1[i]+a2[j] <= M do
c:={op(c),a1[i]+a2[j]};
                                                od;
                                       od;
c:=sort(convert(c,list));
Nc:=nops(c);
   for i from 1 to Nc do
      if b[c[i]+1] = -1 then b[c[i]+1]:= ip; fi;
                      od;
a2:=c; N2:=Nc;
                   od;
[seq(b[i],i=2..M+1)];
end;
Q:=[seq((m^3+5*m)/6,m=0..20)];
LAGRANGE(Q,8,120);

A227781 Least number of squares which add to -1 mod n.

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 4, 1, 1, 2, 3, 1, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 4, 1, 1, 2, 3, 2, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2

Offset: 1

Charles R Greathouse IV, Jul 31 2013

nonn

Pfister proved that a(p) <= 2 for all primes p; then a(p) is called the stufe of the field Z/pZ.

Conjecture: a(n) = 4 if and only if n is divisible by 8 and a(n) = 3 if and only if n is 4 mod 8. Together with A008784 this would completely define the sequence.

			a(3) = 2: 1^2 + 1^2 = -1 mod 3.
a(15) = 2: 2^2 + 5^2 = -1 mod 15.

Albert Pfister, Zur Darstellung von -1 Als Summe von Quadraten in einem Korper, J. London Math. Society, 40 (1965), pp. 159-165.
A. R. Rajwade, Squares, Cambridge Univ. Press, 1983.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A008784, A002828.

isA008784(n)=if(n%2==0, if(n%4, n/=2, return(0))); n==1||vecmax(factor(n)[, 1]%4)==1
a(n)=if(isA008784(n),return(n>1)); if(isprime(n), return(2)); if(n%8==0, return(4)); my(N, cur, new, k=1); for(i=1,n\2,cur=N=bitor(1<<(i^2%n),N)); while(!bittest(cur,n-1), new=0; for(i=1,n\2, t=cur<<(i^2%n); t=bitor(bitand(t,(1<>n); new=bitor(new,t)); k++; cur=new); k

a(n) <= A002828(n-1) <= 4.

a(n) = 1 if and only if n > 1 is in A008784. a(4n) >= 3 for all n.

A277889 After zero, excess of the odd terms in A276573 (the infinite trunk of least squares beanstalk) over the even terms.

Original entry on oeis.org

0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, -2, -1, -2, -3, -2, -1, -2, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -1, -2, -1, -2, -1, -2, -3, -2, -1, -2, -1, -2, -3, -2, -3, -4, -3, -2, -3, -2, -1, -2, -1, -2, -3, -4, -3, -4, -3, -2, -3, -2, -1, -2, -1, -2, -1, -2, -3, -2, -1, -2, -1, -2, -1, 0, -1, 0, 1, 2, 1, 0

Offset: 0

Antti Karttunen, Nov 13 2016

sign

Is a(2358) = 0 the last zero of this sequence? See also comments in A277891.

Antti Karttunen, Table of n, a(n) for n = 0..41014

Cf. A002828, A276573, A277890, A277891.

a(0) = 0; for n >= 1, a(n) = a(n-1) - (-1)^A276573(n).

A338480 Least number of heptagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 1, 2, 3, 3, 4, 5, 6, 2, 3, 4, 4, 5, 6, 7, 3, 4, 1, 2, 2, 3, 4, 4, 5, 2, 3, 3, 4, 5, 5, 6, 3, 4, 4, 5, 2, 3, 3, 1, 2, 3, 4, 3, 4, 4, 2, 3, 4, 5, 4, 5, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 5, 5, 3, 1, 2, 3, 4, 5, 3, 4, 2, 2, 3, 3, 4, 4, 5, 3, 3, 4, 4, 2, 3, 4

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000566, A002828, A061336, A100878, A338479, A338481.

A338484 Least number of centered square numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 5

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Centered Square Number
Index entries for sequences related to centered polygonal numbers

Cf. A001844, A002828, A101412, A234533, A338482.

A338493 Least number of square pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 2, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 3, 4, 3, 4, 2, 3, 4, 4, 5, 3, 4, 5, 5, 6, 4, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 3, 4, 5, 5, 2, 3, 4, 4, 5, 3, 4, 5, 5, 6, 4, 5, 6, 6, 3, 4, 2, 3, 4, 4, 5, 3, 1, 2, 3, 4, 4, 2, 3, 4, 3, 4, 3

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A002828, A104246, A338484, A338494, A338495.

cp's OEIS Frontend

A278498 a(n) = A065338(A276573(n)).

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A278499 a(n) = A276573(n) modulo 4.

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A096436 a(n) = the number of squared primes and 1's needed to sum to n.

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A188462 Least number of 5th powers needed to represent n.

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A193101 Minimal number of numbers of the form (m^3+5m)/6 (see A004006) needed to sum to n.

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Maple

A227781 Least number of squares which add to -1 mod n.

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PARI

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A277889 After zero, excess of the odd terms in A276573 (the infinite trunk of least squares beanstalk) over the even terms.

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A338480 Least number of heptagonal numbers needed to represent n.

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A338484 Least number of centered square numbers needed to represent n.

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