A078135 -id:A078135 - cp's OEIS frontend

A134755 Minimal number such that all greater numbers can be written as sums of squares of primes in more than n ways.

23, 39, 55, 64, 68, 73, 80, 84, 91, 96, 100, 105, 109, 113, 114, 118, 122, 123, 127, 131, 132, 136, 140, 140, 144, 145, 145, 149, 149, 153, 154, 156, 158, 160, 163, 164, 167, 168, 168, 172, 172, 176, 176, 176, 180, 180, 181, 181, 185, 185, 185, 189, 189, 190

Offset: 0

Hieronymus Fischer, Nov 11 2007

nonn

The sequence is well-defined, in that a(n) exists for all n>=0. Proof by induction: a(0) exists. We set b(j):=number of ways to write j as sum of squares of primes (=A090677). If a(n) exists, then b(j)>n for all j>a(n). Setting m:=a(n)+1, we find that there are n+1 sum of squares of primes B(0,i), 1<=i<=n+1, with m=B(0,i).
Further there are n+1 such sum expressions B(1,i), B(2,i) and B(3,i), 1<=i<=n+1, representing m+1, m+2 and m+3, respectively. For all j>a(n) we have j=m+4*floor((j-m)/4)+(j-m) mod 4. Thus j=m+r+s*2^2, where r=0,1,2 or 3. Hence n can be written B(r,i)+s*2^2 and there are n+1 such representations.
Let q be the maximal prime number (to be squared) occurring as a term within those sum expressions B(r,i), 0<=r<=3,1<=i<=n+1. We select a prime number p>q and we set c:=a(n)+p^2. For j>c, we have the n+1 representations B(r(j),i)+s(j)*2^2. Additionally, for j-p^2 (which is >a(n)) there are also n+1 representations B(r_p,i)+s_p*2^2, where r_p:=r(j-p^2), s_p:=s(j-p^2).
Thus j can be written B(r(j),i)+s(j)*2^2, 1<=i<=n+1 and B(r_p,i)+s_p*2^2+p^2, 1<=i<=n+1. By choice of p all these sum representations of j are different, which implies, that there are 2n+2 such representations. It follows b(j)>2n+2>n+1 for all j>c, which implies, that a(n+1) exists.

			a(0)=23, since numbers >23 can be written as sum of squares of primes.
a(1)=39, since there are at least two ways, to write a number >39 as a sum of squares of primes.

Cf. A078134, A078135, A078136, A078139, A090677, A078137, A134754.

a(n)=min( m | A090677(j)>n for all j>m).

A033183 a(n) = number of pairs (p,q) such that 4p + 9q = n.

Original entry on oeis.org

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

Offset: 0

Michel Tixier (tixier(AT)dyadel.net)

nonn

From Reinhard Zumkeller, Nov 07 2009: (Start)
In other words: number of partitions into 4 or 9;
a(n) <= A078134(n); a(A078135(n)) = 0;
a(A167632(n)) = n and a(m) < n for m < A167632(n). (End)

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

Cf. A033182.

CoefficientList[Series[1/((1-x^4)(1-x^9)),{x,0,80}],x] (* or  *) LinearRecurrence[{0,0,0,1,0,0,0,0,1,0,0,0,-1}, {1,0,0,0,1,0,0,0,1,1,0,0,1}, 80] (* Harvey P. Dale, Oct 13 2012 *)

a(n) = [ 7 n/9 ]+1+[ -3 n/4 ].
G.f.: 1/((1-x^4)*(1-x^9)). - Vladeta Jovovic, Nov 12 2004
a(n) = a(n-4) + a(n-9) - a(n-13). - R. J. Mathar, Dec 04 2011

A078129 Numbers which cannot be written as sum of cubes > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 82, 84, 85, 87, 90, 92, 93, 95, 98, 100, 101, 103, 106, 109, 111, 114, 117, 119, 122, 127, 130, 138, 146, 154

Offset: 1

Reinhard Zumkeller, Nov 19 2002

A078128(a(n))=0.
The sequence is finite because every number greater than 181 can be represented using just 8 and 27. - Franklin T. Adams-Watters, Apr 21 2006
More generally, the numbers which are not the sum of k-th powers larger than 1 are exactly those in [1, 6^k - 3^k - 2^k] but not of the form 2^k*a + 3^k*b + 5^k*c with a,b,c nonnegative. This relies on the following fact applied to m=2^k and n=3^k: if m and n are relatively prime, then the largest number which is not a linear combination of m and n with positive integer coefficients is mn - m - n. - Benoit Jubin, Jun 29 2010

			181 is not in the list since 181 = 7*2^3 + 5^3.

Eric Weisstein's World of Mathematics, Cubic Number.
Index entries for sequences related to sums of cubes

Cf. A000578, A078131, A078133, A078130, A078135.

terms = 83; A078131 = (Exponent[#, x]& /@ List @@ Normal[1/Product[1-x^j^3, {j, 2, Ceiling[(3 terms)^(1/3)]}] + O[x]^(3 terms)])[[2 ;; terms+1]];
Complement[Range[Max[A078131]], A078131] (* Jean-François Alcover, Aug 04 2018 *)

Sequence completed by Franklin T. Adams-Watters, Apr 21 2006

Edited by R. J. Mathar and N. J. A. Sloane, Jul 06 2010

A134754 Minimal number such that all greater numbers can be written as sums of squares >1 in more than n ways.

Original entry on oeis.org

23, 39, 39, 55, 55, 55, 59, 59, 63, 71, 71, 71, 71, 75, 75, 75, 75, 79, 79, 87, 87, 87, 87, 87, 91, 91, 91, 91, 91, 91, 95, 95, 95, 95, 95, 96, 96, 99, 99, 103, 103, 103, 103, 103, 103, 103, 107, 107, 107, 107, 107, 107, 107, 107, 111, 111, 111, 111, 111, 111, 111

Offset: 0

Hieronymus Fischer, Nov 11 2007

nonn

The sequence is well-defined, in that a(n) exists for all n>=0. For the reasoning see A078134.

			a(0)=23, since numbers >23 can be written as sum of squares >1.
a(2)=39, since there are at least three ways, to write a number >39 as a sum of squares >1.

Cf. A078134, A078135, A078136, A090677, A078137, A134755.

a(n)=min( m | A078134(j)>n for all j>m).

A122615 Largest integer which cannot be written as a sum of n-th powers of primes.

Original entry on oeis.org

0, 1, 23, 154, 1199, 5314, 34928, 256117, 1565279, 6519069, 49304891, 362617861, 1121432591, 13059091501, 34313897584, 202096681135, 1912393561610, 6341902873937, 54356644026512, 175476300288281, 1352729779867857, 5937475586243116, 39152549345560551

Offset: 0

Jonathan Vos Post, Sep 20 2006

nonn

The powers of only 3 primes are needed, namely 2^n, 3^n and 5^n, which leads to an ultra-fast O(n) execution time. I executed the algorithm in Greenberg (1988) with a PARI/GP program in only a few seconds for 1001 terms. - Mike Oakes, Aug 16 2016

Equivalent definition for this same sequence is "Largest integer which cannot be written as a sum of n-th powers of integers greater than 1". - Mike Oakes, Aug 17 2016

			a(0) = 0 because all positive integers can be written as a sum of 0th powers of primes, i.e. as sums of 1.
a(1) = 1 because 2^1 = 2, 3^1 = 3, hence all positive integers 2 or larger can be written as a*2 + b*3 for a,b nonnegative integers [2 = 2, 3 = 3, 4 = 2+2, 5 = 2+3, 6 = 2+2+2 = 3+3, 7 = 2+2+3, ...].
a(2) = 23 because all integers 24 or larger can be written as a sum of squares and in fact as a sum of squares of primes.
a(3) = 154 because all integers 155 or larger can be written as a sum of cubes of primes.

Mike Oakes, Table of n, a(n) for n = 0..1000
H. Greenberg, Solution to a linear diophantine equation for nonnegative integers, Journal of Algorithms, 9 (1988), 343-353

Cf. A078135, A275743, A275746, A275747.

a[0] = 0; a[n_] := Block[{k = 4, f}, While[Prime[k]^n <= (f = FrobeniusNumber[ Prime[ Range@ k]^n]), k++]; f]; a /@ Range[0, 10] (* Giovanni Resta, Jun 13 2016 *)

a(4)-a(22) from Giovanni Resta, Jun 12 2016

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23

Offset: 1

Gus Wiseman, Dec 14 2018

			24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 79

Offset: 1

Gus Wiseman, Dec 14 2018

			48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).

Cf. A000607, A002095, A005117, A023893, A023894, A064573, A078135, A101417, A246655, A322546.

nn=100;
ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A179101 Numbers which are not the sum of exactly one positive square and a sum of squares-minus-1.

Original entry on oeis.org

2, 3, 5, 6, 8, 11, 14

Offset: 1

Benoit Jubin, Jun 29 2010

Sequence motivated by the study of certain replicate tilings, where each tile can be replaced by a square number of tiles.

Adding multiples of 3=2^2-1 to the numbers 1=1^2, 9=3^2 and 17=3^2+(3^2-1), one obtains all the integers not in the sequence.

Cf. A000290 (squares), A005563 (squares-minus-1), A123120, A078135.

A294574 Numbers having exactly one representation as sum of squares of primes.

Original entry on oeis.org

4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 39

Offset: 1

Ilya Gutkovskiy, Nov 02 2017

Index entries for sequences related to sums of squares

Cf. A001248, A078135, A078136, A090677, A134755.

Flatten[Position[Rest[CoefficientList[Series[Product[1/(1 - x^(Prime[k]^2)), {k, 1, 5}], {x, 0, 40}], x]], 1]]

A090677(a(n)) = 1.

A321936 Number of integer partitions of n containing no 1's, prime powers, or squarefree numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 2, 0, 3, 1, 1, 0, 7, 0, 2, 0, 5, 0, 5, 0, 7, 1, 3, 0, 12, 0, 4, 2, 10, 1, 8, 0, 14, 2, 6, 0, 22, 1, 10, 3, 20, 1, 15, 0, 26, 5, 12, 2

Offset: 0

Gus Wiseman, Dec 11 2018

nonn

Number of integer partitions of n using elements of A126706.

			The a(56) = 7 partitions:
  (56)
  (28,28)
  (36,20)
  (44,12)
  (20,18,18)
  (24,20,12)
  (20,12,12,12)

Cf. A000607, A000961, A002095, A005117, A023893, A023894, A078135, A114374, A126706, A246655, A321346, A321347, A321378, A321665.

nn=100;
ser=Product[If[n==1||PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

cp's OEIS Frontend

A134755 Minimal number such that all greater numbers can be written as sums of squares of primes in more than n ways.

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A033183 a(n) = number of pairs (p,q) such that 4*p + 9*q = n.

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A078129 Numbers which cannot be written as sum of cubes > 1.

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A134754 Minimal number such that all greater numbers can be written as sums of squares >1 in more than n ways.

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A122615 Largest integer which cannot be written as a sum of n-th powers of primes.

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A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

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A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

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A179101 Numbers which are not the sum of exactly one positive square and a sum of squares-minus-1.

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A294574 Numbers having exactly one representation as sum of squares of primes.

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A033183 a(n) = number of pairs (p,q) such that 4p + 9q = n.