A132399 -id:A132399 - cp's OEIS frontend

A117054 Number of ordered ways of writing n = i + j, where i is a prime and j is of the form k*(k+1), k > 0.

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

Offset: 0

N. J. A. Sloane, Jan 15 2009

Based on a posting by Zhi-Wei Sun to the Number Theory Mailing List, Mar 23 2008, where he conjectures that a(2n+1) > 0 for n >= 2.
Zhi-Wei Sun has offered a monetary reward for settling this conjecture.
No counterexample below 10^10. - D. S. McNeil

Zhi-Wei Sun, Posing to Number Theory List (1)
Zhi-Wei Sun, Posting to Number Theory List (2)
Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers, arxiv:0803.3737 and Journal of Combinatorics and Number Theory, 1 (2009), no.1, 65-76

Cf. A132399, A144590.

t0:=array(0..300); for n from 0 to 300 do t0[n]:=0; od:
t1:=[seq(ithprime(i),i=1..70)]; t2:=[seq(n*(n+1),n=1..30)];
for i from 1 to 70 do for j from 1 to 30 do k:=t1[i]+t2[j]; if k <= 300 then t0[k]:=t0[k]+1; fi; od: od:
t3:=[seq(t0[n],n=1..300)];

A137996 Numbers which are not the sum of a triangular number and zero or a prime = 1 (mod 4).

Original entry on oeis.org

2, 4, 7, 9, 12, 22, 24, 25, 31, 46, 48, 70, 75, 80, 85, 87, 93, 121, 126, 135, 148, 162, 169, 186, 205, 211, 213, 216, 220, 222, 246, 255, 315, 331, 357, 375, 396, 420, 432, 441, 468, 573, 588, 615, 690, 717, 735, 738, 750, 796, 879, 924, 1029, 1038, 1080, 1155

Offset: 1

M. F. Hasler, Mar 24 2008

nonn

Zhi-Wei SUN conjectured that n=216 is the only number not of the form n = p + k(k+1)/2 with p a prime or zero and more precisely that the numbers given in this sequence A137996(1..123) = 2,...,88956 are the only numbers which cannot be written in this form with p=0 or p=1 (mod 4).

Zhi-Wei SUN, A new conjecture: n=p+x(x+1)/2, Mar 23, 2008.

Cf. A137997, A132399, A000040, A000217.

{ for( n=1,10^8, t=sqrtint(2*n); if( 0 >= p =n-t*(t+1)/2, !p & next, isprime(p) & p%4==1 & next); until( !t--, isprime( p+=t ) & p%4==1 & next(2)); print1(n","))}

A137997 Numbers which are not the sum of a triangular number and zero or a prime = 3 (mod 4).

Original entry on oeis.org

2, 5, 16, 27, 30, 42, 54, 61, 63, 90, 96, 129, 144, 165, 204, 216, 225, 285, 288, 309, 333, 340, 345, 390, 405, 423, 426, 448, 462, 525, 540, 556, 624, 651, 705, 801, 813, 876, 945, 960, 1056, 1230, 1371, 1380, 1470, 1491, 1827, 2085, 2157, 2181, 2220, 2355

Offset: 1

M. F. Hasler, Mar 24 2008

nonn

Zhi-Wei SUN conjectured that n=216 is the only number not of the form n = p + k(k+1)/2 with p a prime or zero and more precisely that the numbers given in this sequence A137997(1..112) = 2,...,90441 are the only numbers which cannot be written in this form with p=0 or p=3 (mod 4).

Zhi-Wei SUN, A new conjecture: n=p+x(x+1)/2, Mar 23, 2008.

Cf. A137996, A132399, A000040, A000217.

for( n=1,10^8, t=sqrtint( 2*n ); if( 0>= p = n-t*(t+1)/2, !p && next, p -= t++ ); until( !t--, isprime( p+=t ) || next; p%4==3 && next(2)); print1( n"," ))

A187785 Number of ways to write n=x+y (x,y>=0) with {6x-1,6x+1} a twin prime pair and y a triangular number.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 06 2013

nonn

Conjecture: a(n)>0 for all n>48624 not equal to 76106.
We have verified this for n up to 2*10^8. It seems that 723662 is the unique n>76106 which really needs y=0 in the described representation.
Compare the conjecture with another Sun's conjecture associated with A132399.

			a(9)=1 since 9=3+3(3+1)/2 with 6*3-1 and 6*3+1 both prime.

Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no. 1, 65-76.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A001097, A000217, A132399, A187759, A187757, A219157.

a[n_]:=a[n]=Sum[If[PrimeQ[6(n-k(k+1)/2)-1]==True&&PrimeQ[6(n-k(k+1)/2)+1]==True,1,0],{k,0,(Sqrt[8n+1]-1)/2}]
Do[Print[n," ",a[n]],{n,1,100}]

A256558 Number of ways to write n = p + floor(k*(k+1)/4), where p is a prime and k is a positive integer.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 01 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) For any integer m > 4 not equal to 12, each integer n > 1 can be written as p + floor((k^2-1)/m), where p is a prime and k is a positive integer.
We also have some other conjectures on representations n = p + floor(k*(k+1)/m) with m > 4.

			 a(15) = 1 since 15 = 5 + floor(6*7/4) with 5 prime.
a(420) = 1 since 420 = 419 + floor(2*3/4) with 419 prime.
a(945) = 1 since 945 = 877 + floor(16*17/4) with 877 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.

Cf. A000040, A000217, A132399, A256544, A256555.

Do[r=0;Do[If[PrimeQ[n-Floor[k(k+1)/4]],r=r+1],{k,1,(Sqrt[16n+1]-1)/2}];Print[n," ",r];Continue,{n,1,100}]

A238733 Number of primes p < n such that floor((n-p)/3) = (q-1)*(q-3)/8 for some prime q.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 3, 2, 2, 2, 4, 3, 4, 3, 4, 2, 3, 1, 3, 3, 4, 2, 3, 1, 2, 2, 3, 1, 2, 1, 4, 5, 5, 3, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 6, 5, 5, 4, 5, 3, 4, 2, 3, 3, 4, 2, 3, 3, 5, 5, 5, 2, 2, 1, 4, 4, 4, 3, 4, 3, 4, 4, 5, 4, 4, 1, 2

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: (i) For any integers m > 2 and n > 2, there is a prime p < n such that floor((n-p)/m) has the form (q-1)*(q-3)/8 with q an odd prime.
(ii) If m > 2 and n > m + 1,  then there is a prime p < n such that floor((n-p)/m) has the form (q^2 - 1)/8 with q an odd prime, except for the case m = 3 and n = 19.
Note that (q-1)*(q-3)/8 = r*(r+1)/2 with r = (q-3)/2. It seems that a(n) = 1 only for n = 3, 19, 25, 29, 31, 67, 79, 95, 96, 331, 373, 409.

			a(25) = 1 since floor((25-23)/3) = 0 = (3-1)*(3-3)/8 with 23 and 3 both prime.
a(96) = 1 since floor((96-11)/3) = 28 = (17-1)*(17-3)/8 with 11 and 17 both prime.
a(409) = 1 since floor((409-379)/3) = 10 = (11-1)*(11-3)/8 with 379 and 11 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no.1, 65-76. arXiv:0803.3737.
Zhi-Wei Sun, Problems on combinatorial properties of primes, 1402.6641, 2014.

Cf. A000040, A000217, A132399, A144590, A238732.

TQ[n_]:=PrimeQ[Sqrt[8n+1]+2]
t[n_,k_]:=TQ[Floor[(n-Prime[k])/3]]
a[n_]:=Sum[If[t[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

has(x)=issquare(8*x+1,&x) && isprime(x+2)
a(n)=my(s); forprime(p=2,n-1,s+=has((n-p)\3)); s \\ Charles R Greathouse IV, Mar 03 2014

A335641 Number of ordered ways to write 2n+1 as p + x*(9x+7) with p prime and x an integer.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 03 2020

nonn

Conjecture 1: a(n) > 0 for all n > 0. Also, a(n) = 1 only for n = 1, 4, 5, 7, 8, 18, 24, 25, 42, 68, 70, 85, 117, 118, 196, 238, 287, 497, 628, 677, 732.
We have verified a(n) > 0 for all n = 1..2*10^8.
Conjecture 2: Let f(x) be any of the polynomials x*(3x+1), x*(5x+1), 2x*(3x+1), 2x*(3x+2). Then, each odd integer greater than one can be written as p + f(x) with p prime and x an integer.

			a(68) = 1, and 2*68+1 = 137 + 0*(9*0+7) with 137 prime.
a(117) = 1, and 2*117+1 = 233 + (-1)*(9*(-1)+7) with 233 prime.
a(238) = 1, and 2*238+1 = 461 + 1*(9*1+7) with 461 prime.
a(287) = 1, and 2*287+1 = 293 + (-6)*(9*(-6)+7) with 293 prime.
a(732) = 1, and 2*732+1 = 673 + 9*(9*9+7) with 673 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. and Number Theory 1(2009), no. 1, 65-76. See also arXiv:0803.3737 [math.NT], 2008.

Cf. A000040, A132399, A144590, A256071.

tab={};Do[r=0;Do[If[PrimeQ[2n+1-x*(9*x+7)],r=r+1],{x,-Floor[(Sqrt[36(2n+1)+49]+7)/18],(Sqrt[36(2n+1)+49]-7)/18}];
tab=Append[tab,r],{n,1,100}];Print[tab]

A155190 Non-triangular numbers having a unique representation as p + t, where p is prime and t is a triangular number.

Original entry on oeis.org

2, 4, 7, 9, 16, 25, 27, 42, 54, 61, 63, 70, 75, 87, 90, 121, 126, 129, 135, 186, 211, 246, 315, 333, 345, 396, 405, 540, 556, 690, 717, 801, 924, 960, 1080, 1863, 2376, 2826, 3900, 6210, 8316

Offset: 1

T. D. Noe, Jan 21 2009

nonn

Conjectured to be finite. The prime terms are in A065397.

			n=p+t: 2=2+0; 4=3+1; 7=7+0; 9=3+6; 16=13+3; 25=19+6; 27=17+10; 42=41+1; 54=53+1

A132399, A154752

cp's OEIS Frontend

A117054 Number of ordered ways of writing n = i + j, where i is a prime and j is of the form k*(k+1), k > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A137996 Numbers which are not the sum of a triangular number and zero or a prime = 1 (mod 4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A137997 Numbers which are not the sum of a triangular number and zero or a prime = 3 (mod 4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A187785 Number of ways to write n=x+y (x,y>=0) with {6x-1,6x+1} a twin prime pair and y a triangular number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A256558 Number of ways to write n = p + floor(k*(k+1)/4), where p is a prime and k is a positive integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A238733 Number of primes p < n such that floor((n-p)/3) = (q-1)*(q-3)/8 for some prime q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A335641 Number of ordered ways to write 2n+1 as p + x*(9x+7) with p prime and x an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A155190 Non-triangular numbers having a unique representation as p + t, where p is prime and t is a triangular number.

Views

Author

Keywords

Comments

Examples

Crossrefs