A303540 -id:A303540 - cp's OEIS frontend

A303429 Number of ordered pairs (k, m) of nonnegative integers such that n - 3^k - 5^m can be written as the sum of two squares.

0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 3, 4, 1, 4, 4, 6, 3, 6, 4, 5, 5, 6, 4, 8, 4, 6, 5, 5, 4, 7, 5, 7, 5, 6, 4, 5, 3, 4, 6, 5, 5, 7, 5, 3, 6, 4, 4, 8, 3, 6, 5, 5, 4, 6, 4, 7, 6, 4, 4, 5, 4, 4, 5, 4, 5, 8, 4, 4, 5, 6, 4, 8, 2, 9, 7, 5, 5, 6

Offset: 1

Zhi-Wei Sun, Apr 28 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This is equivalent to the author's conjecture in A303656.
It has been verified that a(n) > 0 for all n = 2..10^9.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000244, A000290, A000351, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656.

a(5) = 1 with 5 - 3^1 - 5^0 = 0^2 + 1^2.
a(25) = 1 with 25 - 3^1 - 5^1 = 1^2 + 4^2.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[n-3^k-5^m],r=r+1],{k,0,Log[3,n]},{m,0,If[n==3^k,-1,Log[5,n-3^k]]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A303997 Number of ways to write 2n as p + 3^k + binomial(2m,m), where p is a prime, and k and m are nonnegative integers.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 4, 4, 4, 5, 4, 6, 6, 4, 6, 8, 6, 5, 8, 5, 5, 8, 5, 6, 10, 4, 4, 7, 5, 5, 7, 6, 4, 8, 4, 6, 11, 6, 5, 10, 8, 7, 9, 11, 7, 10, 7, 4, 11, 9, 9, 9, 10, 8, 12, 9, 9, 11, 9, 5, 8, 8, 4, 11, 8, 7, 8, 8, 7, 10, 8, 7, 6, 7, 5, 10, 9, 7, 12, 8, 5, 7, 9, 8, 9, 8, 6, 8, 11

Offset: 1

Zhi-Wei Sun, May 04 2018

nonn

502743678 is the first value of n > 1 with a(n) = 0.

			a(2) = 1 since 2*2 = 2 + 3^0 + binomial(2*0,0) with 2 prime.
a(3) = 2 since 2*3 = 3 + 3^0 + binomial(2*1,1) = 2 + 3^1 + binomial(2*0,0) with 3 and 2 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000224, A000984, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A303934, A303998.

c[n_]:=c[n]=Binomial[2n,n];
tab={};Do[r=0;k=0;Label[bb];If[c[k]>=2n,Goto[aa]];Do[If[PrimeQ[2n-c[k]-3^m],r=r+1],{m,0,Log[3,2n-c[k]]}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,90}];Print[tab]

A303998 Number of ways to write 2n+1 as p + 2^k + binomial(2m,m), where p is a prime, and k and m are positive integers.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 5, 3, 6, 5, 6, 8, 7, 5, 7, 7, 6, 8, 11, 5, 8, 9, 5, 10, 8, 7, 8, 7, 5, 7, 10, 6, 9, 9, 5, 11, 12, 8, 13, 12, 9, 8, 15, 9, 11, 12, 11, 7, 10, 9, 10, 14, 9, 12, 12, 11, 11, 12, 9, 9, 12, 8, 5, 13, 9, 10, 14, 10, 13, 9, 15, 10, 12, 9, 12, 11, 9, 11, 13

Offset: 1

Zhi-Wei Sun, May 04 2018

nonn

Conjecture: a(n) > 0 for all n > 2.

This has been verified for n up to 10^9.

			a(3) = 1 since 2*3+1 = 3 + 2^1 + binomial(2*1,1) with 3 prime.
a(4) = 2 since 2*4+1 = 3 + 2^2 + binomial(2*1,1) = 5 + 2^1 + binomial(2*1,1) with 3 and 5 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000079, A000984, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A303934, A303997, A304031.

c[n_]:=c[n]=Binomial[2n,n];
tab={};Do[r=0;k=1;Label[bb];If[c[k]>2n,Goto[aa]];Do[If[PrimeQ[2n+1-c[k]-2^m],r=r+1],{m,1,Log[2,2n+1-c[k]]}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]

A305048 Number of ordered pairs (k, m) of nonnegative integers such that 5^k + 10^m is not only a primitive root modulo prime(n) but also smaller than prime(n).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 24 2018

nonn

Conjecture: a(n) > 0 for all n > 4. In other words, any prime p > 7 has a primitive root g < p of the form 5^k + 10^m with k and m nonnegative integers.
We have verified this for any prime p > 7 not exceeding 10^9.
It seems that a(n) = 1 only for n = 2, 3, 11, 14, 53, 63, 64, 82, 99, 101, 111, 129, 344, 369, 391, 795, 1170, 1587, 5629, 5718, 6613, 430516.
See also A305048 for similar conjectures.

			a(14) = 1 with 5^2 + 10^0 = 26 a primitive root modulo prime(14) = 43.
a(101) = 1 with 5^0 + 10^0 = 2 a primitive root modulo prime(101) = 547.
a(111) = 1 with 5^2 + 10 = 35 a primitive root modulo prime(111) = 607.
a(5718) = 1 with 5^0 + 10^3 = 1001 a primitive root modulo prime(5718) = 56401.
a(6613) = 1 with 5^1 + 10^3 = 1005 a primitive root modulo prime(6613) = 66301.
a(430516) = 1 with 5^5 + 10^1 = 3135 a primitive root modulo prime(430516) = 6276271.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

A000040, A000351, A011557, A303540, A239957, A241476, A241504, A241516, A305030.

p[n_]:=p[n]=Prime[n];
Dv[n_]:=Dv[n]=Divisors[n];
gp[g_,p_]:=gp[g,p]=Mod[g,p]>0&&Sum[Boole[PowerMod[g,Dv[p-1][[k]],p]==1],{k,1,Length[Dv[p-1]]-1}]==0;
tab={};Do[r=0;Do[If[gp[5^a+10^b,p[n]],r=r+1],{a,0,Log[5,p[n]-1]},{b,0,Log[10,p[n]-5^a]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A305232 Number of ordered ways to write 2n+1 as p + binomial(2k,k) + 2binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 3, 4, 3, 5, 4, 5, 6, 5, 4, 4, 6, 6, 4, 5, 4, 6, 6, 6, 7, 6, 6, 4, 5, 6, 6, 8, 5, 5, 6, 5, 7, 9, 8, 5, 8, 9, 6, 9, 7, 8, 6, 6, 4, 7, 8, 7, 7, 4, 8, 10, 9, 7, 8, 9, 5, 7, 6, 5, 7, 7, 7, 3, 6, 7, 7, 9, 6, 9, 6, 9, 9, 7, 7, 8, 9, 6, 5, 8, 10, 10, 6, 8, 7, 9

Offset: 1

Zhi-Wei Sun, May 27 2018

nonn

The first value of n > 2 with a(n) = 0 is 15212443837. Neither 2*15212443837 + 1 = 30424887675 nor 2*15657981007 + 1 = 31315962015 can be written as the sum of a prime, a central binomial coefficient and twice a central binomial coefficient.

			a(3) = 1 since 2*3 + 1 = 7 = 3 + binomial(2*1,1) + 2*binomial(2*0,0) with 3 an odd prime.
a(368233372) = 1 since 2*368233372 + 1 = 736466745 = 735761311 + binomial(2*11,11) + 2*binomial(2*0,0) with 735761311 an odd prime.
a(5274658504) = 1 since 2*5274658504 + 1 = 10549317009 = 10549316083 + binomial(2*6,6) + 2*binomial(2*0,0) with 10549316083 an odd prime.
a(8722422187) = 1 since 2*8722422187 + 1 = 17444844375 = 17444844367 + binomial(2*2,2) + 2*binomial(2*0,0) with 17444844367 an odd prime.
a(10296844792) = 1 since 2*10296844792 + 1 = 20593689585 = 20593688659 + binomial(2*6,6) + 2*binomial(2*0,0) with 20593688659 an odd prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000984, A303540, A303656, A303702, A303821, A303934, A304034, A304081, A305030.

tab={};Do[r=0;k=0;Label[aa];k=k+1;If[Binomial[2k,k]>=2n+1`,Goto[cc]];m=0;Label[bb];If[2*Binomial[2m,m]>=2n+1-Binomial[2k,k],Goto[aa]]; If[PrimeQ[2n+1-Binomial[2k,k]-2*Binomial[2m,m]],r=r+1];m=m+1;Goto[bb];Label[cc];tab=Append[tab,r],{n,1,90}];Print[tab]

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A303429 Number of ordered pairs (k, m) of nonnegative integers such that n - 3^k - 5^m can be written as the sum of two squares.

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A303997 Number of ways to write 2*n as p + 3^k + binomial(2*m,m), where p is a prime, and k and m are nonnegative integers.

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A303998 Number of ways to write 2*n+1 as p + 2^k + binomial(2*m,m), where p is a prime, and k and m are positive integers.

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A305048 Number of ordered pairs (k, m) of nonnegative integers such that 5^k + 10^m is not only a primitive root modulo prime(n) but also smaller than prime(n).

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A305232 Number of ordered ways to write 2*n+1 as p + binomial(2k,k) + 2*binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.

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A303997 Number of ways to write 2n as p + 3^k + binomial(2m,m), where p is a prime, and k and m are nonnegative integers.

A303998 Number of ways to write 2n+1 as p + 2^k + binomial(2m,m), where p is a prime, and k and m are positive integers.

A305232 Number of ordered ways to write 2n+1 as p + binomial(2k,k) + 2binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.