A263020 -id:A263020 - cp's OEIS frontend

A262999 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)/2) + pi(m^2), where pi(x) denotes the number of primes not exceeding x.

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

Offset: 1

Zhi-Wei Sun, Oct 07 2015

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 3, 5, 7, 35, 37, 217, 7439, 10381.
We have verified this for n up to 120000.
See also A262995, A263001 and A263020 for similar conjectures.

			a(2) = 2 since 2 = pi(1*2/2) + pi(2^2) = pi(2*3/2) + pi(1^2).
a(3) = 1 since 3 = pi(3*4/2) + pi(1^2).
a(5) = 1 since 5 = pi(3*4/2) + pi(2^2).
a(7) = 1 since 7 = pi(3*4/2) + pi(3^2).
a(35) = 1 since 35 = pi(13*14/2) + pi(6^2).
a(37) = 1 since 37 = pi(3*4/2) + pi(12^2).
a(217) = 1 since 217 = pi(17*18/2) + pi(33^2).
a(590) = 1 since 590 = 58 + 532 = pi(23*24/2) + pi(62^2).
a(7439) = 1 since 7439 = 3854 + 3585 = pi(269*270/2) + pi(183^2).
a(10381) = 1 since 10381 = 1875 + 8506 = pi(179*180/2) + pi(296^2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A000720, A038107, A111208, A262995, A263001, A263020.

s[n_]:=s[n]=PrimePi[n^2]
t[n_]:=t[n]=PrimePi[n(n+1)/2]
Do[r=0;Do[If[s[k]>n,Goto[bb]];Do[If[t[j]>n-s[k],Goto[aa]];If[t[j]==n-s[k],r=r+1];Continue,{j,1,n-s[k]+1}];Label[aa];Continue,{k,1,n}];Label[bb];Print[n," ",r];Continue,{n,1,100}]

A263001 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k(k+1)) + pi(m(m+1)/2), where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 07 2015

nonn

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 1, 4, 6.
We have verified this for n up to 10^5.
See also A262995, A262999 and A263020 for similar conjectures.

			a(1) = 1 since 1 = pi(1*2) + pi(1*2/2).
a(4) = 1 since 4 = pi(1*2) + pi(3*4/2).
a(6) = 1 since 6 = pi(2*3) + pi(3*4/2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000720, A002378, A111208, A262995, A262999, A263020.

s[n_]:=s[n]=PrimePi[n(n+1)]
t[n_]:=t[n]=PrimePi[n(n+1)/2]
Do[r=0;Do[If[s[k]>n,Goto[bb]];Do[If[t[j]>n-s[k],Goto[aa]];If[t[j]==n-s[k],r=r+1];Continue,{j,1,n-s[k]+1}];Label[aa];Continue,{k, 1, n}];Label[bb];Print[n," ",r];Continue,{n,1,100}]

A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 09 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 28.
(ii) Any integer n > 0 can be written as pi(k^2) + pi((m^2+1)/2) with k and m positive integers.
(iii) Each n = 1,2,3,... can be written as pi(k^2/2) + pi((m^2+1)/2) with k and m positive integers.
See also A262995, A262999, A263001 and A263020 for similar conjectures.

			a(1) = 1 since 1 = 0 + 1 = pi(1^2) + pi(2^2/2).
a(3) = 1 since 3 = 2 + 1 = pi(2^2) + pi(2^2/2).
a(28) = 1 since 28 = 11 + 17 = pi(6^2) + pi(11^2/2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000720, A038107, A262995, A262999, A263001, A263020.

s[n_]:=s[n]=PrimePi[n^2]
t[n_]:=t[n]=PrimePi[n^2/2]
Do[r=0; Do[If[s[k]>n, Goto[bb]]; Do[If[t[j]>n-s[k], Goto[aa]]; If[t[j]==n-s[k], r=r+1]; Continue, {j, 1, n-s[k]+1}]; Label[aa]; Continue, {k, 1, n}];
Label[bb]; Print[n, " ", r]; Continue, {n,1,100}]

A263107 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2/2) + pi(3*m^2/2), where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 09 2015

nonn

Conjecture: a(n) > 0 for all n > 0.
We have verified this for n up to 3*10^5.
It seems that a(n) = 1 only for n = 1, 2, 3, 6, 19, 22, 48, 60, 396, 1076, 3033, 3889, 4741, 6804, 7919, 9604, 16938, 19169, 32533, 59903, 100407.

			a(1) = 1 since 1 = 1 + 0 = pi(2^2/2) + pi(3*1^2/2).
a(2) = 1 since 2 = 2 + 0 = pi(3^2/2) + pi(3*1^2/2).
a(3) = 1 since 3 = 0 + 3 = pi(1^2/2) + pi(3*2^2/2).
a(6) = 1 since 6 = 0 + 6 = pi(1^2/2) + pi(3*3^2/2).
a(19) = 1 since 19 = 7 + 12 = pi(6^2/2) + pi(3*5^2/2).
a(22) = 1 since 22 = 1 + 21 = pi(2^2/2) + pi(3*7^2/2).
a(48) = 1 since 48 = 1 + 47 = pi(2^2/2) + pi(3*12^2/2).
a(60) = 1 since 60 = 25 + 35 = pi(14^2/2) + pi(3*10^2/2).
a(396) = 1 since 396 = 334 + 62 = pi(67^2/2) + pi(3*14^2/2).
a(1076) = 1 since 1076 = 47 + 1029 = pi(21^2/2) + pi(3*74^2/2).
a(3033) = 1 since 3033 = 7 + 3026 = pi(6^2/2) + pi(3*136^2/2).
a(3889) = 1 since 3889 = 1808 + 2081 = pi(176^2/2) + pi(3*110^2/2).
a(4741) = 1 since 4741 = 4699 + 42 = pi(301^2/2) + pi(3*11^2/2).
a(6804) = 1 since 6804 = 6047 + 757 = pi(346^2/2) + pi(3*62^2/2).
a(7919) = 1 since 7919 = 4049 + 3870 = pi(277^2/2) + pi(3*156^2/2).
a(9604) = 1 since 9604 = 4754 + 4850 = pi(303^2/2) + pi(3*177^2/2).
a(16938) = 1 since 16938 = 2223 + 14715 = pi(198^2/2) + pi(3*327^2/2).
a(19169) = 1 since 19169 = 6510 + 12659 = pi(361^2/2) + pi(3*301^2/2).
a(32533) = 1 since 32533 = 1768 + 30765 = pi(174^2/2) + pi(3*490^2/2).
a(59903) = 1 since 59903 = 59210 + 693 = pi(1213^2/2) + pi(3*59^2/2).
a(100407) = 1 since 100407 = 7554 + 92853 = pi(392^2/2) + pi(3*894^2/2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000720, A262995, A262999, A263001, A263020, A263100.

s[n_]:=s[n]=PrimePi[3n^2/2]
t[n_]:=t[n]=PrimePi[n^2/2]
Do[r=0;Do[If[s[k]>n, Goto[bb]];Do[If[t[j]>n-s[k],Goto[aa]];If[t[j]==n-s[k],r=r+1];Continue,{j, 1, n-s[k]+1}];Label[aa];Continue, {k, 1, n}];Label[bb];Print[n," ",r]; Continue,{n,1,100}]

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A262999 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)/2) + pi(m^2), where pi(x) denotes the number of primes not exceeding x.

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A263001 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)) + pi(m*(m+1)/2), where pi(x) denotes the number of primes not exceeding x.

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A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.

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A263107 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2/2) + pi(3*m^2/2), where pi(x) denotes the number of primes not exceeding x.

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A263001 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k(k+1)) + pi(m(m+1)/2), where pi(x) denotes the number of primes not exceeding x.