A256071 -id:A256071 - cp's OEIS frontend

A256119 Least number p that is zero or an odd prime, such that n - p is a generalized pentagonal number.

0, 0, 0, 3, 3, 0, 5, 0, 3, 7, 3, 11, 0, 11, 7, 0, 11, 5, 3, 7, 5, 19, 0, 11, 17, 3, 0, 5, 13, 3, 23, 5, 17, 7, 19, 0, 29, 11, 3, 13, 0, 19, 7, 3, 29, 5, 11, 7, 13, 23, 43, 0, 17, 13, 3, 29, 5, 0, 7, 19, 3, 59, 5, 23, 7, 43, 31, 41, 11, 29, 0, 31, 37, 3, 17, 5, 19, 0, 43, 53, 3, 11, 5, 13, 7, 59, 29, 17, 11, 19, 13, 79, 0, 23, 17, 3, 19, 5, 41, 7, 0

Offset: 0

Zhi-Wei Sun, Mar 15 2015

nonn

By the conjecture in A256071, a(n) always exists.

			a(21) = 19 since 21 is not a generalized pentagonal number, and 19 is the least odd prime p with 21 - p a generalized pentagonal number.
a(26) = 0 since 26 = (-4)*(3*(-4)-1)/2 is a generalized pentagonal number.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000040, A001318, A256071.

Pen[n_]:=IntegerQ[Sqrt[24n+1]]
Do[If[Pen[n],Print[n," ",0];Goto[aa]];Do[If[Pen[n-Prime[k]],Print[n," ",Prime[k]];Goto[aa]],{k,2,PrimePi[n]}];Label[aa];Continue,{n,0,100}]

A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 26 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 54, 84, 87, 109, 174, 252, 344, 1234, 1439, 2924.

This implies the Twin Prime Conjecture.

			a(1439) = 1 since 1439 = 1424 + 15 = floor(4274/3) + (-3)*(3*(-3)-1)/2 with {3*4274-1,3*4274+1} = {12821,12823} a twin prime pair.
a(2924) = 1 since 2924 = 2334 + 590 = floor(7004/3) + 20*(3*20-1)/2 with {3*7004-1, 3*7004+1} = {21011,21013} a twin prime pair.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.

Cf. A001318, A014574, A257121, A256071, A256707, A257317, A257474.

TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
SQ[n_]:=IntegerQ[Sqrt[24n+1]]
Do[m=0;Do[If[PQ[x]&&SQ[n-x],m=m+1],{x,0,n-1}];
Print[n," ",m];Continue,{n,1,100}]

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]

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A256119 Least number p that is zero or an odd prime, such that n - p is a generalized pentagonal number.

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A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

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

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Keywords

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Mathematica