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A276830 Number of ways to write n as ((p-1)/2)^2 + P_2, where p is an odd prime and P_2 is a product of at most two primes. 2
0, 1, 1, 1, 2, 2, 2, 2, 1, 3, 3, 2, 2, 3, 3, 2, 1, 3, 2, 2, 1, 2, 3, 2, 1, 4, 3, 2, 2, 4, 2, 3, 1, 3, 4, 2, 2, 5, 4, 4, 2, 5, 3, 3, 2, 3, 5, 3, 1, 5, 3, 2, 2, 2, 3, 3, 2, 4, 4, 3, 2, 5, 3, 2, 3, 5, 3, 4, 3, 4, 5, 2, 3, 5, 4, 2, 3, 5, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 9, 17, 21, 25, 33, 49, 109, 169, 189, 361, 841, 961, 12769, 19321.

See also A276825 for a similar conjecture involving cubes, and some comments on x^2 + P_2.

LINKS

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

EXAMPLE

a(2) = 1 since 2 = ((3-1)/2)^2 + 1 with 3 prime.

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

a(4) = 1 since 4 = ((3-1)/2)^2 + 3 with 3 prime.

a(9) = 1 since 9 = ((5-1)/2)^2 + 5 with 5 prime.

a(17) = 1 since 17 = ((5-1)/2)^2 + 13 with 5 and 13 both prime.

a(21) = 1 since 21 = ((5-1)/2)^2 + 17 with 5 and 17 both prime.

a(25) = 1 since 25 = ((5-1)/2)^2 + 3*7 with 5, 3 and 7 all prime.

a(33) = 1 since 33 = ((5-1)/2)^2 + 29 with 5 and 29 both prime.

a(49) = 1 since 49 = ((13-1)/2)^2 + 13 with 13 prime.

a(109) = 1 since 109 = ((13-1)/2)^2 + 73 with 13 and 73 both prime.

a(169) = 1 since 169 = ((13-1)/2)^2 + 7*19 with 13, 7 and 19 all prime.

a(189) = 1 since 189 = ((5-1)/2)^2 + 5*37 with 5 and 37 both prime.

a(361) = 1 since 361 = ((37-1)/2)^2 + + 37 with 37 prime.

a(841) = 1 since 841 = ((37-1)/2)^2 + 11*47 with 37, 11 and 47 all prime.

a(961) = 1 since 961 = ((61-1)/2)^2 + 61 with 61 prime.

a(12769) = 1 since 12769 = ((109-1)/2)^2 + 59*167 with 109, 59 and 167 all prime.

a(19321) = 1 since 19321 = ((277-1)/2)^2 + 277 with 277 prime.

MATHEMATICA

PP[n_]:=PP[n]=PrimeQ[Sqrt[n]]||(SquareFreeQ[n]&&Length[FactorInteger[n]]<=2)

Do[r=0; Do[If[PP[n-((Prime[k]-1)/2)^2], r=r+1; If[r>1, Goto[aa]]], {k, 2, PrimePi[2*Sqrt[n]+1]}]; Print[n, " ", r];

Label[aa]; If[Mod[n, 50000]==0, Print[n]]; Continue, {n, 10^5, 1000000}]

CROSSREFS

Cf. A000040, A000290, A037143, A235645, A276711, A276825.

Sequence in context: A118144 A136691 A231168 * A064741 A281659 A114294

Adjacent sequences:  A276827 A276828 A276829 * A276831 A276832 A276833

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 20 2016

STATUS

approved

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Last modified June 26 04:11 EDT 2019. Contains 324369 sequences. (Running on oeis4.)