login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A254610 Number of decompositions of 2n into sums of two primes p1 <= p2 such that the smallest |k*p1-p2| = 2^m+b, where |b|<=2. 1
0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 4, 6, 4, 4, 7, 4, 5, 8, 5, 4, 9, 4, 4, 7, 2, 5, 7, 4, 5, 8, 5, 6, 9, 5, 5, 11, 4, 5, 8, 3, 5, 7, 5, 4, 7, 6, 6, 8, 5, 4, 9, 3, 6, 8, 4, 7, 9, 4, 5, 11, 7, 5, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(1)=0 is the only zero term up to n=200000.

It is hypothesized that a(1)=0 is the only zero term of this sequence.

The histogram for 1<=n<=60000 of this sequence shows the shape of a distribution with mode=10, and it has a regional maximum at 20.

LINKS

Lei Zhou, Table of n, a(n) for n = 1..10000

Lei Zhou, Histogram for 1<=n<=60000

Index entries for sequences related to Goldbach conjecture

EXAMPLE

For n=1, 2n=2, which cannot be decomposed into the sum of two primes, so a(1)=0.

For n=2, 2n = 4 = 2+2, and 2-2 = 0 = 2^0-1, so the difference from 2^0 is 1, which satisfies the condition. So a(2)=1;

...

For n=5, 2n = 10 = 3+7 = 5+5. |3*2-7| = 1 = 2^0 and |5-5| = 0 = 2^0-1; both satisfy the condition, so a(5)=2.

...

For n=35, 2n = 70 = 3+67 = 11+59 = 17+53 = 23+47 = 29+41. These five Goldbach decompositions make A045917(35)=5. Among these, |3*22-67| = 1 = 2^0; |11*5-59| = 4 = 2^2; |17*3-53| = 2 = 2^1; |23*2-47| = 1 satisfies the condition. However, |29-41| = 12 = 2^3+4 = 2^4-4 does not satisfy the condition. So, a(35)=4 < A045917(35). This is the first term where the two sequences differ.

MATHEMATICA

NumDiff[n1_, n2_] :=  Module[{c1 = n1, c2 = n2}, If[c1 < c2, c1 = c1 + c2; c2 = c1 - c2; c1 = c1 - c2]; k = Floor[c1/c2]; a1 = c1 - k*c2; If[a1 == 0, a2 = 0, a2 = (k + 1) c2 - c1]; Return[Min[a1, a2]]];

Table[e = 2 n; p1 = 1; ct = 0; While[p1 = NextPrime[p1]; p1 <= n, p2 = e - p1; If[PrimeQ[p2], d = NumDiff[p1, p2]; k = Floor[Log[2, d]]; diff1 = d - 2^k; If[diff1 == 0, ct++, diff2 = 2^(k + 1) - d; If[(diff1 <= 2) || (diff2 <= 2), ct++]]]]; ct, {n, 1, 100}]

CROSSREFS

Cf. A045917, A254606

Sequence in context: A225638 A332656 A230443 * A002375 A045917 A240708

Adjacent sequences:  A254607 A254608 A254609 * A254611 A254612 A254613

KEYWORD

nonn,easy

AUTHOR

Lei Zhou, Feb 02 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 05:39 EST 2021. Contains 349530 sequences. (Running on oeis4.)