A103151 -id:A103151 - cp's OEIS frontend

A103506 Smallest prime p such that 2n+1 = 2q + p for some odd prime q, or 0 if no such prime exists.

0, 0, 0, 3, 5, 3, 5, 3, 5, 7, 13, 3, 5, 3, 5, 7, 13, 3, 5, 3, 5, 7, 13, 3, 5, 7, 17, 11, 13, 3, 5, 3, 5, 7, 13, 11, 13, 3, 5, 7, 37, 3, 5, 3, 5, 7, 13, 3, 5, 7, 17, 11, 13, 3, 5, 7, 29, 11, 13, 3, 5, 3, 5, 7, 13, 11, 13, 3, 5, 7, 37, 3, 5, 3, 5, 7, 13, 11

Offset: 1

Lei Zhou, Feb 09 2005

nonn

			For n < 4 there are no such primes, thus a(1)-a(3)=0.
For n=4, 2*4+1 = 9 = 2*3+3, thus a(4)=3.
For n=11, 2*11+1 = 23 = 2*5+13, thus a(11)=13.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

a(n)=0 if A103509(n)=0, otherwise A000040(A103509(n)).

Cf. A103151, A103152, A103153.

Join[{0,0,0}, Table[m=3; While[! (PrimeQ[m] && (((n-m)/2) > 2) && PrimeQ[(n-m)/2]), m=m+2]; m, {n, 9, 299, 2}]]

(define (A103506 n) (let ((ind (A103509 n))) (if (zero? ind) 0 (A000040 ind)))) ;; Antti Karttunen, Jun 19 2007

Edited by Antti Karttunen, Jun 19 2007

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 5, 5, 3, 7, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 5, 5, 3, 7, 3, 3, 5, 5, 3, 7, 3, 19, 5, 3, 7, 5, 11, 3, 11, 3, 3, 5, 3, 3, 5, 3, 7, 5, 11, 7, 11, 11, 3, 11, 3, 13, 5, 3, 3, 5, 5, 7, 7, 3, 3, 5, 5, 3, 7, 5, 3, 7, 3, 13, 5, 3, 7, 5, 3, 3, 5, 5, 7, 7, 3

Offset: 1

Lei Zhou, Feb 09 2005

nonn

			For n < 4 there are no such primes, thus a(1)-a(3)=0. For n=4, 2*4+1 = 9 = 2*3+3, thus a(4)=3. For n=7, 2*7+1 = 15 = 2*5+5, thus a(7)=7.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

a(n)=0 if A103507(n)=0, otherwise A000040(A103507(n)).
Cf. A103151, A103152, A002373.
Cf. A195352 (similar definition, but p=2 is allowed).

Do[m = 3; While[ ! (PrimeQ[m] && ((n - 2*m) > 2) && PrimeQ[n - 2*m]), m = m + 2]; Print[m], {n, 9, 299, 2}]

(define (A103153 n) (let ((ind (A103507 n))) (if (zero? ind) 0 (A000040 ind))))

Edited and Scheme code added by Antti Karttunen, Jun 19 2007

Definition corrected by Hugo Pfoertner, Sep 16 2011

A103152 Smallest odd number which can be written as a sum 2p+q (where p and q are both odd primes, A065091) in exactly n ways, zero if there are no such odd number.

Original entry on oeis.org

9, 13, 17, 29, 45, 51, 69, 81, 99, 93, 105, 135, 153, 201, 195, 165, 231, 237, 321, 297, 225, 363, 725, 393, 285, 315, 471, 483, 435, 405, 465, 561, 555, 495, 609, 783, 675, 867, 849, 963, 645, 525, 693, 897, 795, 915, 987, 735, 855, 825, 765, 1095, 975, 1467

Offset: 1

Lei Zhou, Feb 09 2005

nonn

Conjecture: except for the 2nd, 3rd and 4th terms, all other terms are divisible by 3; See also comments in A103151.
a(23)=725 is also not divisible by 3. [D. S. McNeil, Sep 06 2010]
The only terms a(n) not divisible by 3 for n <= 1450 are a(2),a(3),a(4) and a(23). - Robert Israel, Mar 17 2020

			9 is the smallest odd number with just one such composition: 9 = 3+2*3, thus a(1)=9.
Similarly, 13 is smallest with exactly 2 compositions: 13 = 3+2*5 = 7+2*3, thus a(2)=13.

Robert Israel, Table of n, a(n) for n = 1..1450

Cf. A103151, A001172, A001031.

N:= 2000: # for terms before the first term > N
P:= select(isprime, [seq(i,i=3..N,2)]):
nP:= nops(P):
V:= Vector(N):
for i from 1 while 2*P[i] N then break fi;
    V[k]:= V[k]+1;
od od:
A:= Vector(N):
for i from 1 to N by 2 do if V[i] <> 0 and A[V[i]] = 0 then A[V[i]]:= i fi od:
for i from 1 to N do if A[i] = 0 then break fi od:
seq(A[j],j=1..i-1); # Robert Israel, Mar 17 2020

Array[a, 300]; Do[a[n] = 0, {n, 1, 300}]; n = 9; ct = 0; While[ct < 200, m = 3; ct = 0; While[(m*2) < n, If[PrimeQ[m], cp = n - (2* m); If[PrimeQ[cp], ct = ct + 1]]; m = m + 2]; If[a[ct] == 0, a[ct] = n]; n = n + 2]; Print[a]

(define (A103152 n) (+ 1 (* 2 (first-n-where-fun_n-is-i1 A103151 n))))
(define (first-n-where-fun_n-is-i1 fun i) (let loop ((n 1)) (cond ((= i (fun n)) n) (else (loop (+ n 1)))))) ;; Antti Karttunen, Jun 19 2007

Starting offset changed from 0 to 1 by Antti Karttunen, Jun 19 2007

A103508 a(n) = 1 + 2 * least i such that A103507(i)=n+1, 0 if no such i exists.

Original entry on oeis.org

9, 15, 31, 101, 139, 227, 91, 503, 995, 451, 751, 539, 1819, 1397, 2957, 3461, 1831, 1417, 6023, 3769, 1777, 9587, 5411, 9421, 18653, 8089, 4511, 6541, 10529, 16051, 19049, 13163, 3139, 22937, 23929, 43363, 24919, 43571, 97367, 55571, 14419, 75209

Offset: 1

Lei Zhou, Feb 10 2005

Cf. A103510, A103151, A103152, A103153, A103506, A103507, A025017.

(define (A103508 n) (+ 1 (* 2 (first-n-where-fun_n-is-i1 A103507 (+ 1 n)))))
(define (first-n-where-fun_n-is-i1 fun i) (let loop ((n 1)) (cond ((= i (fun n)) n) (else (loop (+ n 1))))))

Edited and Scheme-code added by Antti Karttunen, Jun 19 2007

A103510 a(n) = 1 + 2 * least i such that A103509(i)=n+1, 0 if no such i exists.

Original entry on oeis.org

9, 11, 21, 57, 23, 55, 245, 241, 115, 833, 83, 523, 437, 193, 447, 733, 167, 689, 1417, 611, 2297, 1081, 2731, 1283, 2755, 5057, 2761, 887, 2719, 9221, 4909, 8179, 4397, 13891, 9557, 2351, 9257, 5869, 10627, 11941, 1487, 2797, 3947, 5899, 11237, 20069

Offset: 1

Lei Zhou, Feb 10 2005

Cf. A103508, A103151, A103152, A103153, A103506, A103509, A025017.

Array[a, 500]; Do[a[n] = 0, {n, 1, 500}]; n = 9; ct = 0; While[ct < 150, m = 3; While[ ! (PrimeQ[m] && (((n - m)/2) > 2) && PrimeQ[(n - m)/2]), m = m + 2]; k = PrimePi[m]; If[a[k] == 0, a[k] = n; ct = ct + 1]; n = n + 2]; Print[a]

(define (A103510 n) (+ 1 (* 2 (first-n-where-fun_n-is-i1 A103509 (+ 1 n)))))
(define (first-n-where-fun_n-is-i1 fun i) (let loop ((n 1)) (cond ((= i (fun n)) n) (else (loop (+ n 1))))))

Edited and Scheme-code added by Antti Karttunen, Jun 19 2007

A276520 a(n) is the number of decompositions of n into unordered form p + c*q, where p, q are terms of A274987, c=1 for even n-s and c=2 for odd n-s.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 0, 3, 3, 1, 2, 4, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 0, 2, 2, 0, 1, 3, 1, 3, 2, 0, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 4, 1, 2, 2, 3, 4, 4, 3, 4

Offset: 1

Lei Zhou, Nov 11 2016

p=q is allowed.
It is conjectured that the primes p, q in A274987 (a subset of all primes) are sufficient to decomposite all numbers into p and c*q (c=1 when n is even, 2 when c is odd) when n > 2551.
This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 3, 4, 5, 7, 32, 52, 55, 61, 128, 194, 214, 244, 292, 334, 388, 782, 902, 992, 1414, 1571, 1712, 1916, 2551}.
There are no more zero terms of a(n) up to n = 100000.

			A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}
For n=6, 6 = 3+3, one case of decomposition, so a(6)=1;
For n=7, 7 < 3+2*3=9, no eligible case could be found, so a(7)=0;
...
For n=17, 17 = 3+2*7 = 7+2*5 = 11+2*3, three cases of decompositions, so a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Lei Zhou, List plot of the first 10000 terms of a(n).

Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443, A276034, A103151, A001031.

p = 3; sp = {p}; Table[l = Length[sp]; While[sp[[l]] < n, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; c = 2 - Mod[n + 1, 2]; ct = 0; Do[If[MemberQ[sp, n - c*sp[[i]]], If[c == 1, If[(2*sp[[i]]) <= n, ct++], ct++]], {i, 1, l}]; ct, {n, 1, 87}]

A147568 a(n) = 2*A000695(n)+3.

Original entry on oeis.org

3, 5, 11, 13, 35, 37, 43, 45, 131, 133, 139, 141, 163, 165, 171, 173, 515, 517, 523, 525, 547, 549, 555, 557, 643, 645, 651, 653, 675, 677, 683, 685, 2051, 2053, 2059, 2061, 2083, 2085, 2091, 2093, 2179, 2181, 2187, 2189, 2211, 2213, 2219, 2221, 2563, 2565, 2571

Offset: 0

Vladimir Shevelev, Nov 07 2008

nonn

Every odd number m>=9 is a unique sum of the form a(k)+2a(l); moreover this sequence is the unique one with such property. In connection with A103151, note that there is no subsequence T of primes such that every odd number m>=9 is expressible as a unique sum of the form m=p+2q, where p and q are in T. One can prove that if one replaces 9 by any integer x_o>9, the statement remains true (see the Shevelev link).

Vladimir Shevelev, On Unique Additive Representations of Positive Integers and Some Close Problems, arXiv:0811.0290 [math.NT], 2008.

Cf. A000695, A062880, A103151.

(* b = A000695 *) b[n_] := If[n==0, 0, If[EvenQ[n], 4 b[n/2] , b[n-1]+1]];
a[n_] := 2 b[n] + 3; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 14 2018 *)

a000695(n) = fromdigits(binary(n), 4);
a(n) = 2*a000695(n)+3; \\ Michel Marcus, Dec 13 2018

More terms from Michel Marcus, Dec 13 2018

A238268 The number of unordered ways that n can be written as the sum of two numbers of the form p or 2p, where p is prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 5, 4, 4, 4, 5, 4, 4, 3, 4, 6, 4, 3, 7, 4, 3, 5, 6, 5, 5, 5, 6, 7, 4, 4, 9, 5, 5, 7, 6, 5, 5, 4, 5, 7, 4, 3, 10, 4, 4, 8, 8, 7, 7, 5, 6, 8, 5, 4, 10, 5, 5, 9, 8, 7, 8, 5, 7, 9, 5, 4, 13, 8, 6, 8, 8, 7

Offset: 4

Lei Zhou, Feb 21 2014

nonn

p and 2p are terms of A001751.
Sequence defined for n >= 4.
It is conjectured that all terms of this sequence are greater than zero.

			n=4, 4=2+2, one case found. So a(4)=1;
...
n=24, 24 = 2+2*11 = 5+19 = 7+17 = 2*5+2*7 = 11+13, 5 cases found.  So a(24)=5;
...
n=33, 33 = 2+31 = 2*2+29 = 7+2*13 = 2*5+23 = 11+2*11 = 2*7+19, 6 cases found.  So a(33)=6.

Lei Zhou, Table of n, a(n) for n = 4..10000

Cf. A001751, A002375, A103151.

Table[ct = 0; Do[If[((PrimeQ[i]) || (PrimeQ[i/2])) && ((PrimeQ[n - i]) || (PrimeQ[(n - i)/2])), ct++], {i, 2, Floor[n/2]}]; ct, {n, 4, 89}]

isp(i) = isprime(i) || (((i % 2) == 0) && isprime(i/2));
a(n) = sum(i=1, n\2, isp(i) && isp(n-i)); \\ Michel Marcus, Mar 07 2014

A152460 Primes p such that there exist positive integer k and prime q with p > q and 3^k = p + 2q or 3^k = q + 2p.

Original entry on oeis.org

3, 5, 11, 13, 17, 23, 29, 31, 37, 43, 47, 59, 67, 71, 97, 101, 103, 107, 109, 113, 137, 149, 157, 181, 197, 229, 233, 239, 251, 263, 269, 271, 281, 283, 307, 311, 313, 331, 347, 349, 353, 359, 367, 383, 431, 467, 503, 523, 563, 571, 587, 607, 643, 647, 683, 691

Offset: 1

Vladimir Shevelev, Dec 05 2008, Dec 12 2008

nonn

a(n) is the greater of primes (p,q) in representations of a power of 3 in Lemoine-Levy's form p+2q (see A046927)

If 3^n=p+2q, then 3^(n-1)<=max(p,q)<3^n. Therefore the sets of greater primes for different powers of 3 do not intersect.

			27=5+2*11=13+2*7=17+2*5=23+2*2, so that 11,13,17 and 23 are in the sequence.

Cf. A103151, A086081, A152451

aa(n)={my(v=[]); forprime(p=2,n\2,q=n-p*2; if(isprime(q),v=concat(v,(max(p,q))))); vecsort(v,,8)};
for(n=2, 7, v=aa(3^n); for(i=1,#v,print1(v[i], ", ")))

If A(x) is the counting function of a(n)<=x, then A(x)=O(xloglogx/(logx)^2).

Program and editing by Charles R Greathouse IV, Nov 02 2009

A238269 Smallest number that can be written in n ways as the sum of two numbers of the form p or 2p, where p is prime.

Original entry on oeis.org

4, 6, 8, 16, 24, 33, 36, 63, 48, 60, 93, 140, 84, 108, 132, 189, 165, 144, 120, 210, 297, 168, 204, 180, 276, 252, 285, 288, 462, 240, 372, 432, 336, 300, 396, 609, 360, 492, 552, 468, 564, 528, 576, 504, 708, 1089, 648, 480, 420, 540, 768, 672, 600, 816, 792

Offset: 1

Lei Zhou, Feb 21 2014

nonn

			4 is the smallest number that can be written in only one way as the sum of two primes or doubled primes, 4=2+2. So a(1)=4;
6 = 2+2*2 = 3+3, two ways, so a(2)=6;
...
33 = 2+31 = 2*2+29 = 7+2*13 = 2*5+23 = 11+2*11 = 2*7+19, 6 ways.  And all numbers smaller than 33 can only be split in 5 or fewer ways.  So a(6)=33.

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

Cf. A001751, A002375, A103151, A238268.

target = 55; n = 3; a = {}; sc = 0; Do[AppendTo[a, 0], {i, 1, target}]; While[sc < target, n++; ct = 0; Do[If[((PrimeQ[i]) || (PrimeQ[i/2])) && ((PrimeQ[n - i]) || (PrimeQ[(n - i)/2])), ct++], {i, 2, Floor[n/2]}]; If[ct<=target,If[a[[ct]] == 0, a[[ct]] = n; sc++]]];a

cp's OEIS Frontend

A103506 Smallest prime p such that 2n+1 = 2q + p for some odd prime q, or 0 if no such prime exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Extensions

A103153 a(n) is the smallest odd prime p such that 2*n+1 = 2*p + A000040(k) for some k>1, or 0 if no such prime exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Extensions

A103152 Smallest odd number which can be written as a sum 2p+q (where p and q are both odd primes, A065091) in exactly n ways, zero if there are no such odd number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Scheme

Extensions

A103508 a(n) = 1 + 2 * least i such that A103507(i)=n+1, 0 if no such i exists.

Views

Author

Keywords

Crossrefs

Programs

Scheme

Extensions

A103510 a(n) = 1 + 2 * least i such that A103509(i)=n+1, 0 if no such i exists.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Scheme

Extensions

A276520 a(n) is the number of decompositions of n into unordered form p + c*q, where p, q are terms of A274987, c=1 for even n-s and c=2 for odd n-s.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A147568 a(n) = 2*A000695(n)+3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A238268 The number of unordered ways that n can be written as the sum of two numbers of the form p or 2p, where p is prime.

Views

Author

Keywords

Comments

Examples

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.