A014092 -id:A014092 - cp's OEIS frontend

A213879 Positive palindromes that are not the sum of two positive palindromes.

1, 111, 131, 141, 151, 161, 171, 181, 191, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10301, 10401, 10501, 10601, 10701, 10801, 10901, 11111, 11211, 11311, 11411, 11511, 11611, 11711, 11811, 11911, 12021, 12121, 12321, 12421, 12521, 12621, 12721, 12821

Offset: 1

Arkadiusz Wesolowski, Jun 23 2012

These numbers do not occur in A035137.

			22 is not a member because 22 = 11 + 11.

Alois P. Heinz, Table of n, a(n) for n = 1..2151 (first 111 terms from N. J. A. Sloane)
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes

Cf. A002113, A014092, A035137, A083142, A088601, A260255, A319477.

# From N. J. A. Sloane, Sep 09 2015: bP is a list of the palindromes
a:={}; M:=400; for n from 3 to M do p:=bP[n];
# is p a sum of two palindromes?
sw:=-1; for i from 2 to n-1 do j:=p-bP[i]; if digrev(j)=j then sw:=1; break; fi;
od;
if sw<0 then a:={op(a),p}; fi; od:
b:=sort(convert(a,list));

lst1 = {}; lst2 = {}; r = 12821; Do[If[FromDigits@Reverse@IntegerDigits[n] == n, AppendTo[lst1, n]], {n, r}]; l = Length[lst1]; Do[s = lst1[[i]] + lst1[[j]]; AppendTo[lst2, s], {i, l - 1}, {j, i}]; Complement[lst1, lst2]
palQ[n_] := Reverse[x = IntegerDigits[n]] == x; t1 = Select[Range[12900], palQ[#] &]; Complement[t1, Union[Flatten[Table[i + j, {i, t1}, {j, t1}]]]] (* Jayanta Basu, Jun 15 2013 *)

({ A002113 } intersect { A319477 }) minus { 0 }. - Alois P. Heinz, Sep 19 2018

A100962 Numbers that can neither be written as the sum nor as the product of two primes.

Original entry on oeis.org

1, 2, 3, 11, 17, 23, 27, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 107, 113, 117, 125, 127, 131, 135, 137, 147, 149, 157, 163, 167, 171, 173, 179, 189, 191, 197, 207, 211, 223, 227, 233, 239, 245, 251, 255, 257, 261, 263, 269, 275, 277, 281, 293, 297

Offset: 1

Reinhard Zumkeller, Nov 24 2004

nonn

Intersection of A014092 and A100959.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084997.

Cf. A064911, A014092, A157931, A100959.

a100962 n = a100962_list !! (n-1)
a100962_list = filter ((== 0) . a064911) a014092_list
-- Reinhard Zumkeller, Oct 15 2014

A133398 Numbers that are not Mersenne primes.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64

Offset: 1

Omar E. Pol, Nov 22 2007, Apr 05 2008

Numbers that are not in A000668.

Terms a(1) to a(2042) coincide with those of A138836, but then a(2043)=2047 <> A138836(2043)=2048.

Cf. A000037, A014092. Mersenne primes: A000668.

Cf. A138836.

A051035 Composite numbers which can be represented as the sum of two primes (i.e., A002808 excluding A025583).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Eric W. Weisstein

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Partition.

Cf. A002808, A025583.

Cf. A010051, subsequence of A014092.

a051035 n = a051035_list !! (n-1)
a051035_list = filter ((== 0) . a010051) a014091_list

r[n_] := Reduce[2 <= p <= q && n == p + q, {p, q}, Primes]; Select[Range[4, 105], r[#] =!= False && ! PrimeQ[#] & ] (* Jean-François Alcover, Oct 29 2012 *)

A062302 Number of ways writing n-th prime as a sum of a prime and a nonprime.

Original entry on oeis.org

0, 1, 0, 1, 4, 3, 6, 5, 8, 9, 8, 11, 12, 11, 14, 15, 16, 15, 18, 19, 18, 21, 22, 23, 24, 25, 24, 27, 26, 29, 30, 31, 32, 31, 34, 33, 36, 37, 38, 39, 40, 39, 42, 41, 44, 43, 46, 47, 48, 47, 50, 51, 50, 53, 54, 55, 56, 55, 58, 59, 58, 61, 62, 63, 62, 65, 66, 67, 68, 67, 70, 71, 72

Offset: 1

Labos Elemer, Jul 05 2001

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A061358, A062602, A000040, A014092, A025584.

Table[c = 0; Do[i = Prime[k]; If[i + j == Prime[n] && ! PrimeQ[j], c = c + 1], {k, n - 1}, {j, Prime[n] - 1}]; c, {n, 73}] (* Jayanta Basu, Apr 22 2013 *)
nn = 100; mx = Prime[nn]; ps = Prime[Range[nn]]; notPs = Complement[Range[mx], ps]; t2 = Table[0, {Range[mx]}]; Do[s = i + j; If[s <= mx, t2[[s]]++], {i, ps}, {j, notPs}];  t2[[ps]] (* T. D. Noe, Apr 23 2013 *)

a(n) = A062602(A000040(n)) = number of [nonprime+prime] partitions of prime(n)

A086860 Numbers in A086473 corresponding to the unique product of two numbers having the unique sum of A086533.

Original entry on oeis.org

52, 244, 1168, 1776, 4672, 4192, 2608, 724, 8128, 916, 1912, 3328, 15424, 9952, 3352, 3592, 53632, 80128, 36352, 51712, 65152, 5272, 20512, 72832, 22432, 111756, 133888, 84352, 6472, 48448, 26272, 172288, 107392, 37480, 187648, 242496

Offset: 1

Lekraj Beedassy, Sep 12 2003

nonn

Related to Martin Gardner's "Impossible Problem".

a(n) is thus a subsequence of A086473, itself a subsequence of A058080. Consider the mapping f:P->S defined thus: S is the sum of a factor pair (both different from 1) of P, where P is a(n). If S is A086533(n) (a subsequence of A014092), then both f and its inverse are injective (but not onto).

Cf. A086533.

Corrected by Ray Chandler, Oct 23 2003

A110673 Numbers that are neither the sum nor the difference of two primes.

Original entry on oeis.org

23, 37, 47, 53, 67, 79, 83, 89, 93, 97, 113, 117, 119, 121, 123, 127, 131, 143, 145, 157, 163, 167, 173, 185, 187, 203, 205, 207, 211, 215, 217, 219, 223, 233, 245, 247, 251, 257, 263, 277, 287, 289, 293, 297, 299, 301, 303, 307, 317, 321, 323, 325, 327, 331

Offset: 1

Eric Angelini, Sep 14 2005

The sequence is obtained by interleaving A099019 and A134797. From Goldbach's conjecture, apparently all terms are odd. - Bob Selcoe, Mar 10 2015

Intersection of A007921 and A014092. - Michel Marcus, Mar 16 2015

Michel Marcus, Table of n, a(n) for n = 1..7596
Oliver Knill, Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes, arXiv preprint arXiv:1606.05958 [math.NT], 2016.
Wikipedia, Goldbach's conjecture

Cf. A007921 (not the difference), A014092 (not the sum).

Cf. also A099019, A134797.

Lim=331; nn=PrimePi[Lim+1]; (* Lim is upper limit of sequence; nn is range of primes to consider *)
dif=Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]; (* differences of two primes *)
sum=Union[Join[Flatten[Total/@Subsets[Prime[Range[nn]],{2}]],Table[2*Prime[n], {n, nn}]]];seq2; (* sums of two primes *)
Complement[Range[Lim],dif,sum] (* neither sum nor difference *) (* James C. McMahon, Jun 10 2024 *)

Corrected and extended by Joshua Zucker, May 04 2006

Offset corrected by Arkadiusz Wesolowski, May 19 2012

A152482 Even numbers which are not the sum of 2 even semiprimes.

Original entry on oeis.org

2, 4, 6, 22, 34, 46, 54, 58, 70, 74, 82, 94, 102, 106, 114, 118, 130, 134, 142, 154, 158, 166, 174, 178, 186, 190, 194, 202, 214, 226, 234, 238, 242, 246, 250, 254, 262, 270, 274, 286, 290, 294, 298, 310, 314, 322, 326, 334, 342, 346, 354, 358, 370, 374, 378

Offset: 1

Donovan Johnson, Dec 06 2008

nonn

Twice A014092. [From Omar E. Pol, Dec 26 2008]

			102 is in this sequence because no 2 even semiprimes sum to 102. 104 is not in this sequence because the sum of 10 (even semiprime) and 94 (even semiprime) is 104.

Cf. A001358, A100484, A152483, A152484.

Cf. A014092. [From Omar E. Pol, Dec 26 2008]

a(n) = A014092(n)*2. [From Omar E. Pol, Dec 26 2008]

A178431 Joint-rank array of the sums of two primes.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 6, 5, 5, 6, 9, 7, 7, 7, 9, 11, 10, 8, 8, 10, 11, 14, 12, 12, 10, 12, 12, 14, 16, 15, 13, 13, 13, 13, 15, 16, 19, 17, 17, 15, 17, 15, 17, 17, 19, 23, 20, 18, 18, 18, 18, 18, 18, 20, 23, 25, 24, 21, 20, 21, 20, 21, 20, 21, 24, 25

Offset: 1

Clark Kimberling, Dec 21 2010

Joint-rank arrays are defined at A157927 for arrays in which duplicates occur (and otherwise, at A182801).

			A corner of the array A178400 of sums of two primes:
4...5...7...9..13...
5...6...8..10..14...
7...8..10..12..14...
9..10..12..14..18...
Each of these P(i,j) is replaced by its rank when all the
numbers P(h,k) are jointly ranked, leaving A178431:
1...2...4...6...9...
2...3...5...7..10...
4...5...7...8..12...
6...7...8..10..13...
The number of distinct sums p+q <=13 is 9.

Cf. A014092, A157927, A178400.

A191837 Least even number m which can be written as sum of 2n primes p(1) < ... < p(2n) < m/2 such that m-p(i) is also prime for i=1,...,2n.

Original entry on oeis.org

48, 108, 204, 324, 624, 630, 1050, 1320, 1590, 2100, 2400, 2730, 3570, 3960, 4830, 5460, 5880, 6930, 7770, 9240, 9450, 11970, 12810, 13020, 14910, 14910, 17430, 18480, 20160, 21630, 23100, 24150, 28770, 28770, 31290, 32760, 32760, 36960, 36960, 39270, 39270, 50190, 51870, 51870

Offset: 2

J. M. Bergot, Jun 17 2011

nonn

Original definition: In the Goldbach partitions of 2n, find the first 2n with four prime elements to sum to it; find the first 2n with six elements summing to 2n; and so for 2k elements.
Whenever there is more than one decomposition of m as sum of primes, it must be odd+odd=even. Then, only an even number of (odd prime) summands can yield m. Moreover, we restrict these summands to be the lesser one of the decompositions p+q=m, therefore we need more than 2 such summands to yield m, and a(1) is undefined.
The integers in this sequence are all congruent to 0 mod 6.
There can be more than one composition of m. E.g., for m=48, 48=5+7+17+19 and 48=7+11+13+17.
Conjecture: For all a(n), a(n)-1 can be found in A014092 (numbers not the sum of two primes), and a(n)+1 can be found in A007921. (numbers not the difference of two primes). - J. Stauduhar, Aug 28 2012
From J. Stauduhar, Aug 22 2011: (Start)
  All a(n) are congruent to 0 mod 6=2*3.
  All a(n) >= a(7)=630 are congruent to 0 mod 30=2*3*5.
  All a(n) >= a(16)=4830 are congruent to 0 mod 210=2*3*5*7.
  All a(n) >= a(279)=3513510 are congruent to 0 mod 2310=2*3*5*7*11.
  All a(n) >= a(1440)=137507370 are congruent to 0 mod 30030=2*3*5*7*11*13.  (End)

			For 48, we have 48=5+43=7+41=17+31=19+29 (ignoring 11+37), and use 5+7+17+19 to give the first even number having four such primes summing to itself.
Similarly, 108 is the least even number with six prime elements summing to itself: 5+103=7+101=11+97=19+89=29+79=37+71 and taking 5+7+11+19+29+37=108.
a(2) = 48 = 5+7+17+19 = 7+11+13+17
a(3) = 108 = 5+7+11+19+29+37
a(9) = 1320 = 13+17+19+23+29+31+37+41+43+61+71+83+89+97+103+107+149+307

J. Stauduhar, Table of n, a(n) for n = 2..2500
J. Stauduhar, C program to generate sequence A191837

nCk[a_, b_]:=Block[{ndx=ns= a, i=rs=b, ct=t=0}, If[(d[[1]]-1)==(ns-rs), For[ct=1, ct<=rs, ct++, t+=s[[d[[ct]]]]]; If[t==m, Print[sm/2, " ", t]; sm+=2; m-=6; Return[False], Return[False]]]; While[d[[i]]==ndx && i>1, --i; --ndx]; d[[i]]+=1; i++; While[i<=rs, d[[i]]=d[[i-1]]+1; ++i;]; For[ct=1, ct<=rs, ct++, t+=s[[d[[ct]]]]; If[t>m, Break[]]]; If[t==m, Print[sm/2, " ", t]; sm+=2; m-=6; Return[False]]; Return[True]]; For[sm=4; m=6, sm<=60, m+=6, s={}; sum=smndct=pct=0; For[p=5, pm, Break[]]; If[smndct++= sm, d=Range[sm]; While[nCk[Length[s], sm]]]]; (* J. Stauduhar, Sep 07 2012*)

a(n)=forstep(m=2,1e9,2,L=[]; forprime(p=1,m\2-1,isprime(m-p)||next;L=concat(L,p)); #L<2*n&next; sum(i=#L-2*n+1,#L,L[i])

a(4)-a(5) from M. F. Hasler, Jun 21 2011
a(2) to a(5) verified; a(6) to a(10) added by S Kolman, Jul 03 2011
a(11) to a(13) added by S Kolman, Jul 04 2011
a(14) to a(14) added by S Kolman, Jul 05 2011
Confirmed a(7). a(6) corrected by J. Stauduhar, Jul 08 2011
Corrected a(8)-a(14) and extended to a(2500). - J. Stauduhar, Jul 12 2011
Edited by J. Stauduhar, Aug 28 2012

cp's OEIS Frontend

A213879 Positive palindromes that are not the sum of two positive palindromes.

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A100962 Numbers that can neither be written as the sum nor as the product of two primes.

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Haskell

A133398 Numbers that are not Mersenne primes.

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A051035 Composite numbers which can be represented as the sum of two primes (i.e., A002808 excluding A025583).

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Haskell

Mathematica

A062302 Number of ways writing n-th prime as a sum of a prime and a nonprime.

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Mathematica

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A086860 Numbers in A086473 corresponding to the unique product of two numbers having the unique sum of A086533.

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Extensions

A110673 Numbers that are neither the sum nor the difference of two primes.

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Mathematica

Extensions

A152482 Even numbers which are not the sum of 2 even semiprimes.

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Formula

A178431 Joint-rank array of the sums of two primes.

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A191837 Least even number m which can be written as sum of 2n primes p(1) < ... < p(2n) < m/2 such that m-p(i) is also prime for i=1,...,2n.

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