cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A025583 Composite numbers that are not the sum of 2 primes.

Original entry on oeis.org

27, 35, 51, 57, 65, 77, 87, 93, 95, 117, 119, 121, 123, 125, 135, 143, 145, 147, 155, 161, 171, 177, 185, 187, 189, 203, 205, 207, 209, 215, 217, 219, 221, 237, 245, 247, 249, 255, 261, 267, 275, 287, 289, 291, 297, 299, 301, 303, 305, 321, 323, 325, 327, 329, 335, 341
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Goldbach conjectured that every integer > 5 is the sum of three primes.
Conjecture: This is the sequence of odd numbers k such that (k mod x) mod 2 != 1, where x is the greatest m <= k such that m, m-1 and m-2 are all composite. Verified for first 10000 terms. - Benedict W. J. Irwin, May 06 2016
Numbers k, such that however many of k coins are placed with heads rather than tails showing, either those showing heads or those showing tails can be arranged in a rectangular pattern with multiple rows and columns. (If the Goldbach conjecture for even numbers is false this comment should be restricted to the odd terms of this sequence, as it might otherwise define a variant sequence). - Peter Munn, May 15 2017

Links

Crossrefs

Cf. A051034, A001031, A002372, A002374, A071335.
Cf. A002808, A000040, A010051.

Programs

  • Haskell
    a025583 n = a025583_list !! (n-1)
a025583_list = filter f a002808_list where
   f x = all (== 0) $ map (a010051 . (x -)) $ takeWhile (< x) a000040_list
-- Reinhard Zumkeller, Oct 15 2014
  • Mathematica
    f[n_] := (p = 0; pn = PrimePi[n]; Do[ If[n == Prime[i] + Prime[k], p = p + 1; If[p > 2, Break[]]], {i, 1, pn}, {k, i, pn}]; p ); Select[Range[2, 400], ! PrimeQ[#] && f[#] == 0 & ] (* Jean-François Alcover, Mar 07 2011 *)
upto=350;With[{c=PrimePi[upto]},Complement[Range[4,upto], Prime[Range[ c]], Union[Total/@Tuples[Prime[Range[c]],{2}]]]] (* Harvey P. Dale, Jul 14 2011 *)
Select[Range[400],CompositeQ[#]&&Count[IntegerPartitions[#,{2}],?(AllTrue[ #,PrimeQ]&)]==0&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Feb 21 2021 *)

A218007 Number of partitions of n into at most three primes (including 1).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 5, 4, 5, 4, 6, 5, 7, 5, 7, 5, 8, 6, 9, 6, 9, 7, 10, 7, 10, 5, 10, 6, 12, 7, 13, 7, 12, 8, 14, 7, 14, 6, 15, 8, 17, 9, 17, 8, 18, 10, 19, 10, 19, 7, 20, 9, 21, 9, 20, 7, 21, 11, 25, 11, 24, 9, 26, 11, 27, 9, 24, 8, 28, 12, 30, 13, 29
Offset: 1

Views

Author

Frank M Jackson, Mar 26 2013

Keywords

Comments

The above sequence relies on the strong Goldbach's conjecture that any positive integer is the sum of at most three distinct terms from {1 union primes}.

Examples

			a(21)=9 as 21 = 1+1+19 = 2+19 = 1+3+17 = 2+2+17 = 1+7+13 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7

Links

Crossrefs

Cf. A002375, A025583, A068307, A071335, A185101.

Programs

  • Mathematica
    primeQ[p0_] := If[p0==1, True, PrimeQ[p0]]; SetAttributes[primeQ, Listable]; goldbachcount[p1_] := (parts=IntegerPartitions[p1, 3]; count=0; n=1; While[n<=Length[parts], If[Intersection[Flatten[primeQ[parts[[n]]]]][[1]] == True, count++]; n++]; count); Table[goldbachcount[i], {i, 1, 100}]
Table[Length[Select[#/.(1->2)&/@IntegerPartitions[n,3],AllTrue[#,PrimeQ]&]],{n,80}] (* Harvey P. Dale, Jan 11 2023 *)

A223893 Number of partitions of n into at most three distinct primes.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 4, 3, 4, 3, 4, 3, 5, 3, 5, 3, 4, 4, 5, 6, 5, 5, 5, 5, 7, 6, 5, 7, 4, 7, 7, 8, 7, 7, 6, 10, 8, 9, 9, 8, 7, 12, 8, 12, 8, 10, 6, 14, 9, 15, 8, 13, 7, 14, 11, 16, 8, 14, 7, 19, 11, 19, 10, 15, 9, 21, 12, 20, 11, 18
Offset: 1

Views

Author

Frank M Jackson, Mar 28 2013

Keywords

Comments

The sequence shows a stronger version of the Goldbach conjecture that for n > 6, n has partitions with at most three distinct primes.

Examples

			a(21)=3 as 21 = 2+19 = 3+5+13 = 3+7+11.

Links

Crossrefs

Cf. A000586, A000607, A071335, A185101, A218007, A218469.

Programs

  • Mathematica
    a[n_] := Length@Select[IntegerPartitions[n, 3, Prime@Range@PrimePi@n],
Sort@#==Union@# &]; Array[a, 100] (* Giovanni Resta, Mar 29 2013 *)

A347578 Number of partitions of n into at most 4 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 9, 11, 11, 13, 11, 15, 12, 16, 15, 16, 15, 18, 16, 20, 17, 23, 18, 24, 20, 26, 22, 26, 23, 31, 23, 33, 26, 35, 26, 39, 27, 41, 32, 41, 31, 46, 31, 48, 34, 51, 34, 54, 36, 58, 40, 58, 42, 64, 41
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Crossrefs

Cf. A000040, A000607, A071335, A117278, A259194, A347552, A347609, A347610.

Formula

a(n) = Sum_{k=1..4} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A347609 Number of partitions of n into at most 5 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 9, 10, 11, 11, 14, 14, 15, 17, 18, 19, 21, 22, 23, 25, 27, 27, 32, 29, 34, 33, 37, 37, 42, 39, 47, 44, 51, 47, 58, 50, 61, 57, 67, 61, 73, 65, 80, 71, 86, 75, 95, 79, 101, 86, 107, 92, 115, 95, 125, 103, 132, 108
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Crossrefs

Cf. A000040, A000607, A071335, A117278, A259195, A347552, A347578, A347610.

Formula

a(n) = Sum_{k=1..5} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A347610 Number of partitions of n into at most 6 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 9, 11, 12, 14, 15, 17, 18, 20, 23, 24, 27, 28, 32, 33, 37, 38, 43, 44, 48, 51, 55, 55, 63, 62, 70, 71, 77, 78, 89, 87, 97, 96, 108, 105, 121, 114, 133, 127, 144, 139, 161, 149, 174, 165, 189, 177, 208, 188, 226, 206
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Crossrefs

Cf. A000040, A000607, A071335, A117278, A259196, A347552, A347578, A347609.

Formula

a(n) = Sum_{k=1..6} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A083338 Number of partitions of odd numbers into three primes and of even numbers into two primes.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 4, 2, 3, 2, 5, 3, 5, 3, 5, 3, 7, 2, 7, 3, 6, 2, 9, 4, 8, 4, 9, 2, 10, 3, 11, 4, 10, 3, 12, 4, 13, 5, 12, 4, 15, 3, 16, 5, 14, 3, 17, 4, 16, 6, 16, 3, 19, 5, 21, 6, 20, 2, 20, 5, 22, 6, 21, 5, 22, 5, 28, 7, 24, 4, 25, 5, 29, 8, 27, 5, 29, 4, 33, 9, 29, 4
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 24 2003

Keywords

Comments

a(n) > 0 for all n iff Goldbach's conjectures hold.

Links

Crossrefs

Cf. A000607, A061358, A068307, A071335, A083339.

Programs

  • Mathematica
    f[n_] := Length@ IntegerPartitions[n, If[ OddQ@ n, {3}, {2}], Prime@ Range@ PrimePi@ n]; Array[f, 92] (* Robert G. Wilson v, Nov 28 2012 *)

Formula

a(n) = if n is even then A045917(n/2) else A054860((n-1)/2).
For even n: a(n) = A061358(n); for odd n: a(n) = A068307(n). - Antti Karttunen, Sep 14 2017

A307857 Number of partitions of n into 1, 2 or 3 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 3, 4, 3, 5, 4, 6, 5, 9, 7, 10, 8, 12, 10, 15, 11, 18, 15, 20, 17, 24, 19, 28, 22, 30, 26, 36, 29, 41, 34, 42, 37, 51, 41, 55, 47, 59, 53, 66, 54, 73, 63, 78, 70, 85, 72, 94, 81, 99, 89, 108, 92, 118, 102, 121, 110, 135, 117, 143, 126
Offset: 1

Views

Author

Wesley Ivan Hurt, May 01 2019

Keywords

Examples

			a(9) = 3, because 9 can be written as the sum of nonprimes with at most 3 parts in three ways: 9 = 8+1 = 4+4+1.
a(10) = 4, because 10 can be written as the sum of nonprimes with at most 3 parts in four ways: 10 = 9+1 = 6+4 = 8+1+1.
a(11) = 3, because 11 can be written as the sum of nonprimes with at most 3 parts in three ways: 10+1 = 9+1+1 = 6+4+1.
a(12) = 5, because 12 can be written as the sum of nonprimes with at most 3 parts in five ways: 12 = 8+4 = 6+6 = 10+1+1 = 4+4+4.

Links

Crossrefs

Cf. A005171, A018252, A071335.

Formula

a(n) = c(n) + ( Sum_{i=1..floor(n/2)} c(i) * c(n-i) ) + ( Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) ), where c = A005171.

A347552 Number of partitions of n into at most 2 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 1, 3, 0, 2, 1, 3, 2, 2, 1, 4, 0, 4, 1, 2, 1, 3, 1, 4, 2, 3, 1, 4, 1, 5, 1, 4, 0, 3, 1, 5, 1, 3, 0, 4, 1, 6, 2, 3, 1, 5, 0, 6, 1, 2, 1, 5, 1, 6, 2, 5, 1, 5, 0, 7, 1, 4, 1, 5, 1, 8, 1, 5
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Crossrefs

Cf. A000040, A000607, A061358, A071335, A117278, A347578, A347609, A347610.

Formula

a(n) = Sum_{k=1..2} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A218469 Number of partitions of n into at most three distinct primes (including 1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 5, 5, 6, 5, 5, 5, 6, 6, 6, 5, 4, 6, 6, 9, 7, 7, 6, 8, 7, 10, 6, 8, 5, 10, 8, 12, 9, 10, 7, 13, 9, 14, 10, 12, 7, 15, 9, 17, 9, 13, 6, 17, 10, 21, 10, 15, 8, 19, 11, 22, 9, 16, 8, 24, 12, 25, 12, 19, 10, 26, 12
Offset: 1

Views

Author

Frank M Jackson, Mar 26 2013

Keywords

Comments

Using {1 union primes} as the base, the above sequence relies on the strong Goldbach's conjecture that any positive integer is the sum of at most three distinct terms.

Examples

			a(21)=5 as 21 = 2+19 = 1+3+17 = 1+7+13 = 3+5+13 = 3+7+11.

Links

Crossrefs

Cf. A002375, A025583, A068307, A071335, A185101, A218007.

Programs

  • Mathematica
    primeQ[p0_] := If[p0==1, True, PrimeQ[p0]]; SetAttributes[primeQ, Listable]; goldbachcount[p1_] := (parts=IntegerPartitions[p1, 3]; count=0; n=1; While[n<=Length[parts], If[Intersection[Flatten[primeQ
  [parts[[n]]]]][[1]]&&Total[Intersection[parts[[n]]]]==Total[parts
  [[1]]], count++]; n++]; count); Table[goldbachcount[i], {i, 1, 100}]
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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