A050936 -id:A050936 - cp's OEIS frontend

A067379 Primes expressible as the sum of (at least two) consecutive primes in at least 3 ways.

311, 863, 1151, 1367, 1951, 2393, 2647, 2689, 3389, 4957, 5059, 5153, 7451, 7901, 8819, 10499, 10859, 10949, 12329, 12641, 12713, 13127, 13297, 14369, 14699, 14759, 14951, 15091, 15329, 15527, 16223, 16249, 16829, 18089, 18311, 18401

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
Patrick De Geest, WON plate 122
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A050936, A067372-A067381, A307610.

m=2*6!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[PrimeQ[p]&&pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

Prime(n) such that A307610(n) > 3. - Ray Chandler, Sep 21 2023

A084146 Integers that have exactly one representation as a sum of two or more consecutive primes.

Original entry on oeis.org

5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 39, 42, 48, 49, 52, 53, 56, 58, 59, 67, 68, 71, 75, 77, 78, 84, 88, 95, 97, 98, 101, 102, 109, 121, 124, 127, 128, 129, 131, 132, 139, 144, 150, 155, 156, 158, 159, 160, 161, 162, 168, 169, 172, 173, 181, 184, 186, 192

Offset: 1

Eric W. Weisstein, May 15 2003

nonn

More fundamental than A050936, which gives integers having 1 *or more* such representations

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A050936, A084143, A337095 (subset of primes).

# uses code of A084143
isA084146 := proc(n::integer)
    if A084143(n) = 1 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 300 do
    if isA084146(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Aug 19 2020

A084143(a(n)) = 1. - Ray Chandler, Sep 20 2023

More terms from Matthew Conroy, May 25 2003

A067373 Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways.

Original entry on oeis.org

240, 287, 311, 340, 371, 510, 660, 803, 863, 864, 931, 961, 990, 1012, 1060, 1099, 1104, 1151, 1164, 1236, 1313, 1320, 1367, 1392, 1524, 1643, 1650, 1710, 1788, 1793, 1854, 1951, 1956, 2040, 2303, 2304, 2387, 2393, 2436, 2507, 2556, 2586, 2647, 2670

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			240 = (113 + 127) = (53 + 59 + 61 + 67) = (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43) or (#2,113) (#4,53) (#8,17).

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A050936, A067372-A067381, A054998.

m=3*5!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

A084143(a(n)) > 2. - Ray Chandler, Sep 20 2023

Offset corrected by Donovan Johnson, Nov 14 2013

A067374 Integers expressible as the sum of (at least two) consecutive primes in at least 4 ways.

Original entry on oeis.org

311, 863, 1164, 1320, 1650, 1854, 2856, 2867, 3198, 3264, 3754, 4200, 4920, 5100, 5770, 5999, 6504, 8152, 10134, 10320, 10536, 10649, 11058, 12294, 12438, 12762, 12820, 12954, 12990, 14369, 14699, 14826, 15329, 15610, 15762, 16199, 16277

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			E.g., 311 = 101 + 103 + 107 = 53 + 59 + 61 + 67 + 71 = 31 + 37 + 41 + 43 + 47 + 53 + 59 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46

Cf. A050936, A067372-A067381, A054999.

Clear[lst,lst1,m,n,p,a,b] m=2*6!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

A084143(a(n)) > 3. - Ray Chandler, Sep 20 2023

Offset corrected by Donovan Johnson, Nov 14 2013

A067375 Integers expressible as the sum of (at least two) consecutive primes in at least 5 ways.

Original entry on oeis.org

16277, 20272, 25416, 28500, 34421, 41074, 45101, 46660, 50560, 53424, 59068, 68787, 70104, 70692, 71548, 78756, 85433, 85481, 88453, 94350, 98881, 105827, 117907, 120151, 121847, 125952, 130638, 130789, 131420, 132539, 133367, 134376, 135918, 139853, 158810

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			E.g. 16277 = (#7,2297) (#11,1451) (#13,1213) (#35,359) (#37,331).

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46

Cf. A050936, A067372-A067381, A055000.

t={};Do[p=Prime[m];Do[p=p+Prime[n];If[p<200000,AppendTo[t,p]],{n,m+1,7001}],{m,1,7000}];t=Sort@t;f5[l_]:=Module[{t={}},Do[If[l[[n]]==l[[n+4]],AppendTo[t,l[[n]]]],{n,Length[l]-4}];t];Union@f5[t] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

A084143(a(n)) > 4. - Ray Chandler, Sep 20 2023

Offset and a(35) corrected by Donovan Johnson, Nov 14 2013

A087072 Numbers having no partitions into a sum of two or more consecutive primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 16, 19, 20, 21, 22, 25, 27, 29, 32, 33, 34, 35, 37, 38, 40, 43, 44, 45, 46, 47, 50, 51, 54, 55, 57, 61, 62, 63, 64, 65, 66, 69, 70, 73, 74, 76, 79, 80, 81, 82, 85, 86, 87, 89, 91, 92, 93, 94, 96, 99, 103, 104, 105, 106, 107, 108, 110, 111

Offset: 1

Reinhard Zumkeller, Aug 08 2003

nonn

A084143(a(n)) = 0; complement of A050936.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A050936, A084143.

A367021 Numbers that can be written as both the sum of two or more consecutive nonprimes and the sum of two or more consecutive primes.

Original entry on oeis.org

5, 10, 17, 18, 23, 26, 28, 31, 36, 39, 41, 49, 53, 58, 59, 60, 67, 68, 71, 75, 77, 78, 83, 84, 90, 95, 97, 101, 102, 109, 112, 121, 124, 127, 128, 129, 131, 132, 138, 139, 143, 150, 152, 155, 156, 158, 159, 160, 161, 162, 168, 169, 172, 173, 180, 181, 184, 187, 192

Offset: 1

Tamas Sandor Nagy, Nov 01 2023

nonn

It seems that more than one consecutive number set from one kind or the other may exist for a term. Also, for some terms, an equal number of addends from each kind may correspond.

			5 is a term because 5 = 1 + 4 = 2 + 3, which is the sum of two consecutive nonprimes and also the sum of two consecutive primes.
17 is a term because 17 = 8 + 9 = 2 + 3 + 5 + 7, the sum of two consecutive nonprimes and also the sum of four consecutive primes.

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A018252, A050936, A366976.

from sympy import isprime
primes = [x for x in range(2,3000) if isprime(x)]
comps = [x for x in range(1,3000) if not isprime(x)]
psums = set(sum(primes[p:p+pn]) for pn in range(2,100) for p in range(len(primes)-pn))
csums = set(sum(comps[c:c+cn]) for cn in range(2,100) for c in range(len(comps)-cn))
terms = sorted(list(psums.intersection(csums)))
print(terms)
# David Consiglio, Jr., Dec 18 2023

More terms from David Consiglio, Jr., Dec 18 2023

A067376 Smallest integer expressible as the sum of (at least two) consecutive primes in n ways.

Original entry on oeis.org

5, 36, 240, 311, 16277, 130638, 218918, 9186778, 274452156, 4611108324, 12941709050

Offset: 1

Patrick De Geest, Feb 04 2002

a(10)-a(11) found by Wilfred Whiteside in 2007 (see Rivera link). - Michael S. Branicky, Jul 27 2022

			In n=7 ways: 218918 = (#12,18199) (#16,13619) (#22,9851) (#28,7691) (#38,5623) (#46,4561) (#62,3301).

Patrick De Geest, WONplate 122
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A050936, A067372-A067381, A054859.

Offset corrected and a(8)-a(9) from Donovan Johnson, Mar 14 2010

a(10) confirmed and a(10)-a(11) entered by Michael S. Branicky, Jul 27 2022

A135363 Sums of two or more consecutive semiprimes.

Original entry on oeis.org

10, 15, 19, 24, 25, 29, 33, 36, 39, 43, 47, 48, 50, 51, 54, 58, 59, 60, 67, 68, 69, 72, 73, 75, 77, 79, 82, 83, 84, 85, 91, 93, 94, 95, 97, 100, 101, 102, 106, 107, 109, 112, 115, 116, 118, 120, 122, 123, 126, 127, 128, 133, 134, 140, 142, 143, 146, 148, 151, 152

Offset: 1

Jonathan Vos Post, Dec 09 2007

This is to A050936 as A001358 is to A000040.

			a(1) = 10 = 4 + 6.
a(2) = 15 = 6 + 9.
a(3) = 19 = 9 + 10 = 4 + 6 + 9.
a(4) = 24 = 10 + 14.
a(5) = 25 = 6 + 9 + 10.
a(6) = 29 = 14 + 15 = 4 + 6 + 9 + 10.
a(7) = 33 = 9 + 10 + 14.
a(8) = 36 = 15 + 21.
a(9) = 39 = 10 + 14 + 15.
a(10) = 43 = 21 + 22.

Cf. A000040, A001358, A050936.

isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true; else false ; fi ; end: A001358 := proc(n) option remember ; local a; if n <= 3 then op(n,[4,6,9]) ; else a := A001358(n-1)+1 ; while not isA001358(a) do a := a+1 ; od ; RETURN(a) ; fi ; end: isA135363 := proc(n) local frst,lst, psum ; for frst from 1 do if A001358(frst) >= n then RETURN(false) ; fi ; for lst from frst+1 do psum := add(A001358(k),k=frst..lst) ; if psum = n then RETURN(true) ; elif psum > n then break ; fi ; od: od: end: for n from 4 to 200 do if isA135363(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Dec 11 2007

okQ[n_] := With[{SP = Select[Range[n], PrimeOmega[#] == 2 &]}, Select[IntegerPartitions[n, {2, Infinity}, SP], SequencePosition[SP, Reverse@#] != {}&]] != {};
Reap[For[k = 10, k < 200, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jan 29 2024 *)

Corrected and extended by R. J. Mathar, Dec 11 2007

A163246 Squares which can be represented as the sum of consecutive primes in more than one way.

Original entry on oeis.org

36, 100, 576, 841, 961, 1764, 1849, 2209, 2304, 7056, 22801, 24649, 25600, 30276, 31684, 32400, 36481, 39601, 40000, 47524, 48400, 48841, 57600, 58081, 66564, 69169, 69696, 76729, 77284, 80089, 85849, 93636, 94864, 96721, 112896, 119716, 128164, 134689, 138384, 140625, 142884, 147456

Offset: 1

Gaurav Kumar, Jul 23 2009

nonn

			6^2 = 36 = 5 + 7 + 11 + 13 = 17 + 19.

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A000290, A034707, A050936, A067372, A084143, A163244.

{ A000290 } intersect { A034707 }.
{ A000290 } intersect { A050936 }.
k^2 such that A084143(k^2) > 1. - Georg Fischer, Jul 08 2022

Offset corrected by Arkadiusz Wesolowski, Mar 28 2012

Duplicate 2304 removed and some missing terms inserted by Georg Fischer, Jul 08 2022

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A067379 Primes expressible as the sum of (at least two) consecutive primes in at least 3 ways.

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A084146 Integers that have exactly one representation as a sum of two or more consecutive primes.

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A067373 Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways.

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A067374 Integers expressible as the sum of (at least two) consecutive primes in at least 4 ways.

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A067375 Integers expressible as the sum of (at least two) consecutive primes in at least 5 ways.

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A087072 Numbers having no partitions into a sum of two or more consecutive primes.

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A367021 Numbers that can be written as both the sum of two or more consecutive nonprimes and the sum of two or more consecutive primes.

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A067376 Smallest integer expressible as the sum of (at least two) consecutive primes in n ways.

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