A034963 -id:A034963 - cp's OEIS frontend

A133537 Sum of sixth powers of two consecutive primes.

793, 16354, 133274, 1889210, 6598370, 28964378, 71183450, 195081770, 742859210, 1482327002, 3453230090, 7315830650, 11071467290, 17100578378, 32943576458, 64344894770, 93700908002, 141978756530, 218558666090, 279434510210

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=793 because 2^6+3^6=793.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133538.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a, {n, 1, 100}]

a(n) = A030516(n) + A030516(n+1). - Michel Marcus, Nov 09 2013

A127339 Numbers that are the sum of 12 consecutive primes.

Original entry on oeis.org

197, 236, 276, 318, 364, 412, 460, 510, 562, 612, 662, 714, 766, 822, 880, 936, 990, 1040, 1092, 1152, 1212, 1276, 1336, 1402, 1464, 1524, 1586, 1650, 1716, 1786, 1854, 1918, 1980, 2040, 2100, 2162, 2234, 2304, 2370, 2436, 2502, 2564, 2634, 2700, 2770

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of coefficient of x^11 of the polynomial Product_{j=0..11} (x - prime(n+j)) of degree 12; the roots of this polynomial are prime(n), ..., prime(n+11).

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338.

[&+[ NthPrime(n+k): k in [0..11] ]: n in [1..100] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 11}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],12,1] (* Harvey P. Dale, May 05 2018 *)

{m=45;k=12;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=45;k=12;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

Edited by Klaus Brockhaus, Jan 13 2007

A102655 Numbers that are the arithmetic mean of four successive primes.

Original entry on oeis.org

9, 12, 15, 18, 22, 30, 38, 42, 46, 55, 60, 68, 81, 87, 102, 105, 108, 114, 120, 127, 139, 144, 149, 155, 165, 175, 181, 186, 195, 200, 215, 228, 232, 241, 247, 253, 260, 265, 270, 278, 291, 306, 312, 318, 333, 341, 352, 357, 363, 381, 387, 399, 420, 426, 431

Offset: 1

Giovanni Teofilatto, Feb 02 2005

			a(1) = 9 because (5+7+11+13)/4=9;
a(2) = 12 because (7+11+13+17)/4=12;
a(3) = 15 because (11+13+17+19)/4=15.

Cf. A034963.

Select[ Table[ Sum[ Prime[i], {i, n, n + 3}]/4, {n, 83}], IntegerQ[ # ] &] (* Robert G. Wilson v, Feb 04 2005 *)
Select[MovingAverage[Prime[Range[500]],4],IntegerQ] (* Harvey P. Dale, Aug 10 2012 *)

4*n = A000040(i) + A000040(i+1) + A000040(i+2) + A000040(i+3) for some i>=1.

Edited by Robert G. Wilson v and W. Neville Holmes, Feb 04 2005

A133533 Sum of sixth powers of three consecutive primes.

Original entry on oeis.org

16418, 134003, 1904835, 6716019, 30735939, 76010259, 219219339, 789905091, 1630362891, 4048053411, 8203334331, 13637193699, 21850682619, 39264939507, 75124110099, 115865269131, 184159290171, 270079040451, 369892892379

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=16418 because 2^6+3^6+5^6=16418.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532.

L:= [seq(ithprime(i)^6,i=1..100)]:
L[1..-3]+L[2..-2]+L[3..-1]; # Robert Israel, Jun 28 2018

a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a, {n, 1, 100}]
Total/@(Partition[Prime[Range[25]],3,1]^6)  (* Harvey P. Dale, Mar 29 2011 *)

a(n) = A133537(n) + A030516(n+2). - Michel Marcus, Nov 09 2013

A131019 Semiperimeters of quadrilaterals whose sides are 4 consecutive odd primes.

Original entry on oeis.org

13, 18, 24, 30, 36, 44, 51, 60, 69, 76, 84, 92, 101, 110, 120, 129, 136, 145, 153, 162, 174, 185, 195, 204, 210, 216, 228, 240, 254, 267, 278, 288, 298, 310, 319, 330, 341, 350, 362, 372, 381, 390, 400, 415, 430, 445, 456, 464, 471, 482, 494, 506, 520, 530

Offset: 1

Jonathan Vos Post, Jun 09 2007

nonn

(2+3+5+7)/2 = 8.5, not an integer. Hence we restrict to odd primes. The cyclic quadrilaterals whose areas, rounded, are prime are given in A131020. The prime semiperimeters begin: a(1) = 13, a(13) = 101. This arises in the cyclic quadrilateral analog of A106171.

			a(1) = (3 + 5 + 7 + 11)/2 = 13.

Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta's Formula", Sect. 3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.

J. L. Coolidge, A Historically Interesting Formula for the Area of a Quadrilateral, Amer. Math. Monthly 46, 345-347, 1939.
Eric Weisstein's World of Mathematics, Brahmagupta's Formula.

Cf. A106171, A131020.

Cf. A034963.

A131019 := proc(n) local i ; add( ithprime(n+i),i=1..4)/2 ; end: for n from 1 to 180 do printf("%d, ",A131019(n)) : od:

Plus@@@Partition[Prime[Range[2,6! ]],4,1]/2 (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

a(n) = (prime(n) + prime(n+1) + prime(n+2) + prime(n+3))/2 for n>1.

a(n) = (prime(n+1) + prime(n+2) + prime(n+3) + prime(n+4))/2 = A034963(n)/2.

Edited by R. J. Mathar, Jun 12 2007

A133539 Sum of third powers of five consecutive primes.

Original entry on oeis.org

1834, 4023, 8909, 15643, 27467, 50525, 78119, 123859, 185921, 253261, 332695, 451781, 606507, 764567, 985823, 1239911, 1480051, 1767711, 2112517, 2516723, 3071485, 3712769, 4312457, 4965713, 5555773, 6085997, 7104079, 8259443

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=1834 because 2^3+3^3+5^3+7^3+11^3=1834.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133540, A133541, A133542, A133543.

a = 3; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total[#^3]&/@Partition[Prime[Range[40]],5,1] (* Harvey P. Dale, May 01 2013 *)

a(n) = A133525(n) + A030078(n+4). - Michel Marcus, Nov 09 2013

A133543 Sum of seventh powers of five consecutive primes.

Original entry on oeis.org

20391154, 83139543, 493476029, 1387269643, 4791271547, 22021660685, 49471526279, 143993064739, 337853466881, 606267252541, 1095640496695, 2242839022421, 4636558630107, 7584547192247, 13373440186463

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=20391154 because 2^7+3^7+5^7+7^7+11^7=20391154

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133541, A133542.

a = 7; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[20]]^7,5,1] (* Harvey P. Dale, Mar 05 2022 *)

A133541 Sum of fifth powers of five consecutive primes.

Original entry on oeis.org

181258, 552519, 1972133, 4445107, 10864643, 31214741, 59472599, 127396699, 240776801, 381348901, 590182759, 979749101, 1625329443, 2354069543, 3557186207, 5132070551, 6786946651, 9149078751, 12243523093, 16477457435

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=181258 because 2^5+3^5+5^5+7^5+11^5=181258.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133542, A133543.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[30]]^5,5,1] (* Harvey P. Dale, Dec 02 2017 *)

a(n) = A133527(n) + A050997(n+4). - Michel Marcus, Nov 09 2013

A133542 Sum of sixth powers of five consecutive primes.

Original entry on oeis.org

1905628, 6732373, 30869213, 77899469, 225817709, 818869469, 1701546341, 4243135181, 8946193541, 15119520701, 25303912709, 46580770157, 86195577389, 132965847509, 217102866629, 334423935221, 463593800381, 664500722261

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=1905628 because 2^6+3^6+5^6+7^6+11^6=1905628.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133541, A133543.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@(Partition[Prime[Range[30]],5,1]^6)  (* Harvey P. Dale, Mar 13 2011 *)

a(n) = A133528(n) + A030516(n+4). - Michel Marcus, Nov 09 2013

A245577 Numbers k such that k^4 is a sum of 4 consecutive primes.

Original entry on oeis.org

12, 90, 208, 212, 234, 242, 314, 366, 404, 410, 416, 486, 540, 590, 750, 888, 908, 1152, 1418, 1444, 1500, 1524, 1658, 1666, 1736, 1798, 1814, 1874, 1940, 1942, 2094, 2138, 2266, 2496, 2584, 3058, 3062, 3206, 3660, 4034, 4080, 4208, 4368, 4422, 4606, 4872

Offset: 1

Zak Seidov, Nov 29 2014

nonn

			12^4 = 20736 = prime(689) + prime(689 + 1) + prime(689 + 2) + prime(689 + 3) = 5171 + 5179 + 5189 + 5197.

Robert G. Wilson v, Table of n, a(n) for n = 1..5100 (first 1189 terms from Zak Seidov)

Cf. A034963, A051395, A206280.

fQ[n_] := MemberQ[ Total@# & /@ Partition[ Table[ NextPrime[n^4/4, i], {i, {-3, -2, -1, 1, 2, 3}}], 4, 1], n^4]; Select[ Range@ 5000, fQ] (* Robert G. Wilson v, Dec 03 2014 *)

isscpn(n) = {np = n^4; p = precprime(np\4); for (i=1, 3, p = precprime(p-1);); while(1, q = nextprime(p+1); r = nextprime(q+1); s = nextprime(r+1); if ((v=p+q+r+s) == np, return (1)); if (v > np, return (0)); p = q;);} \\ Michel Marcus, Nov 30 2014

cp's OEIS Frontend

A133537 Sum of sixth powers of two consecutive primes.

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A127339 Numbers that are the sum of 12 consecutive primes.

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A102655 Numbers that are the arithmetic mean of four successive primes.

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A133533 Sum of sixth powers of three consecutive primes.

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A131019 Semiperimeters of quadrilaterals whose sides are 4 consecutive odd primes.

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A133539 Sum of third powers of five consecutive primes.

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A133543 Sum of seventh powers of five consecutive primes.

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A133541 Sum of fifth powers of five consecutive primes.

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