2,6,8,12,16 - OEIS Search Results

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Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.

+20
9

2, 6, 8, 12, 16, 20, 26, 30, 32, 36, 42, 48, 50, 56, 60, 66, 70, 76, 80, 84, 90, 94, 100, 110, 114, 120, 126, 130, 136, 140, 146, 152, 156, 158, 162, 168, 176, 182, 186, 188, 196, 200, 210, 212, 216, 226, 236, 240, 246, 252, 254, 264, 270, 272, 278 (list; graph; listen)

	OFFSET	`2,1`
	COMMENT	`Tony Forbes defines a prime k-tuplet (distinguished from a prime k-tuple) to be a maximally possible dense cluster of primes (a prime constellation) which will necessarily involve consecutive primes whereas a prime k-tuple is a prime cluster which may not necessarily be of maximum possible density (in which case the primes are not necessarily consecutive.)` `a(1) would be 0 (for a prime 1-tuplet.)`
	REFERENCES	`R. K. Guy, "Unsolved Problems in Number Theory", lists a number of relevant papers in Section A8.` `G. H. Hardy and J.E. Littlewood, "Partitio Numerorum III", Acta Math. 44 (1922) 1-70, see final section.` `John Leech, "Groups of primes having maximum density", Math. Tables Aids to Comput., 12 (1958) 144-145.`
	LINKS	`T. D. Noe, Table of n, a(n) for n=2..672 (from Engelsma's data)` `Thomas J. Engelsma, Permissible Patterns` `Tony Forbes, k-tuplets` `Eric Weisstein's World of Mathematics, Prime Constellation.`
	FORMULA	`s(k), k >= 2, is smallest s such that there exist B = {b_1, b_2, ..., b_k} with s = b_k - b_1 and such that for all primes p <= k, not all residues modulo p are represented by B.`
	CROSSREFS	`Equals A020497 - 1.` `Sequence in context: A084724 A111051 A077561 this_sequence A111224 A139718 A135311` `Adjacent sequences: A008404 A008405 A008406 this_sequence A008408 A008409 A008410`
	KEYWORD	`nonn,nice`
	AUTHOR	`T. Forbes (anthony.d.forbes(AT)googlemail.com)`
	EXTENSIONS	`Correction from weidhaas(AT)wotan.llnl.gov (Pat Weidhaas) Jun 15 1997.` `Edited by Daniel Forgues (squid(AT)zensearch.com), Aug 13 2009`

A136513

Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter n.

+20
5

0, 0, 2, 2, 6, 8, 12, 16, 26, 30, 38, 44, 56, 60, 74, 82, 96, 108, 128, 138, 154, 166, 188, 196, 220, 238, 262, 278, 304, 324, 344, 366, 398, 416, 452, 468, 506, 526, 562, 588, 616, 644, 686, 714, 754, 780, 824, 848, 894, 930, 976, 1008, 1056, 1090, 1134, 1170 (list; graph; listen)

	OFFSET	`1,3`
	COMMENT	`As n -> infinity, lim a(n)/(n^2) -> pi/8`
	FORMULA	`a(n) = 2 * Sum(floor(sqrt((n/2)^2 - k^2))), k = 1 ... floor(n/2)`
	EXAMPLE	`a(3) = 2 because a circle centered at the origin and of radius 3/2 encloses (-1,1) and (1,1) in the upper half plane`
	MATHEMATICA	`Table[2*Sum[Floor[Sqrt[(n/2)^2 - k^2]], {k, 1, Floor[n/2]}], {n, 1, 100}]`
	CROSSREFS	`Cf. Alternating merge of A136514 and A136515 a(n) = 2 * A136483 = 1/2 * A136485.` `Sequence in context: A033748 A033736 A033760 this_sequence A054153 A000673 A129383` `Adjacent sequences: A136510 A136511 A136512 this_sequence A136514 A136515 A136516`
	KEYWORD	`easy,nonn`
	AUTHOR	`Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008`

A084724

Beginning with 2, the smallest even number greater than the previous term such that every partial product + 1 is a prime.

+20
4

2, 6, 8, 12, 16, 18, 20, 22, 26, 36, 42, 44, 100, 120, 124, 162, 168, 174, 192, 218, 272, 278, 338, 364, 380, 392, 502, 512, 532, 560, 594, 614, 698, 790, 814, 838, 922, 938, 1072, 1082, 1092, 1102, 1146, 1182, 1256, 1354, 1360, 1484, 1508, 1566, 1662, 1690 (list; graph; listen)

	OFFSET	`1,1`
	CROSSREFS	`Cf. A083566, A084723, A084725.` `Sequence in context: A064796 A083769 A057656 this_sequence A111051 A077561 A008407` `Adjacent sequences: A084721 A084722 A084723 this_sequence A084725 A084726 A084727`
	KEYWORD	`nonn`
	AUTHOR	`Amarnath Murthy and Meenakshi Srikanth (amarnath_murthy(AT)yahoo.com), Jun 13 2003`
	EXTENSIONS	`More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jan 03 2005`

A111051

Numbers n such that 3*n^2 + 1 is prime.

+20
4

2, 6, 8, 12, 16, 20, 22, 26, 34, 36, 40, 58, 64, 68, 78, 82, 84, 86, 98, 112, 120, 126, 142, 146, 148, 152, 156, 160, 168, 188, 190, 194, 196, 208, 216, 218, 222, 238, 240, 244, 246, 254, 264, 272, 282, 286, 294, 300, 302, 306, 308, 316, 320, 330, 338, 344, 348 (list; graph; listen)

	OFFSET	`1,1`
	COMMENT	`The resulting primes are cuban primes of the form p = (x^3 - y^3 )/(x - y), x=y+2). - Jani Melik (jani_melik(AT)hotmail.com), Jul 18 2007`
	LINKS	`Zak Seidov, Table of n, a (n) for n = 1..1000`
	EXAMPLE	`a(1)=2 because p = 1+32^2 = 13 is prime.` `a(2)=6 because p = 1+36^2 = 109 is prime.` `a(3)=8 because p = 1+38^2 = 193 is prime.` `If n=98 then (3n^2) + 1 = 28813 (prime).`
	MAPLE	`ts_kubpra_ind:=proc(n) local i, tren, ans; ans:=[ ]: for i from 0 to n do tren:=1+3*i^2: if (isprime(tren)='true') then ans:=[ op(ans), i ] fi od: RETURN(ans); end: ts_kubpra_ind(2000); - Jani Melik (jani_melik(AT)hotmail.com), Jul 18 2007`
	CROSSREFS	`Cf. A002648.` `Sequence in context: A083769 A057656 A084724 this_sequence A077561 A008407 A111224` `Adjacent sequences: A111048 A111049 A111050 this_sequence A111052 A111053 A111054`
	KEYWORD	`nonn`
	AUTHOR	`Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Oct 06 2005`
	EXTENSIONS	`More terms from Jani Melik (jani_melik(AT)hotmail.com), Jul 18 2007` `Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 28 2007`

A057656

Number of points (x,y) in square lattice with (x-1/2)^2+y^2 <= n.

+20
2

2, 6, 8, 12, 16, 16, 22, 26, 26, 30, 34, 38, 40, 44, 44, 48, 56, 56, 60, 60, 62, 70, 74, 74, 78, 82, 82, 86, 90, 94, 96, 104, 104, 104, 108, 108, 116, 120, 124, 128, 128, 128, 134, 138, 138, 142, 150, 150, 154, 158, 158, 166, 166, 166, 166, 174 (list; graph; listen)

	OFFSET	`0,1`
	REFERENCES	`J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.`
	CROSSREFS	`Partial sums of A004020. Cf. A057656, A057961, A057962.` `Sequence in context: A138626 A064796 A083769 this_sequence A084724 A111051 A077561` `Adjacent sequences: A057653 A057654 A057655 this_sequence A057657 A057658 A057659`
	KEYWORD	`nonn`
	AUTHOR	`N. J. A. Sloane (njas(AT)research.att.com), Oct 15 2000`

A077561

Indices of terms of A025487 which divide the terms. Numbers n such that A025487(n) is a multiple of n.

+20
2

1, 2, 6, 8, 12, 16, 20, 24, 32, 45, 48, 66, 84, 90, 96, 108, 120, 140, 144, 150, 154, 162, 168, 175, 180, 192, 198, 200, 216, 220, 224, 240, 252, 264, 280, 288, 300, 315, 324, 336, 360, 375, 390, 396, 432, 486, 504, 525, 570, 576, 594 (list; graph; listen)

	OFFSET	`1,2`
	CROSSREFS	`Cf. A025487, A077562.` `Sequence in context: A057656 A084724 A111051 this_sequence A008407 A111224 A139718` `Adjacent sequences: A077558 A077559 A077560 this_sequence A077562 A077563 A077564`
	KEYWORD	`nonn`
	AUTHOR	`Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 10 2002`
	EXTENSIONS	`More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Aug 24 2003`

A083769

Rearrangement of even numbers such that every partial product + 1 is a prime.

+20
2

2, 6, 8, 12, 16, 10, 4, 30, 26, 22, 24, 14, 50, 42, 18, 64, 46, 60, 32, 36, 20, 34, 28, 108, 48, 44, 68, 282, 90, 54, 76, 62, 180, 66, 132, 86, 74, 38, 58, 106, 120, 52, 244, 94, 100, 82, 138, 156, 98, 72, 172, 150, 248, 154, 166, 114, 162, 126, 124, 208, 222, 324, 212 (list; graph; listen)

	OFFSET	`1,1`
	COMMENT	`Conjecture: every even number is a member.`
	EXAMPLE	`2+1=3, 2*6+1=13, etc. are primes.`
	CROSSREFS	`Cf. A083770.` `Sequence in context: A120227 A138626 A064796 this_sequence A057656 A084724 A111051` `Adjacent sequences: A083766 A083767 A083768 this_sequence A083770 A083771 A083772`
	KEYWORD	`nonn`
	AUTHOR	`Amarnath Murthy and Meenakshi Srikanth (amarnath_murthy(AT)yahoo.com), May 06 2003`
	EXTENSIONS	`More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Nov 23 2004`

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Last modified February 9 11:24 EST 2010. Contains 172296 sequences.

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