A140755 -id:A140755 - cp's OEIS frontend

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 73303, 73361, 73421, 73483, 3443897, 3071069, 3071137, 15949847, 76553693, 365462323, 365462399, 2204597, 9721, 1842719, 246407633, 246407719, 246407807, 246407897, 246407989

Offset: 0

M. F. Hasler, May 26 2016

nonn

All terms are prime, since this is necessary and sufficient to get a prime for x = 0.
The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, ....
Sequence A273595 is the subsequence of terms for which 2n+1 is prime.
For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.
The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)

Don Reble, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index to the OEIS, Entries related to primes produced by polynomials.

Cf. A273595, A273770.
Cf. also A002837 (n such that n^2-n+41 is prime), A007634 (n such that n^2+n+41 is composite), A005846 (primes of form n^2+n+41), A097823, A144051, A187057 ... A187060, A190800, A191456 ff.
The first line of data coincides with that of A202018, A107448, A155884 (and also A140755, A142719, except for some initial terms), which are all related.

A273756(n,p=2*n+1,L=10^(5+n\10),m=0,Q)={forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2)&& next; x>m&& [Q=q,m=x]; break));Q}

Edited, following a remark by Don Reble, by M. F. Hasler, Jan 23 2018

a(27) corrected and more terms from Don Reble, Feb 15 2018

A140754 Array T(n,k) = 19n^2 + 10k^2, read by rows, with 14 columns and 14 rows.

Original entry on oeis.org

29, 59, 109, 179, 269, 379, 509, 659, 829, 1019, 1229, 1459, 1709, 1979, 86, 116, 166, 236, 326, 436, 566, 716, 886, 1076, 1286, 1516, 1766, 2036, 181, 211, 261, 331, 421, 531, 661, 811, 981, 1171, 1381, 1611, 1861, 2131, 314, 344, 394, 464, 554, 664, 794

Offset: 1

Aldrich Stevens (aldrichstevens(AT)msn.com), May 27 2008

			The array in full:
    29   59  109  179  269  379  509  659  829 1019 1229 1459 1709 1979
    86  116  166  236  326  436  566  716  886 1076 1286 1516 1766 2036
   181  211  261  331  421  531  661  811  981 1171 1381 1611 1861 2131
   314  344  394  464  554  664  794  944 1114 1304 1514 1744 1994 2264
   485  515  565  635  725  835  965 1115 1285 1475 1685 1915 2165 2435
   694  724  774  844  934 1044 1174 1324 1494 1684 1894 2124 2374 2644
   941  971 1021 1091 1181 1291 1421 1571 1741 1931 2141 2731 2621 2891
  1226 1256 1306 1376 1466 1576 1706 1856 2026 2216 2426 2656 2906 3176
  1549 1579 1629 1699 1789 1899 2029 2179 2349 2539 2749 2979 3229 3499
  1910 1940 1990 2060 2150 2260 2390 2540 2710 2900 3110 3340 3590 3860
  2309 2339 2389 2459 2549 2669 2789 2939 3109 3299 3509 3739 3989 4259
  2746 2776 2826 2896 2986 3096 3226 3376 3546 3736 3946 4176 4426 4696
  3221 3251 3301 3371 3461 3571 3701 3851 4021 4211 4421 4651 4901 5171
  3734 3764 3814 3884 3974 4084 4214 4364 4534 4724 4934 5164 5414 5684

Aldrich Stevens, Table of n, a(n) for n = 1..196

Cf. A140340, A140755.

[19*n^2+10*k^2: k in [1..14], n in [1..14]]; // G. C. Greubel, Oct 22 2023

Table[19*n^2 +10*k^2, {n,14}, {k,14}]//Flatten (* G. C. Greubel, Oct 22 2023 *)

flatten([[19*n^2+10*k^2 for k in range(1,15)] for n in range(1,15)]) # G. C. Greubel, Oct 22 2023

Edited with better definition by Omar E. Pol, Jan 05 2009

A140340 Square array read by rows: T(n,k) = 19n^2+10k^2-(n-1)(20(k-1)+10), with 14 columns.

Original entry on oeis.org

29, 59, 109, 179, 269, 379, 509, 659, 829, 1019, 1229, 1459, 1709, 1979, 76, 86, 116, 166, 236, 326, 436, 566, 716, 886, 1076, 1286, 1516, 1766, 161, 151, 161, 191, 241, 311, 401, 511, 641, 791, 961, 1151, 1361, 1591, 284, 254, 244, 254, 284, 334, 404, 494, 604, 734, 884, 1054, 1244, 1454

Offset: 1

Aldrich Stevens (aldrichstevens(AT)msn.com), May 29 2008

			Array begins:
   29,  59, 109, 179, 269, 379, 509, 659, 829, 1019, 1229, 1459, 1709, 1979;
   76,  86, 116, 166, 236, 326, 436, 566, 716,  886, 1076, 1286, 1516, 1766;
  161, 151, 161, 191, 241, 311, 401, 511, 641,  791,  961, 1151, 1361, 1591;
  284, 254, 244, 254, 284, 334, 404, 494, 604,  734,  884, 1054, 1244, 1454;
  445, 395, 365, 355, 365, 395, 445, 515, 605,  715,  845,  995, 1165, 1355;
...

Cf. A140754 (a very similar array), A140755.

T:=(n,k) -> 19*n^2+10*k^2-(n-1)*(20*(k-1)+10);

Edited by Omar E. Pol and N. J. A. Sloane, Jan 11 2009

A140350 This sequence splices 36x^2 - 810x + 2753 and the first of its transforms 9x^2 - 423x + 3167 into a prime chain 91 terms long.

Original entry on oeis.org

3167, 2753, 2753, 1979, 2357, 1277, 1979, 647, 1619, 89, 1277, 397

Offset: 1

Aldrich Stevens (aldrichstevens(AT)msn.com), May 29 2008

Cf. A140755.

procedure Ndegrees; var a : array[0..16] of extended; aa, ct,bb,cc: longint; n,nh,i,j : integer; ab1,ab2 : extended; begin for i := 0 to 16 do a[i] := 0; N := 7; a[0] := 3167{ FIRST TERM OF PRIME CHAIN}; nh := 1; a[1] := 2753 ;a[2] := 2753 ;a[3] := 1979 ;a[4] := 2357 ;a[5] := 1277 ;a[6] := 1979 ; a[7] := 647 ;a[8] := 1619 ; repeat for i := N downto nh do BEGIN a[i] := a[i] - a[i-1] ; IF NH = 5 THEN A[I] := ABS(A[I]); {******} END; nh := nh + 1; until nh = n + 2; ct := 0; repeat ct := ct + 1; ab1 := a[n] + a[n-1]; for i := N-1 downto 1 do begin IF I = 5 THEN IF ODD(ct) THEN A[i] := -A[i];{******} ab2 := a[i] + a[i-1] ; a[i] := ab1; ab1 := ab2; end; WRITELN(AB1); READLN; until 1<0; END;

cp's OEIS Frontend

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

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A140754 Array T(n,k) = 19n^2 + 10k^2, read by rows, with 14 columns and 14 rows.

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A140340 Square array read by rows: T(n,k) = 19n^2+10k^2-(n-1)(20(k-1)+10), with 14 columns.

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Maple

Extensions

A140350 This sequence splices 36x^2 - 810x + 2753 and the first of its transforms 9x^2 - 423x + 3167 into a prime chain 91 terms long.

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Pascal

cp's OEIS Frontend

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A140754 Array T(n,k) = 19*n^2 + 10*k^2, read by rows, with 14 columns and 14 rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Extensions

A140340 Square array read by rows: T(n,k) = 19*n^2+10*k^2-(n-1)*(20*(k-1)+10), with 14 columns.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A140350 This sequence splices 36x^2 - 810x + 2753 and the first of its transforms 9x^2 - 423x + 3167 into a prime chain 91 terms long.

Views

Author

Keywords

Crossrefs

Programs

Pascal

A140754 Array T(n,k) = 19n^2 + 10k^2, read by rows, with 14 columns and 14 rows.

A140340 Square array read by rows: T(n,k) = 19n^2+10k^2-(n-1)(20(k-1)+10), with 14 columns.