A371001 -id:A371001 - cp's OEIS frontend

A371002 a(n) are the records of the number of square addends producing a sequence of consecutive composites starting at A371001(n).

0, 2, 3, 6, 8, 11, 14, 17, 26, 44, 68, 92, 113, 128, 158, 209, 224, 293, 356, 401, 413, 431, 473, 584, 629, 650, 743, 854, 1007, 1010, 1106, 1112, 1148, 1280, 1295, 1319, 1358, 1364, 1388, 1616, 1643, 1664, 1799, 1859, 1973, 2000, 2162, 2234, 2294, 2393, 2501

Offset: 1

Hugo Pfoertner, Mar 07 2024

nonn

			a(1) = 0: A371001(1) = 4 is the least composite; 4+1^2 = 5 is prime;
a(2) = 2: A371001(2) = 8; 2 composites 8+1^2, 8+2^2; 8+3^2 is prime.
a(3) = 3: A371001(3) = 21; 3 composites 21+1^2, 21+2^2, 21+3^2; 21+4^2 is prime.
a(4) = 6: A371001(4) = 24; 6 composites 24+1^2, 24+2^2, 24+3^3, 24+4^2, 24+5^2, 24+6^2; 24+7^2=73 is prime.
a(51) = 2501 corresponds to A371001(51) = 1470353892839 =
13*59*103*18611839, followed by the 2501 composites 1470353892839+1,
1470353892839+4, ..., 1470353892839+2501^2 = 2^7*3^2*5*23*139*79857;
1470353892839+2502^2 = 1470360152843 is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..56

Cf. A000290, A002808, A371001.

PARI
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See A371001.
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a(45)-a(50) confirmed by data in A371001 from Martin Ehrenstein, Mar 08 2024

A370999 Least composite number k such that the number m of consecutive composite sums k + j*(j+1)/2, j = 1, ..., m is a new maximum.

Original entry on oeis.org

4, 8, 9, 15, 24, 90, 105, 114, 225, 264, 300, 945, 5349, 7035, 11739, 17280, 35475, 46914, 190365, 351645, 603054, 1209900, 3146220, 3279864, 6407664, 26447649, 115192665, 408291345, 1080430119, 1298351109, 6459163344, 7731193299, 8096124894, 8884256514, 18927105834

Offset: 1

Karl-Heinz Hofmann and Hugo Pfoertner, Mar 09 2024

nonn

a(60) > 10^15.

Karl-Heinz Hofmann and Hugo Pfoertner, Table of n, a(n) for n = 1..59
IBM Research, Composite Sequences, Ponder This Challenge March 2024.

A371000 gives the corresponding counts m.

Cf. A000217, A371001, A371002.

a(43) in b-file corrected by Kebbaj Mohamed Reda, May 29 2024

A371000 a(n) are the records of the number of addends of the form j*(j+1)/2 (A000217) producing a sequence of consecutive composites starting at A370999(n).

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 15, 21, 27, 33, 45, 87, 99, 102, 123, 156, 246, 273, 282, 330, 429, 465, 477, 561, 681, 891, 1050, 1206, 1338, 1443, 1449, 1479, 1656, 1701, 1836, 1899, 1941, 2151, 2199, 2280, 2301, 2517, 2592, 2595, 2709, 2724, 2751, 2760, 2934, 3045, 3240, 3333

Offset: 1

Karl-Heinz Hofmann and Hugo Pfoertner, Mar 09 2024

nonn

The "Ponder This Challenge" of March 2024 asked for the smallest start value of sequences of at least 1000 (or 2024 to gain the bonus) composite numbers in a progression defined by A000217.

Karl-Heinz Hofmann and Hugo Pfoertner, Table of n, a(n) for n = 1..59
IBM Research, Composite Sequences, Ponder This Challenge March 2024.

Cf. A000217, A370999, A371001, A371002.

a371000(upto) = {my(m=0); forcomposite (k=4, upto, for (j=1, oo, if (isprime(k+(j*(j+1))/2), if (j>m, print1(j-1, ", "); m=j); break)))};
a371000(10^7)

cp's OEIS Frontend

A371002 a(n) are the records of the number of square addends producing a sequence of consecutive composites starting at A371001(n).

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A370999 Least composite number k such that the number m of consecutive composite sums k + j*(j+1)/2, j = 1, ..., m is a new maximum.

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Author

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Links

Crossrefs

Extensions

A371000 a(n) are the records of the number of addends of the form j*(j+1)/2 (A000217) producing a sequence of consecutive composites starting at A370999(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI