A281256 Runs of consecutive integers in A270877, which is produced by a decaying trapezoidal modification of the sieve of Eratosthenes.

8, 13, 1, 19, 16, 4, 32, 64, 22, 49, 34, 166, 27, 71, 38, 44, 172, 59, 302, 1984, 46771, 56, 178, 94, 346, 4925, 59492, 188357, 68, 205, 352, 617, 7408, 113492, 371918, 881212, 80, 211, 382, 939, 9110, 114602, 964583, 6671161, 24365591, 89, 214, 581, 1011, 11090, 207938, 1008362

Offset: 1

Author

Peter Munn, Jan 18 2017

Keywords

• nonn
• tabl

Comments

Square table T, read by ascending antidiagonals, where T(n,m) gives the least integer in the n-th occurrence of a run of exactly m consecutive integers in the ordered sequence A270877.

A270877 is sifted from the positive integers by modifying the sieve of Eratosthenes: instead of eliminating integers that would enumerate a rectangular area dot pattern with one side held at a constant length (equal to each surviving integer in turn), the sieve eliminates those enumerating a trapezoidal area dot pattern with the constant length being the trapezoid's longest side. Given this geometric relationship, it is considered worth looking for qualities that A270877 may have in common with the sequence of primes, potentially influenced by related causes such as the effect of prime factors on A270877.

The columns of this sequence, listing the runs of m consecutive integers within A270877, merit comparative examination with equivalent sequences for prime k-tuples. For m=5, the notably larger ratio between T(1,5) and T(2,5) resembles early large ratio gaps in the occurrence sequences of k-tuples such as A022008 (sextuples), whereas columns m<5 are more comparable with those for shorter k-tuples such as A001359 (twin primes) and A007530 (quadruples), each having a relatively low-valued first term (less than 60) and without such a large ratio gap. In comparison, the columns for runs m>5 appear more like the sequences for some longer k-tuples such as A027570 (a 10-tuple sequence). Row 1 merits comparative examination with A186702 for primes.

The author conjectures that T(n,m) exists for all n>=1, m>=1.

Examples

			4, 5 and 6 occur in A270877, but 3 and 7 do not. This is the first run of exactly 3 consecutive integers in A270877, so T(1,3) = 4.
Square table T(n,m) begins:
   8,   1,   4,   49,    38,  46771,  188357,   881212, ...
  13,  16,  22,   71,  1984,  59492,  371918,  6671161, ...
  19,  64,  27,  302,  4925, 113492,  964583,  8799769, ...
  32, 166,  59,  346,  7408, 114602, 1008362, 13579777, ...
  34, 172,  94,  617,  9110, 207938, 1094293, 14874616, ...
  44, 178, 352,  939, 11090, 291712, 1156214, 15974752, ...
  56, 205, 382, 1011, 13007, 323716, 1239046, 20585962, ...
  68, 211, 581, 1080, 13216, 429915, 1433918, 20745838, ...
  80, 214, 599, 1091, 14710, 442807, 1702694, 24321313, ...
  89, 223, 624, 1151, 15052, 457220, 1712927, 25634557, ...
		

Crossrefs

This is an analysis of A270877.

Cf. A001359, A007530, A022008, A027570, A186702.