A226833 -id:A226833 - cp's OEIS frontend

A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

4, 4, 6, 10, 10, 201, 133, 133, 635, 697, 2215, 2215, 4979, 2995, 13561, 22903, 1691, 5951, 72697, 72697, 72697, 172151, 172151, 1782371, 1782371, 3660743, 3660743, 3660743, 3660743, 13298267, 2235913, 41963249

Offset: 1

T. D. Noe, Jun 28 2013

Smallest number in row A226833(n).

Cf. A096003 (largest semiprime in row), A097824 (gaps).

A226835 Triangle whose n-th row has the smallest n 3-almost primes in an arithmetic progression.

Original entry on oeis.org

8, 8, 12, 12, 20, 28, 20, 44, 68, 92, 20, 44, 68, 92, 116, 402, 410, 418, 426, 434, 442, 266, 370, 474, 578, 682, 786, 890, 266, 370, 474, 578, 682, 786, 890, 994, 1270, 1414, 1558, 1702, 1846, 1990, 2134, 2278, 2422, 1394, 1586, 1778, 1970, 2162, 2354, 2546, 2738, 2930, 3122

Offset: 1

T. D. Noe and Jonathan Vos Post, Jun 30 2013

Note that this triangle (at least for all n <= 29) is twice A226833, which is the similar triangle of semiprimes.

			Triangle:
8,
8,    12,
12,   20,   28,
20,   44,   68,   92,
20,   44,   68,   92,   116,
402,  410,  418,  426,  434,  442,
266,  370,  474,  578,  682,  786,  890,
266,  370,  474,  578,  682,  786,  890,  994,
1270, 1414, 1558, 1702, 1846, 1990, 2134, 2278, 2422,
1394, 1586, 1778, 1970, 2162, 2354, 2546, 2738, 2930, 3122

Cf. A226833 (similar triangle of semiprimes).

TriPrimeQ[n_Integer] := If[Abs[n] < 2, False, (3 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; p3 = Select[Range[4000], TriPrimeQ]; nn = Length[p3]; t = {}; n = 0; last = 1; While[n++; found = False; last = n; While[k = last - 1; p3Short = Take[p3, last]; While[d = p3[[last]] - p3[[k]]; nums = Table[p3[[last]] - i*d, {i, 0, n - 1}]; int = Intersection[nums, p3Short]; nums[[-1]] > 0 && Length[int] < n, k--]; nums[[-1]] <= 0 && last < nn, last++]; If[last < nn, AppendTo[t, Reverse[nums]]]; last < nn]; t

A227067 Least n-prime p such that the number of even n-primes (<= p) equals the number of odd n-primes (<= p).

Original entry on oeis.org

3, 51, 32151, 300863849741

Offset: 1

T. D. Noe, Jul 03 2013

An n-prime is a number having n prime factors (counted multiply). For any n, the ratio of even n-primes to odd n-primes tends to decrease with the magnitude of the numbers. This may explain why the initial terms in A226835 are all even. The a(4) term is greater than 10^9.

There is only one other semiprime such that half of the previous semiprimes are odd: 62. For 3-primes, there are three other numbers: 32158, 32163, and 32170.

			The first such prime is 3 because up to 3 there are an equal number of even and odd primes. The first such semiprime is 51 because there are 9 evens and 9 odds: 4, 6, 10, 14, 22, 26, 34, 38, 46 and 9, 15, 21, 25, 33, 35, 39, 49, 51.

Cf. A226833, A226835, A227069.

nn = 3; Table[p = 1; odds = 0; evens = 0; While[odds*evens == 0 || odds != evens, p++; If[PrimeOmega[p] == n, If[OddQ[p], odds++, evens++]]]; p, {n, nn}]

a(4) from Donovan Johnson, Aug 13 2013

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A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

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A226835 Triangle whose n-th row has the smallest n 3-almost primes in an arithmetic progression.

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A227067 Least n-prime p such that the number of even n-primes (<= p) equals the number of odd n-primes (<= p).

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