A219996 -id:A219996 - cp's OEIS frontend

A190639 Centuries whose prime pattern repeats in the next century.

473267, 726760, 1773439, 1808828, 1919128, 2131583, 2165420, 2339971, 2390652, 2518488, 2802591, 2844914, 2982584, 2996184, 3183263, 3193175, 3250986, 3418185, 3428241, 3633472, 3909324, 3953449, 4280455, 4303819, 4373399, 4658285, 4728653, 4978360, 5165402, 5254365

Offset: 1

M. F. Hasler, May 15 2011

Alternate definition: Numbers x such that for all N in [100x,100x+99], N is prime iff N+100 is prime.
Contains in particular the first of two consecutive prime-free centuries, i.e., N such that there is no prime in [100 N,100 (N+2)], cf. A181098.
x belongs to this sequence if and only if the primality character of (100 * x) + k is the same as (100 * (x+1)) + k for all k = 0..99. - V. Raman, Dec 09 2012

			The first value refers to two consecutive prime-free centuries (cf. A181098); the second value is such that 100*a(2)+17 and 100*a(2)+117 are the only primes between 100*a(2) and 100*(a(2)+2). See the link for more examples.

Donovan Johnson, Table of n, a(n) for n = 1..1000
J. K. Andersen, in reply to R. Wood, Re: First repetition of prime pattern within "centuries", Yahoo group "primenumbers", May 15, 2011.
Jens Kruse Andersen, Phil Carmody, Maximilian Hasler, First repetition of prime pattern within "centuries", digest of 11 messages in primenumbers Yahoo group, May 15, 2011.

Cf. A181098.

Cf. A219996 (upper century).

a(n) ~ n. In particular there are x - 200x/log x + O(x/log^2 x) members of this sequence below x. - Charles R Greathouse IV, Dec 09 2012

a(n) = A219996(n) - 1. - V. Raman, Dec 09 2012

a(1)-a(5) computed by J. K. Andersen, May 15 2011

a(6)-a(30) from Donovan Johnson, May 15 2011

A220063 Decades whose semiprime pattern is the same as semiprime pattern in the previous decade.

Original entry on oeis.org

104, 389, 435, 438, 529, 658, 884, 1110, 1183, 1533, 1548, 1557, 1669, 1799, 1824, 1825, 1915, 1993, 2011, 2076, 2085, 2153, 2313, 2355, 2372, 2617, 2628, 2648, 2673, 3204, 3234, 3258, 3280, 3295, 3373, 3415, 3513, 3601, 3636, 3906, 3931, 3936, 4125, 4154

Offset: 1

Jonathan Vos Post, Dec 10 2012

This is to 10 and semiprimes A001358 as A219996 is to 100 and primes A000040. The first decade (1,2,3,4,5,6,7,8,9,10) has a unique pattern, as no decade ending with a multiple k*10 for k>1 ends with a semiprime; so it does not matter whether 10 is at the beginning or the end of a decade.

			a(1) = 104 because the decade (1030..1039) has the same semiprime pattern as the previous decade: (1020..1029), namely that each has only a single semiprime, respectively 1027 = 13 * 79 and 1037 = 17 * 61. [corrected by _Bobby Jacobs_, Sep 28 2016]

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001358, A219996, A277459, A277460.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nn = 50000; s = Table[SemiPrimeQ[n], {n, nn}]; t = Partition[s, 10]; t2 = {}; Do[If[t[[i]] == t[[i - 1]], AppendTo[t2, i]], {i, 2, Length[t]}]; t2 (* T. D. Noe, Dec 11 2012 *)
semiPrimeQ[n_] := PrimeOmega@ n == 2; f[n_] := semiPrimeQ@# & /@ (10 n + Range@9); a = f[0]; k = 1; lst = {}; While[k < 10001, b = f[k]; If[a == b, AppendTo[lst, k]]; a = b; k++]; lst (* Robert G. Wilson v, Dec 11 2012 *)

a(n) ~ n. In particular there are x - 200x log log x/log x + O(x/log x) members of this sequence below x. - Charles R Greathouse IV, Dec 11 2012

a(n) = A277459(n) + 2 = A277460(n) + 1. - Bobby Jacobs, Oct 27 2016

All terms from T. D. Noe, Dec 11 2012, and (with 1 already added to each) all terms after the first from Robert G. Wilson v, by email to Jonathan Vos Post.

A258275 a(n) = smallest number k > n such that the interval k100 to k100+99 has exactly the same prime pattern as the interval n100 to n100+99 (or 0 if no such term is known).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4812895043702, 0, 38905562023, 0, 2406071834559, 0, 834998571515, 15367548589719, 274894696197322, 0, 3339850458, 0, 0, 90345210525, 127636130731, 0, 0, 7916673590887, 498009080381, 1128063679395, 616923037, 301998772331

Offset: 1

Tim Johannes Ohrtmann, Jun 25 2015

			a(13) = 38905562023 because the primes between 1300 and 1399 are 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381 and 1399 and 38905562023 is the least century>13 that has exactly the same prime pattern: 3890556202301, 3890556202303, 3890556202307, 3890556202319, 3890556202321, 3890556202327, 3890556202361, 3890556202367, 3890556202373, 3890556202381, 3890556202399.

Cf. A164987, A190639, A219996.

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A190639 Centuries whose prime pattern repeats in the next century.

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A220063 Decades whose semiprime pattern is the same as semiprime pattern in the previous decade.

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A258275 a(n) = smallest number k > n such that the interval k100 to k100+99 has exactly the same prime pattern as the interval n100 to n100+99 (or 0 if no such term is known).

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cp's OEIS Frontend

A190639 Centuries whose prime pattern repeats in the next century.

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Author

Keywords

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Examples

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A220063 Decades whose semiprime pattern is the same as semiprime pattern in the previous decade.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A258275 a(n) = smallest number k > n such that the interval k*100 to k*100+99 has exactly the same prime pattern as the interval n*100 to n*100+99 (or 0 if no such term is known).

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Author

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A258275 a(n) = smallest number k > n such that the interval k100 to k100+99 has exactly the same prime pattern as the interval n100 to n100+99 (or 0 if no such term is known).