A070865 -id:A070865 - cp's OEIS frontend

A255909 Second difference sequence of A070865.

1, 4, 2, 2, 2, 6, 2, 2, 4, 4, 4, 2, 6, 2, 2, 4, 2, 8, 6, 6, 2, 2, 2, 2, 2, 20, 6, 6, 2, 2, 8, 2, 4, 4, 2, 6, 2, 4, 4, 4, 4, 4, 4, 4, 12, 10, 2, 12, 12, 6, 8, 4, 4, 2, 8, 16, 10, 2, 18, 6, 6, 6, 2, 2, 4, 8, 2, 18, 2, 14, 4, 4, 4, 4, 10, 10, 4, 2, 10, 4, 4, 2

Offset: 1

Clark Kimberling, Apr 14 2015

Does 2 occur infinitely many times? If so, does every even positive integer occur infinitely many times? More generally, suppose that p < q are primes, and let p(1) = p, p(2) = q, and, for n > 2, let p(n) = least prime h such that h - p(n-1) > p(n-1) - p(n-2). Does every even positive integer occur infinitely many times in the second difference sequence of (p(n))?

			A070865 = (2,3,5,11,19,29,41,59,79,...)
1st differences:  1,2,6,8,10,12,18,20,...
2nd differences:  1,4,2,2,2,6,2,...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A070865, A000040.

d = 0; p = 2; t = {p}; Do[d = NextPrime[p + d] - p; AppendTo[t, p += d], {200}]; t
Differences[t,2]  (* A255909 *)
(* uses Vladimir Joseph Stephan Orlovsky's program at A070865 *)

A070866 Smallest prime such that the difference of successive terms is nondecreasing.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 29, 37, 47, 59, 71, 83, 97, 113, 131, 149, 167, 191, 223, 257, 293, 331, 373, 419, 467, 521, 577, 641, 709, 787, 877, 967, 1061, 1163, 1277, 1399, 1523, 1657, 1801, 1949, 2099, 2251, 2411, 2579, 2749, 2927, 3109, 3299, 3491, 3691, 3907

Offset: 1

Amarnath Murthy, May 16 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A070865.

using Primes
function A070866(bound)
    a, b = 2, 3
    P = [a, b]
    while true
        p = nextprime(b + (b - a))
        p > bound && break
        push!(P, p)
        a, b = b, p
    end
P end
A070866(100000) |> println # Peter Luschny, Dec 23 2019

d = 2; p = 2; t = {2, 3}; Do[p = NextPrime[p + d - 1]; d = p - t[[-1]]; AppendTo[t, p], {98}]; t (* T. D. Noe, Nov 21 2011 *)

s=1; t=1; for(n=1,100,s=s+t; while(isprime(s+t)==0,t++); print1(s+t,","))

a(1)=2, a(2)=3, a(n) = A007918(2*a(n-1) - a(n-2)). - Reinhard Zumkeller, Jul 08 2004

More terms from Benoit Cloitre, May 20 2002

A257559 The slowest increasing sequence of semiprimes with strictly increasing difference of successive terms.

Original entry on oeis.org

4, 6, 9, 14, 21, 33, 46, 62, 82, 106, 133, 161, 194, 235, 278, 323, 371, 422, 478, 535, 597, 662, 731, 802, 878, 955, 1037, 1121, 1207, 1294, 1382, 1473, 1565, 1658, 1754, 1851, 1954, 2059, 2165, 2279, 2395, 2513, 2638, 2771, 2906, 3043

Offset: 1

Zak Seidov, Apr 30 2015

nonn

Semiprime analog of A070865.

Heuristically a(n) is around n^2 log n/log log n. - Charles R Greathouse IV, May 01 2015

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001358, A070865.

issemi(n)=bigomega(n)==2
t=0; print1(last=4); while(1, n=last+t; while(!issemi(n++),); print1(", "n); t=n-last; last=n) \\ Charles R Greathouse IV, Apr 30 2015

a(n) >> n^2. - Charles R Greathouse IV, Apr 30 2015

A284188 a(1)=2; thereafter a(n+1) = a(n)+i if a(n) is a prime and a(1),...,a(n) contains i primes, or a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i primes.

Original entry on oeis.org

2, 3, 5, 8, 5, 9, 5, 10, 5, 11, 18, 11, 19, 28, 19, 29, 40, 29, 41, 54, 41, 55, 41, 56, 41, 57, 41, 58, 41, 59, 78, 59, 79, 100, 79, 101, 124, 101, 125, 101, 126, 101, 127, 154, 127, 155, 127, 156, 127, 157, 188, 157, 189, 157, 190, 157, 191, 226, 191, 227, 264, 227, 265

Offset: 1

Bob Selcoe, Mar 21 2017

nonn

Without repeated terms, the primes appear in order as A070865.
Variant of A284172; the difference is that in A284172, a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i composites (rather than i primes).
For n >= 3: When a(n) = prime p it is followed by an even number j at a(n+1); p repeats k-j times (where k is the smallest prime > j), appearing at a(n+2m) {m=1..k-j}. a(n+2m+1) = p+m until p+m = k (immediately following the final p); k now becomes "new p" immediately followed by a "new j", and the process repeats.

			a(10) = 11; there are 7 primes in the sequence up to and including a(10) so a(11) = 11+7 = 18. 18 is composite so a(12) = 18-7 = 11.  Now there are 8 primes in the sequence; and since 11 is prime, a(13) = 11+8 = 19 (the 9th prime in the sequence), so a(14) = 28.

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^16.
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^12, showing a(n) in red and A284172(n) in blue, accentuating the first 12 terms of A284172(n) by enlargement.

Cf. A070865, A284172.

c:= proc(n) option remember; `if`(n<1, 0,
      `if`(isprime(a(n)), 1, 0)+c(n-1))
    end:
a:= proc(n) option remember; `if`(n=1, 2, (m->
      `if`(isprime(m), 1, -1)*c(n-1)+m)(a(n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 15 2017

Block[{c = 1, m = 2, n}, {2}~Join~Reap[Do[If[PrimeQ[m], Set[n, m + c]; c++, Set[n, m - c + 1]]; Sow[n]; m = n, 63]][[-1, -1]]] (* Michael De Vlieger, Oct 20 2021 *)

lista(nn) = {print1(a=2, ", "); nbp = 1; for (n=2, nn, if (isprime(a), a += nbp, a -= nbp); print1(a, ", "); if (isprime(a), nbp++););} \\ Michel Marcus, Mar 24 2017

A178547 Composite numbers; start sequence with composite number and end with prime, length of successive sequences are strictly increasing.

Original entry on oeis.org

4, 6, 12, 20, 30, 42, 60, 80, 102, 128, 158, 192, 228, 270, 314, 360, 410, 462, 522, 588, 660, 734, 810, 888, 968, 1050, 1152, 1260, 1374, 1490, 1608, 1734, 1862, 1994, 2130, 2268, 2412, 2558, 2708, 2862, 3020, 3182, 3348, 3518, 3692, 3878, 4074, 4272, 4482

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 29 2010

nonn

			{004, 005} {006, 007, 008, 009, 010, 011} {012, 013, 014, 015, 016, 017, 018, 019} {020, 021, 022, 023, 024, 025, 026, 027, 028, 029} {030, 031, 032, 033, 034, 035, 036, 037, 038, 039, 040, 041}..

Cf. A070865

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; d=0;p=2;lst={};Do[d=PrimeNext[p+d]-p;p+=d;AppendTo[lst,p+1],{n,0,5!}];lst

A073847 a(0) = 1, c(0) = 1, a(n) for n > 0 is the smallest prime a(n-1) + c(n), where c(n) is composite and larger than c(n-1).

Original entry on oeis.org

1, 5, 11, 19, 29, 41, 59, 79, 101, 127, 157, 191, 227, 269, 313, 359, 409, 461, 521, 587, 659, 733, 809, 887, 967, 1049, 1151, 1259, 1373, 1489, 1607, 1733, 1861, 1993, 2129, 2267, 2411, 2557, 2707, 2861, 3019, 3181, 3347, 3517, 3691, 3877, 4073, 4271, 4481

Offset: 0

Amarnath Murthy, Aug 14 2002

nonn

Same as A070865 after first term. - David Wasserman, Jun 27 2005

			a(6) = 59 since 59 = a(5) + c(6) = 41 + 18, 18 is composite and larger than c(5) = 12 and the composite numbers 14, 15, 16 do not lead to a prime when added to 41.

a[n_] := a[n] = NextPrim[a[n - 1] + a[n - 1] - a[n - 2]]; a[0] = 1; a[1] = 5; Table[ a[n], {n, 0, 50}]

{print1(a=1,","); c=1; for(n=1,48,c++; while(isprime(c)||!isprime(a+c),c++); a=a+c; print1(a,","))}

Edited and extended by Klaus Brockhaus and Robert G. Wilson v, Aug 14 2002

A085626 Partial sums of A051935.

Original entry on oeis.org

2, 5, 11, 19, 29, 41, 59, 79, 101, 127, 157, 191, 227, 269, 313, 359, 409, 461, 521, 587, 659, 733, 809, 887, 967, 1049, 1151, 1259, 1373, 1489, 1607, 1733, 1861, 1993, 2129, 2267, 2411, 2557, 2707, 2861, 3019, 3181, 3347, 3517, 3691, 3877, 4073, 4271, 4481

Offset: 1

Patrick Capelle, Jul 10 2003

nonn

Same as A070865 after first term. - David Wasserman, Jun 27 2005

			a(3) = 11 because it is the sum of the first 3 terms of A051935: 2+3+6 = 11.

Cf. A051935, A070865.

A178549 a(n) is a composite number at the start of an interval of consecutive integers, ending in a prime, and non-overlapping with and at least as long as the interval addressed by a(n-1).

Original entry on oeis.org

4, 6, 8, 12, 18, 24, 30, 38, 48, 60, 72, 84, 98, 114, 132, 150, 168, 192, 224, 258, 294, 332, 374, 420, 468, 522, 578, 642, 710, 788, 878, 968, 1062, 1164, 1278, 1400, 1524, 1658, 1802, 1950, 2100, 2252, 2412, 2580, 2750, 2928, 3110, 3300, 3492, 3692, 3908

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 29 2010

nonn

Non-overlapping intervals of the integers of nondecreasing length, starting with a composite number and ending with a prime number, define an intermediate table (as below), the first column of which defines the sequence.
   4,  5
   6,  7
   8,  9, 10, 11
  12, 13, 14, 15, 16, 17
  18, 19, 20, 21, 22, 23
  24, 25, 26, 27, 28, 29
  30, 31, 32, 33, 34, 35, 36, 37
  38, 39, 40, 41, 42, 43, 44, 45, 46, 47
  48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59
  60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Cf. A070865, A178547.

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; d=0;p=2;lst={}; Do[d=PrimeNext[p+d]-p;p=p+=d;d--;AppendTo[lst,p+1],{n,0,5!}];lst

a(n) = 1 + A070866(n+1). - R. J. Mathar, Jun 07 2010

Definition rephrased by R. J. Mathar, Jun 07 2010

cp's OEIS Frontend

A255909 Second difference sequence of A070865.

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A070866 Smallest prime such that the difference of successive terms is nondecreasing.

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Julia

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A257559 The slowest increasing sequence of semiprimes with strictly increasing difference of successive terms.

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A284188 a(1)=2; thereafter a(n+1) = a(n)+i if a(n) is a prime and a(1),...,a(n) contains i primes, or a(n+1) = a(n)-i if a(n) is composite and a(1),...,a(n) contains i primes.

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Maple

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A178547 Composite numbers; start sequence with composite number and end with prime, length of successive sequences are strictly increasing.

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A073847 a(0) = 1, c(0) = 1, a(n) for n > 0 is the smallest prime a(n-1) + c(n), where c(n) is composite and larger than c(n-1).

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A085626 Partial sums of A051935.

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A178549 a(n) is a composite number at the start of an interval of consecutive integers, ending in a prime, and non-overlapping with and at least as long as the interval addressed by a(n-1).

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