A211005 -id:A211005 - cp's OEIS frontend

A069754 Counts transitions between prime and nonprime to reach the number n.

0, 1, 1, 2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 10, 11, 12, 13, 14, 14, 14, 15, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20, 20, 21, 22, 22, 22, 23, 24, 25, 26, 26, 26, 27, 28, 28, 28, 28, 28, 29, 30, 30, 30, 30, 30, 31, 32, 33, 34, 34, 34, 34, 34, 35, 36, 36, 36, 37, 38, 39

Offset: 1

T. D. Noe, May 02 2002

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

			a(6) = 4 because there are 4 transitions: 1 to 2, 3 to 4, 4 to 5 and 5 to 6.

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

Cf. A000720 (pi).
Cf. A211005 (run lengths).
Same parity: A010051, A061007, A035026, A071574.

a069754 1 = 0
a069754 2 = 1
a069754 n = 2 * a000720 n - 2 - (toInteger $ a010051 $ toInteger n)
-- Reinhard Zumkeller, Dec 04 2012

For[lst={0}; trans=0; n=2, n<100, n++, If[PrimeQ[n]!=PrimeQ[n-1], trans++ ]; AppendTo[lst, trans]]; lst
(* Second program: *)
pts[n_]:=Module[{c=2PrimePi[n]},If[PrimeQ[n],c-3,c-2]]; Join[{0,1},Array[ pts,80,3]] (* Harvey P. Dale, Nov 12 2011 *)
Accumulate[If[Sort[PrimeQ[#]]=={False,True},1,0]&/@Partition[ Range[ 0,80],2,1]] (* Harvey P. Dale, May 06 2013 *)

When n is prime, a(n) = 2*pi(n) - 3. When n is composite, a(n) = 2*pi(n) - 2. pi(n) is the prime counting function A000720.

For n > 2: a(n) = 2*A000720(n) - 2 - A010051(n). - Reinhard Zumkeller, Dec 04 2012

A211006 Pair (n,p) where n is the sum of adjacent nonprimes and p is the sum of adjacent primes.

Original entry on oeis.org

1, 5, 4, 5, 6, 7, 27, 11, 12, 13, 45, 17, 18, 19, 63, 23, 130, 29, 30, 31, 170, 37, 117, 41, 42, 43, 135, 47, 250, 53, 280, 59, 60, 61, 320, 67, 207, 71, 72, 73, 380, 79, 243, 83, 430, 89, 651, 97, 297, 101, 102, 103, 315, 107, 108, 109, 333, 113, 1560, 127, 387, 131

Offset: 1

Omar E. Pol, Aug 11 2012

			-----------------------------------------
.     Array from
n      A000027                      a(n)
-----------------------------------------
1         1;                          1
2         2, 3;                       5
3         4;                          4
4         5;                          5
5         6;                          6
6         7;                          7
7         8, 9, 10;                  27
8        11;                         11
9        12;                         12
10       13;                         13
11       14, 15, 16;                 45
12       17;                         17
13       18;                         18
14       19;                         19
15       20, 21, 22;                 63
16       23;                         23
17       24, 25, 26, 27, 28;        130

Partial sums give A211007.
A variant of A109921.
Cf. A000040, A018252, A054265, A211005.

a(n) = A109921(n+2), n >= 3.

A211007 Surface area of the first n faces of the structure mentioned in A211006.

Original entry on oeis.org

1, 6, 10, 15, 21, 28, 55, 66, 78, 91, 136, 153, 171, 190, 253, 276, 406, 435, 465, 496, 666, 703, 820, 861, 903, 946, 1081, 1128, 1378, 1431, 1711, 1770, 1830, 1891, 2211, 2278, 2485, 2556, 2628, 2701, 3081, 3160, 3403, 3486, 3916, 4005, 4656, 4753, 5050

Offset: 1

Omar E. Pol, Aug 12 2012

nonn

All members of this sequence are triangular numbers A000217.

It appears that this is the same as A147846 without 3.

Partial sums of A211006.

Cf. A000040, A018252, A000217, A147876, A211005.

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A069754 Counts transitions between prime and nonprime to reach the number n.

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A211006 Pair (n,p) where n is the sum of adjacent nonprimes and p is the sum of adjacent primes.

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A211007 Surface area of the first n faces of the structure mentioned in A211006.

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