A006508 -id:A006508 - cp's OEIS frontend

A064960 The prime then composite recurrence; a(2n) = a(2n-1)-th prime and a(2n+1) = a(2n)-th composite and a(1) = 1.

1, 2, 6, 13, 22, 79, 108, 593, 722, 5471, 6290, 62653, 69558, 876329, 951338, 14679751, 15692307, 289078661, 305618710, 6588286337, 6908033000, 171482959009, 178668550322, 5040266614919, 5225256019175, 165678678591359, 171068472492228, 6039923990345039

Offset: 1

Robert G. Wilson v, Oct 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..31
Andrew R. Booker, The Nth Prime Page
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A006508 & A064961, see also A057450, A057451, A057452, A057453, A057456 & A057457 and A049076, A049077, A049078, A049079, A049080 & A049081.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a = {1}; b = 1; Do[ If[ !PrimeQ[b], b = Prime[b], b = Composite[b]]; a = Append[a, b], {n, 1, 23}]; a

from functools import cache
from sympy import prime, composite
@cache
def A064960(n): return 1 if n == 1 else composite(A064960(n-1)) if n % 2 else prime(A064960(n-1)) # Chai Wah Wu, Jan 01 2022

a(26)-a(28) from Chai Wah Wu, May 07 2018

A064961 Composite-then-prime recurrence; a(2n) = a(2n-1)-th composite and a(2n+1) = a(2n)-th prime and a(1) = 1.

Original entry on oeis.org

1, 4, 7, 14, 43, 62, 293, 366, 2473, 2892, 26317, 29522, 344249, 376259, 5429539, 5831545, 101291779, 107457490, 2198218819, 2310909505, 54720307351, 57128530327, 1543908890351, 1603146693999, 48871886538151, 50527531769529, 1720466016680911, 1772475453490311

Offset: 1

Robert G. Wilson v, Oct 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..32
Andrew R. Booker, The Nth Prime Page
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A006508 & A064960, see also A057450, A057451, A057452, A057453, A057456 & A057457 and A049076, A049077, A049078, A049079, A049080 & A049081.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a = {1, 4}; b = 4; Do[ If[ !PrimeQ[b], b = Prime[b], b = Composite[b]]; a = Append[a, b], {n, 1, 23}]; a

a(24)-a(26) corrected and a(27)-a(28) added by Chai Wah Wu, Aug 22 2018

A072415 a(1) = 2; a(n) = a(n-1)-th even nontotient number.

Original entry on oeis.org

2, 26, 182, 878, 3626, 13508, 46922, 155156, 494468, 1529590, 4615074, 13633310, 39548078, 112918434, 317920082, 883996002

Offset: 1

Labos Elemer, Jun 17 2002

Cf. A006508, A071255, A005277, A072416.

a = Table[0, {10^6}]; Do[b = EulerPhi[n]/2; If[b < 10^6 + 1, a[[b]] = 1], {n, 1, 2 10^7}]; b = Select[ Range[10^6], a[[ # ]] == 0 &]; c = 1; Do[ Print[c]; c = 2b[[c]], {n, 0, 8}]

a(3)=182 is the 26th = a(2)-th term in A005277.

Edited by Robert G. Wilson v, Jun 20 2002

a(11)-a(16) from Donovan Johnson, Oct 29 2010

A072416 a(1) = 2; a(n) = half of the a(n-1)-th even nontotient number.

Original entry on oeis.org

2, 13, 49, 149, 373, 835, 1727, 3383, 6341, 11419, 19966, 34067, 56967, 93578, 151313, 241281, 379934, 591413, 910879, 1389289, 2099905, 3147743, 4682362, 6914234, 10141072, 14779023, 21408820, 30837491, 44179913, 62974570, 89329823

Offset: 1

Labos Elemer, Jun 17 2002

nonn

Cf. A006508, A071255, A005277, A072415.

a = Table[0, {10^6}]; Do[b = EulerPhi[n]/2; If[b < 10^6 + 1, a[[b]] = 1], {n, 1, 2 10^7}]; b = Select[ Range[10^6], a[[ # ]] == 0 &]; c = 2; Do[ Print[c]; c = b[[c]], {n, 0, 17}]

a(5) = 373 = 746/2, where 746 is the a(4) = 149th term in A005277.

Edited by Robert G. Wilson v, Jun 20 2002

a(18)-a(31) from Donovan Johnson, Oct 29 2010

A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

Original entry on oeis.org

1, 1, 12, 2, 1, 1, 3, 11, 1, 1, 7, 9, 1, 2, 10, 4, 2, 1, 1, 6, 8, 3, 3, 1, 9, 3, 1, 1, 18, 3, 1, 5, 7, 2, 2, 1, 4, 8, 2, 14, 1, 1, 6, 17, 2, 6, 1, 4, 6, 1, 1, 2, 2, 3, 7, 1, 13, 6, 1, 4, 16, 5, 16, 1, 5, 31, 35, 3, 5, 2, 1, 2, 3, 1, 1, 2, 6, 1, 1, 12, 5, 1, 2

Offset: 1

Matthew Campbell, Sep 25 2015

nonn

a(n) is also the smallest value of k at which b(k, n+1) - b(k, n) > 1.

			When n = 3, writing Composite(x) for A002808(x):
1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.
2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.
3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.
4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.
5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.
6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.
7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.
8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.
9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.
10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.
11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.
12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.
--------------------------------------------------------------
Since the composite function was applied 12 times, a(3)=12.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Primes and nonprimes: A000040, A002808, A008578, A018252.
a(1) = p, a(n+1) = a(n)-th composite number: A006508, A022450, A022451, A025010, A025011, A059407, A059408.
Composites with order n > 1: A050435, A050436, A050438, A050439, A050440.
Composites with order n = b, n >= 1: A022449.
Composites with prime subscripts: A065858.
Composites without prime subscripts: A175251.
Order of compositeness: A059981, A236536.
Prime(n)-1: A006093.

c = Select[Range[10^5], CompositeQ]; Table[k = 1; While[! PrimeQ[Nest[c[[#]] &, n, k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, Jul 15 2016 *)

Terms from a(12) onward from Jon E. Schoenfield, Sep 27 2015

A328661 If n is the k-th composite number then a(n) = a(k), otherwise a(n) = n.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 3, 1, 5, 11, 2, 13, 7, 3, 1, 17, 5, 19, 11, 2, 13, 23, 7, 3, 1, 17, 5, 29, 19, 31, 11, 2, 13, 23, 7, 37, 3, 1, 17, 41, 5, 43, 29, 19, 31, 47, 11, 2, 13, 23, 7, 53, 37, 3, 1, 17, 41, 59, 5, 61, 43, 29, 19, 31, 47, 67, 11, 2, 13, 71, 23, 73

Offset: 1

Rémy Sigrist, Oct 24 2019

nonn

			a(42) = a(28) = a(18) = a(10) = a(5) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A288469 and A328018 for similar sequences.

Cf. A002808, A006508.

k=0; for (n=1, #a=vector(73), print1 (a[n] = if (bigomega(n)>1, a[k++], n) ", "))

a(n) = 1 iff n belongs to A006508.

A356398 a(1) = 4, and for any n > 1, a(n+1) is the a(n)-th squarefree number.

Original entry on oeis.org

4, 5, 6, 7, 10, 14, 21, 33, 53, 85, 138, 222, 366, 599, 985, 1613, 2651, 4357, 7169, 11795, 19401, 31913, 52487, 86347, 142021, 233615, 384277, 632091, 1039741, 1710305, 2813358, 4627790, 7612435, 12521926, 20597674, 33881799, 55733298, 91677666, 150803687

Offset: 1

Rémy Sigrist, Aug 05 2022

nonn

See A071255 for a similar sequence.

The ratio a(n+1)/a(n) tends to Pi^2/6 (A013661) as n tends to infinity.

			a(1) = 4 and the fourth squarefree number is 5, so a(2) = 5.

Rémy Sigrist, C program

Cf. A005117, A006508, A007097, A013661, A071255.

C
```
See Links section.
```

a(n+1) = A005117(a(n)).

A377181 Rectangular array, by antidiagonals: (row 1) = r(1) = A002808 (composite numbers); (row n) = r(n) = A002808(r(n-1)) for n>=1.

Original entry on oeis.org

4, 6, 9, 8, 12, 16, 9, 15, 21, 26, 10, 16, 25, 33, 39, 12, 18, 26, 38, 49, 56, 14, 21, 28, 39, 55, 69, 78, 15, 24, 33, 42, 56, 77, 94, 106, 16, 25, 36, 49, 60, 78, 105, 125, 141, 18, 26, 38, 52, 69, 84, 106, 140, 164, 184, 20, 28, 39, 55, 74, 94, 115, 141, 183, 212, 236

Offset: 1

Clark Kimberling, Oct 19 2024

			 corner:
   4     6     8     9    10    12    14    15    16    18
   9    12    15    16    18    21    24    25    26    28
  16    21    25    26    28    33    36    38    39    42
  26    33    38    39    42    49    52    55    56    60
  39    49    55    56    60    69    74    77    78    84
  56    69    77    78    84    94   100   105   106   115
  78    94   105   106   115   125   133   140   141   152

Cf. A002808 (row 1), A050545 (row 2), A280327 (row 3), A006508 (column 1), A022450 (column 2), A023451 (column 3), A059981, A236356, A280327 (principal diagonal), A377173, A114577 (dispersion of the composite numbers).

c[n_] := c[n] = Select[Range[500], CompositeQ][[n]]
r[0] = Table[c[n], {n, 1, 10}]
r[n_] := r[n] = c[r[n - 1]]
Grid[Table[r[n], {n, 0, 6}]]  (* array *)
p[n_, k_] := r[n][[k]];
Table[p[n - k + 1, k], {n, 0, 9}, {k, n + 1, 1, -1}] // Flatten  (* sequence *)

A059981(n) = number of appearances of A002808(n).

A065962 a(1) = 1, a(n) = a(n - 1) + pi(a(n - 1)) + 1.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 26, 36, 48, 64, 83, 107, 136, 169, 209, 256, 311, 376, 451, 539, 639, 755, 889, 1044, 1220, 1420, 1644, 1904, 2196, 2524, 2894, 3313, 3780, 4307, 4898, 5553, 6286, 7104, 8015, 9025, 10147, 11393, 12769, 14293, 15971, 17832

Offset: 1

Labos Elemer, Dec 08 2001

Labos came up with this sequence when trying to write a Mathematica program for A006508. The entire loop "While[ k - PrimePi[ k ] - 1, k++ ]" is meaningless; all the function g[n] really does is add up n + pi(n) + 1 and then NestList makes the recurrence happen. - Alonso del Arte, Oct 25 2011

			a(4) = 7 because a(3) = 4 and 4 + pi(4) + 1 = 4 + 2 + 1 = 7.
a(5) = 12 because a(4) = 7 and 7 + pi(7) + 1 = 7 + 4 + 1 = 12.

Cf. A000720.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 1, 50 ]
NestList[#+PrimePi[#]+1&,1,50] (* Harvey P. Dale, Feb 13 2016 *)

A207573 a(1)=1, a(n+1) = prime(a(n)+1) - a(n).

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 29, 84, 355, 2038, 15723, 156920, 1959177, 29788564, 539206155, 11406258536, 277708694711, 7683630307352, 239005572292955, 8281900782667178, 317172995786425445

Offset: 1

Gerasimov Sergey, Feb 19 2012

nonn

			a(2) = prime(a(1)+1)-a(1) =  3-1 = 2;
a(3) = prime(a(2)+1)-a(2) =  5-2 = 3;
a(4) = prime(a(3)+1)-a(3) =  7-3 = 4;
a(5) = prime(a(4)+1)-a(4) = 11-4 = 7.

Cf. A006508, A000040.

a(19)-a(21) from Charles R Greathouse IV, Feb 20 2012

cp's OEIS Frontend

A064960 The prime then composite recurrence; a(2n) = a(2n-1)-th prime and a(2n+1) = a(2n)-th composite and a(1) = 1.

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A064961 Composite-then-prime recurrence; a(2n) = a(2n-1)-th composite and a(2n+1) = a(2n)-th prime and a(1) = 1.

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A072415 a(1) = 2; a(n) = a(n-1)-th even nontotient number.

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A072416 a(1) = 2; a(n) = half of the a(n-1)-th even nontotient number.

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A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

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A328661 If n is the k-th composite number then a(n) = a(k), otherwise a(n) = n.

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A356398 a(1) = 4, and for any n > 1, a(n+1) is the a(n)-th squarefree number.

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A377181 Rectangular array, by antidiagonals: (row 1) = r(1) = A002808 (composite numbers); (row n) = r(n) = A002808(r(n-1)) for n>=1.

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A065962 a(1) = 1, a(n) = a(n - 1) + pi(a(n - 1)) + 1.