A079000 -id:A079000 - cp's OEIS frontend

A079948 First differences of A079000.

3, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Feb 22 2003

nonn

Alternate description of sequence: start with a(1)=3; apply 1->2, 2->11, 3->21; iterate. - Matthew Vandermast, Mar 08 2003

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
N. J. A. Sloane, Seven Staggering Sequences.

b[1] = 1; b[n_] := (k = Floor[Log[2, (n+3)/6]]; j = n - (9*2^k-3); 12*2^k - 3 + 3*j/2 + Abs[j]/2); Array[b, 106] // Differences (* Jean-François Alcover, Sep 02 2018 *)

After first two terms, a run of length 3*2^k 1's followed by a run of length 3*2^k 2's, for k = 0, 1, ...
a(n) = floor(log_2(8*(floor((n+3)/3))/3)) - floor(log_2(floor((n+3)/3))) for n>2; with a(1)=3 and a(2)=2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003
Also a(n) = A079882(A002264(n+3)) for n>2, where A002264=floor(n/3). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

A079250 Even numbers in A079000.

Original entry on oeis.org

4, 6, 8, 16, 18, 20, 34, 36, 38, 40, 42, 44, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 286, 288, 290, 292, 294, 296, 298, 300

Offset: 1

N. J. A. Sloane, Feb 04 2003

nonn

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

(* b = A079000 *) b[1] = 1; b[n_] := (k = Floor[Log[2, (n + 3)/6]]; j = n - (9*2^k - 3); 12*2^k - 3 + 3*j/2 + Abs[j]/2);
Select[Array[b, 300], EvenQ] (* Jean-François Alcover, Sep 04 2018 *)

{4, 6, 8} and Union_{k >=1} {2i : 9*2^(k-1)-1 <= i <= 3*2^(k+1)-2 }.

A079251 Complement of A079000.

Original entry on oeis.org

2, 3, 5, 10, 12, 14, 22, 24, 26, 28, 30, 32, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 190, 192, 194, 196, 198, 200, 202, 204

Offset: 1

N. J. A. Sloane, Feb 04 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

nmax = 150; b[1] = 1; b[n_] := Module[{k, j}, k = Floor[Log[2, (n+3)/6]]; j = n - 9*2^k + 3; 12*2^k - 3 + 3j/2 + Abs[j]/2];
Complement[Range[b[nmax]], Array[b, nmax]] (* Jean-François Alcover, Nov 28 2018 *)

a(n)=if(n<3,if(n<2,2,3),3*2^floor(log(2/3*(n-1))/log(2))+2*n-4) /* Ralf Stephan */

a(n)=b(n-1)+2, with b(0)=0, b(2n)=2b(n)+1+3[n>1], b(2n+1)=2b(n)+1+5[n>0]. - Ralf Stephan, Oct 07 2003

A079252 Even numbers not in A079000.

Original entry on oeis.org

2, 10, 12, 14, 22, 24, 26, 28, 30, 32, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 190, 192, 194, 196, 198, 200, 202, 204, 206

Offset: 1

N. J. A. Sloane, Feb 04 2003

nonn

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

nmax = 150;
b[1] = 1; b[n_] := (k = Floor[Log[2, (n + 3)/6]]; j = n - (9 2^k - 3); 12 2^k - 3 + 3 j/2 + Abs[j]/2);
Complement[Range[2, b[nmax], 2], Table[b[n], {n, 1, nmax}]] (* Jean-François Alcover, Feb 13 2019 *)

See A079250 for formula.

A079325 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a member of A079000".

Original entry on oeis.org

1, 3, 4, 6, 10, 11, 12, 14, 22, 23, 25, 27, 28, 29, 30, 32, 46, 48, 50, 52, 54, 55, 57, 58, 59, 60, 61, 63, 65, 67, 68, 69, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 129, 130, 131, 133, 135, 137

Offset: 1

Matthew Vandermast, Feb 12 2003

nonn

			a(2) cannot be 2, which would imply that 2 is a member of A079000 (it is not); letting a(2)=3 creates no contradiction, since 3 is not a member of A079000 and the third term (4) is the next A079000 member in the sequence.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Aronson transform of A079000.

A080566 Partial sums of A079000.

Original entry on oeis.org

1, 5, 11, 18, 26, 35, 46, 59, 74, 90, 107, 125, 144, 164, 185, 208, 233, 260, 289, 320, 353, 387, 422, 458, 495, 533, 572, 612, 653, 695, 738, 782, 827, 874, 923, 974, 1027, 1082, 1139, 1198, 1259, 1322, 1387, 1454, 1523, 1593, 1664, 1736, 1809, 1883, 1958

Offset: 1

N. J. A. Sloane, Feb 21 2003

nonn

A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

Original entry on oeis.org

2, 4, 5, 8, 10, 13, 16, 17, 20, 23, 25, 26, 31, 32, 34, 37, 40, 43, 46, 47, 50, 52, 61, 62, 64, 65, 67, 68, 73, 74, 79, 80, 85, 86, 89, 92, 94, 100, 103, 104, 107, 109, 113, 115, 122, 124, 125, 128, 130, 134, 136, 137, 146, 148, 149, 151, 155, 158, 160, 163

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

A self-describing sequence.
The prime indices of m are the numbers k such that prime(k) divides m.
The sequence is monotonically increasing, since once a number is rejected it stays rejected. Sequence is closed under multiplication for a similar reason. - N. J. A. Sloane, Aug 26 2018

			After the initial term 2, the next term cannot be 3 because 3 has prime index 2, and 2 is already in the sequence. The next term could be 10, which has prime indices 1 and 3, but 4 (with prime index 1) is smaller. So a(2) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000002, A001462, A079000, A079254, A214577, A276625, A277098, A280996, A303431.

For first differences see A317963, for primes see A317964.

A:= NULL:
P:= {}:
for n  from 2 to 1000 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    A:= A, n;
    P:= P union {ithprime(n)};
  fi
od:
A; # Robert Israel, Aug 26 2018

gaQ[n_]:=Or[n==0,And@@Cases[FactorInteger[n],{p_,k_}:>!gaQ[PrimePi[p]]]];
Select[Range[100],gaQ]

Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019

A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 19, 21, 27, 29, 33, 37, 39, 43, 47, 49, 53, 57, 59, 61, 63, 71, 77, 79, 81, 83, 87, 89, 91, 97, 99, 101, 107, 111, 113, 117, 121, 127, 129, 131, 133, 139, 141, 143, 147, 149, 151, 159, 163, 169, 171, 173, 177, 179, 181, 183, 189, 193, 197

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  19: {8}
  21: {2,4}
  27: {2,2,2}
  29: {10}
  33: {2,5}
  37: {12}
  39: {2,6}
  43: {14}
  47: {15}
  49: {4,4}
  53: {16}
  57: {2,8}
  59: {17}
  61: {18}
  63: {2,2,4}

Complement of A324694. Prime indices are A304360.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098.
Cf. A324696, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

aQ[n_]:=And@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>!aQ[PrimePi[p]]];
Select[Range[100],aQ]

A109298 Primal codes of finite idempotent functions on positive integers.

Original entry on oeis.org

1, 2, 9, 18, 125, 250, 1125, 2250, 2401, 4802, 21609, 43218, 161051, 300125, 322102, 600250, 1449459, 2701125, 2898918, 4826809, 5402250, 9653618, 20131375, 40262750, 43441281, 86882562, 181182375, 362364750, 386683451, 410338673, 603351125, 773366902, 820677346

Offset: 1

Jon Awbrey, Jul 06 2005

nonn

Finite idempotent functions are identity maps on finite subsets, counting the empty function as the idempotent on the empty set.
From Gus Wiseman, Mar 09 2019: (Start)
Also numbers whose ordered prime signature is equal to the distinct prime indices in increasing order. A prime index of n is a number m such that prime(m) divides n. The ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base. The case where the prime indices are taken in decreasing order is A324571.
Also numbers divisible by prime(k) exactly k times for each prime index k. These are a kind of self-describing numbers (cf. A001462, A304679).
Also Heinz numbers of integer partitions where the multiplicity of m is m for all m in the support (counted by A033461). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of distinct elements of A062457. For example, 43218 = prime(1)^1 * prime(2)^2 * prime(4)^4.
(End)

			Writing (prime(i))^j as i:j, we have the following table of examples:
Primal Codes of Finite Idempotent Functions on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = ` ` 2:2
` ` `18 = 1:1 2:2
` ` 125 = ` ` ` ` 3:3
` ` 250 = 1:1 ` ` 3:3
` `1125 = ` ` 2:2 3:3
` `2250 = 1:1 2:2 3:3
` `2401 = ` ` ` ` ` ` 4:4
` `4802 = 1:1 ` ` ` ` 4:4
` 21609 = ` ` 2:2 ` ` 4:4
` 43218 = 1:1 2:2 ` ` 4:4
`161051 = ` ` ` ` ` ` ` ` 5:5
`300125 = ` ` ` ` 3:3 4:4
`322102 = 1:1 ` ` ` ` ` ` 5:5
`600250 = 1:1 ` ` 3:3 4:4
From _Gus Wiseman_, Mar 09 2019: (Start)
The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2) has prime signature {1,2} and the distinct prime indices are also {1,2}.
       1: {}
       2: {1}
       9: {2,2}
      18: {1,2,2}
     125: {3,3,3}
     250: {1,3,3,3}
    1125: {2,2,3,3,3}
    2250: {1,2,2,3,3,3}
    2401: {4,4,4,4}
    4802: {1,4,4,4,4}
   21609: {2,2,4,4,4,4}
   43218: {1,2,2,4,4,4,4}
  161051: {5,5,5,5,5}
  300125: {3,3,3,4,4,4,4}
  322102: {1,5,5,5,5,5}
  600250: {1,3,3,3,4,4,4,4}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Awbrey, Riffs and Rotes

Cf. A076954, A106177, A108352, A108371, A109297, A109301.
Cf. A001156, A033461, A056239, A062457, A112798, A118914, A124010 (ordered prime signature), A181819, A276078, A304679.
Cf. A324524, A324525, A324570, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Select[Range[10000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]==k]&]

is(n) = my(f = factor(n)); for(i = 1, #f~, if(prime(f[i, 2]) != f[i, 1], return(0))); 1 \\ David A. Corneth, Mar 09 2019

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/prime(n)^n) = 1.6807104966... - Amiram Eldar, Jan 03 2021

Offset set to 1, missing terms inserted and more terms added by Alois P. Heinz, Mar 08 2019

A277098 Finitary primes. Primes of finitary index.

Original entry on oeis.org

2, 3, 5, 11, 13, 29, 31, 41, 47, 79, 101, 109, 113, 127, 137, 167, 179, 211, 257, 271, 293, 313, 317, 397, 401, 421, 449, 487, 491, 547, 599, 601, 617, 677, 709, 733, 773, 811, 823, 829, 907, 929, 977, 991, 1033, 1063, 1109, 1187, 1231, 1259, 1297, 1361, 1429, 1483, 1489, 1559, 1609, 1621, 1741, 1759, 1831, 1871, 1889

Offset: 1

Gus Wiseman, Sep 29 2016

nonn

Definition: prime(n) is a finitary prime iff n is a product of distinct finitary primes, where prime = A000040. This sequence could be described as a "multiplicative Aronson transform" of A005117. (Aronson transforms such as A079000 satisfy "n is in a if and only if a(n) is in b".

The composite bijection (finitary primes -> finitary numbers -> finite sets of finitary primes) can be used to construct a natural linear extension SET : N -> F where F is the partially ordered inverse limit of all finite Boolean algebras of finite sets of positive integers. Then a(n) = prime(Prod_p a(p)) where the product is over SET(n).

			The sequence of all nonempty finite sets of positive integers (a=1 b=2.. *=27) begins:
0,a,b,c,ab,ac,d,e,bc,ad,ae,f,abc,
g,bd,be,h,i,cd,af,ag,ce,abd,abe,
j,ah,bf,bg,ai,k,l,acd,m,bh,n,ace,
o,bi,de,cf,cg,aj,bcd,p,abf,q,abg,
bce,ak,ch,r,al,am,ci,bj,abh,an,s,
t,ao,abi,ade,acf,u,bk,acg,v,w,df,
bl,abcd,ap,bm,dg,aq,ef,bn,abce,
cj,x,y,eg,ach,bo,z,ar,bde,bcf,*

Gus Wiseman, Table of n, a(n) for n = 1..10000

Subsequence of A302491.

Cf. A000040, A005117, A079000, A079254, A276625.

has(p)=if(p<7, 1, my(f=factor(primepi(p))); if(vecmax(f[,2])>1, return(0)); for(i=1,#f~, if(!has(f[i,1]), return(0))); 1)
is(n)=isprime(n) && has(n) \\ Charles R Greathouse IV, Aug 03 2023

a(n) = A000040(A276625(n)).

cp's OEIS Frontend

A079948 First differences of A079000.

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

Formula

A079250 Even numbers in A079000.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A079251 Complement of A079000.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

A079252 Even numbers not in A079000.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A079325 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a member of A079000".

Views

Author

Keywords

Examples

Links

Crossrefs

A080566 Partial sums of A079000.

Views

Author

Keywords

A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A109298 Primal codes of finite idempotent functions on positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula