Chayim Lowen - cp's OEIS frontend

A262149 Rocket sequence 50: a(0)=50, a(n)=A073846(a(n-1)).

50, 97, 68, 139, 94, 211, 140, 349, 222, 607, 378, 1129, 689, 427, 272, 769, 476, 1493, 901, 552, 1783, 1072, 3863, 2268, 9151, 5275, 3077, 1819, 1092, 3931, 2308, 9323, 5370, 24113, 13671, 7825, 4528, 20021, 11385, 6537, 3796, 16363, 9336, 44927, 25250

Offset: 0

Chayim Lowen, Sep 12 2015

nonn

This sequence has been checked up to a(98) = 1078406742163 without reaching 50. It seems to be slowly climbing in value in both the negative and positive directions. Hence, its period is either extremely large or, as I conjecture, infinite. Thus I dubbed the sequence "Rocket" because, as opposed the "Hailstone" sequences, it never seems to "fall".

This sequence, when extended to all integers using a(n-1) = A073898(a(n)), is R#(50), see A073846 for definition. - Chayim Lowen, Jan 25 2016

			a(1) = A073846(a(0)) = A073846(50) = 97.

Chayim Lowen, Table of n, a(n) for n = 0..99

Cf. A073846, A261314, A261621.

s = Module[{p = Prime@ Range@ PrimePi@ #, c}, c = Complement[Range@ #, p]; Riffle[Take[c, Length@ p], p]] &[5*10^5]; NestList[s[[#]] &, 50, 44] (* Michael De Vlieger, Jan 27 2016, after Harvey P. Dale at A073846 *)

a(n+1) = A073846(a(n)).

a(n-1) = A073898(a(n)).

Missing term 1092 inserted by Chayim Lowen, Mar 26 2017

A261621 Rocket Sequence 42: a(0) = 42, a(n) = A073846(a(n-1)).

Original entry on oeis.org

42, 73, 52, 101, 70, 149, 100, 229, 150, 379, 243, 159, 108, 251, 164, 421, 267, 174, 449, 286, 823, 508, 1609, 968, 3461, 2040, 8111, 4689, 2745, 1631, 981, 600, 1987, 1189, 723, 448, 1423, 861, 530, 1697, 1020, 3643, 2146, 8623, 4978, 22193, 12602, 62791

Offset: 0

Chayim Lowen, Sep 09 2015

nonn

This sequence has been checked up to a(108) = 106838266736 without reaching 42. It seems to be slowly climbing in value in both the negative and positive directions. Its period is either extremely large or more probably infinite.

			a(1) = A073846(a(0)) = A073846(42) = 73.

Chayim Lowen, Table of n, a(n) for n = 0..54

Cf. A073846, A073898, A261314, A262149.

a(n+1) = A073846(a(n)).

a(n-1) = A073898(a(n)).

A263570 Smallest positive integer such that n iterations of A073846 are required to reach an even number.

Original entry on oeis.org

2, 3, 17, 31, 163, 353, 721, 1185, 1981, 3363, 5777, 10039, 29579, 52737, 94705, 171147, 311101, 568431, 1043463, 1923619, 3559911, 6611675, 12319517, 23023727, 651267929, 1234823707, 2345409699, 4462239583, 8502848523, 16226083005, 31007327791, 59331187155

Offset: 0

Chayim Lowen, Oct 21 2015

nonn

A number is considered to be its own zeroth iteration.
Is the sequence defined for all n? If so, are there infinitely many composite numbers? If not, are infinitely many a(n) defined?
From Hartmut F. W. Hoft, Apr 05 2016: (Start)
Numbers a(6)...a(11) and a(12)...a(23) each belong to iteration sequences that start with prime numbers 10039 and 23023727, respectively, while the other numbers in the sequences are composite.
For the entire iteration sequences and computation of the additional numbers for this sequence see A271363. (End)
For n>1, a(n) is the least integer k such that the repeated application of x -> A073846(x) strictly decreases exactly n times in a row. - Hugo Pfoertner and Michel Marcus, Mar 11 2021

			a(2)=17 because A073846(17) = 15, A073846(15) = 14; thus it took two steps whereas no smaller positive integer has this property.

Martin Ehrenstein, Table of n, a(n) for n = 0..43

Cf. A073846, A271363.

(* Since A073846(9)=9, search starts with 11 *)
c25000000 = Select[Range[25000000], CompositeQ];
a073846[n_] := c25000000[[Floor[n/2]]]
a073846Nest[n_] := Length[NestWhileList[a073846, n, OddQ]]
a263570[n_] := Module[{list={2, 3}, i, length}, For[i=11, i<=n, i+=2, length=a073846Nest[i]; If[Length[list]Hartmut F. W. Hoft, Apr 05 2016 *)

For n>0, a(n+1) >= A073898(b(a(n))), where b(m) is the smallest odd composite not smaller than m, equality always holds if a(n) is composite.

a(24)-a(31) from Hartmut F. W. Hoft, Apr 05 2016

A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).

Original entry on oeis.org

1, 2, 5, 11, 59, 653, 84761, 2763189059, 377406001499268899, 2638619515495963542360422694651593, 135435890329895562961039215198033899386421965445591860752412324961

Offset: 0

Chayim Lowen, Sep 15 2015

nonn

The product of the first n+1 terms is the smallest deficient multiple of the product of the first n terms.
The product of any finite number of distinct terms of this sequence is deficient.
a(n) for n > 0 is the lexicographically earliest sequence of primes P, such that the asymptotic density of the squarefree numbers (A005117) which are not divisible by any prime in P is 3/Pi^2 (A104141), i.e., half the asymptotic density of all the squarefree numbers. - Amiram Eldar, Nov 30 2020

			a(3) = 11 because A001065(2*5*7) = A001065(70) = 74 > 70, and A001065(2*5*11) = A001065(110) = 106 < 110.
From _M. F. Hasler_, Dec 14 2017: (Start)
Let Q(x) = 1/(2x/sigma(x) - 1), P(n) = Product( a(k), k<n): P(0) = 1 (empty product). Then:
Q(P(0)) = 1, a(0) = nextprime(1) = 2 = P(1).
Q(P(1)) = 3, a(1) = 5. (2*3 is perfect, P(2) = 2*5 is deficient.)
Q(P(2)) = 9, a(2) = 11. (2*5*7 is weird, P(3) = 2*5*11 is deficient.)
Q(P(3)) = 54, a(3) = 59. (P(3)*53 is weird, P(4) = 2*5*11*59 is deficient.)
Q(P(4)) = 648, a(4) = 653. (P(4)*647 is weird, P(5) = 2*5*11*59*653 is deficient.)
Q(P(5)) = 84758.4, a(5) = 84761. (P(5)*84751 is abundant and semiperfect: sum of all proper divisors except {1, 2, 11, 22, 55, 59, 590}; P(6) = 2*5*11*59*653*84761 is deficient.) (End)

Amiram Eldar, Table of n, a(n) for n = 0..14

Cf. A001065, A005100, A005117, A104141, A151800 (nextprime).

Cf. A002975 (primitive weird numbers), A000203 (sigma), A295001 (same definition but a(0) = 4).

a[0]=1; a[n_] := a[n] = NextPrime[1/(2*Product[a[i],{i,1,n-1}]/Product[a[i]+1,{i,1,n-1}]-1)]; Array[a, 11, 0] (* Amiram Eldar, Jun 10 2019 *)

lista(nn) = {print1(p=1, ", "); vp = [p]; for (n=2, nn, np = nextprime(1+floor(1/(2*prod(i=2, n-1, vp[i]/(vp[i]+1))-1))); vp = concat(vp, np); print1(np, ", "););} \\ Michel Marcus, Oct 16 2015

a=List(); m=1; for(n=0, 13, listput(a, p=nextprime(1\(2/sigma(m,-1)-1)+1)); p>default(primelimit)&&addprimes(p); m*=p); a \\ M. F. Hasler, Dec 14 2017

a(n) = A151800(floor(1/(2*(Product_{i=2..n-1} a(i)/(a(i)+1))-1))), where A151800 is the "next larger prime" function.
Lim_{n->infinity} A001065(Product_{i=0..n} a(i))/(Product_{i=0..n} a(i)) = 1. [Corrected by M. F. Hasler, Dec 04 2017]
Conjecture: log(a(n)) ~ e^(an+b) where a and b are approximately 0.6 and -1.6 respectively.

A261314 Rocket Sequence 34: a(0)=34, a(n) = A073846(a(n-1)).

Original entry on oeis.org

34, 59, 44, 79, 56, 107, 75, 54, 103, 72, 151, 102, 233, 153, 104, 239, 156, 397, 253, 165, 112, 263, 171, 116, 271, 176, 457, 290, 829, 512, 1619, 974, 3469, 2044, 8123, 4696, 20879, 11861, 6807, 3952, 17159, 9786, 47459

Offset: 0

Chayim Lowen, Aug 14 2015

nonn

A073846(n) is defined as follows: if n = 2m for some integer m, A073846(n) is the m-th prime, if n = 2m-1 for some integer m, A073846(n) is the m-th nonprime.
Consider the (totally) ordered set {n, A073846(n), A073846(A073846(n))...} and let us append to this the ordered set {...b(b(b(n))),b(b(n)),b(n)} where b(m) = A073898(m) is the inverse of A073846. Let us call the result R#(n). It is clear that if m is a value in R#(n), R#(m) is just R#(n) with a different offset. Therefore, unless there is a need to do otherwise, let us denote each sequence by its lowest value. {a(n)} when extended to all integers (the last few unlisted values are ... 36, 61, 45, 34) is R#(34).
A given sequence c(n) can be one of two kinds. It can either be periodic with c(m) = c(0) for some m, or it can include infinitely many distinct values. R#(n) is finite for all n<34. However, this sequence has been checked up to a(86) = 1091595086717 without reaching 34. Instead it seems to be slowly climbing in value in both the negative and positive directions. Hence, its period is either extremely large or nonexistent (infinite). I conjecture that the latter is the case. Thus I dubbed the sequence "Rocket" because, as opposed to the "Hailstone" sequences, it never seems to "fall".

			a(1) = A073846(a(0)) = A073846(34) = 59.

Chayim Lowen, Table of n, a(n) for n = 0..86

Cf. A073846.

f[n_, lim_] := Block[{p = Prime@ Range@ PrimePi@ lim, c, s, a = {34}}, c = Complement[Range@ lim, p]; s = Riffle[Take[c, Length@ p], p]; Do[AppendTo[a, s[[a[[k]]]]], {k, n}]; a]; f[48, 10000000] (* Michael De Vlieger, Aug 26 2015, after Harvey P. Dale at A073846 *)

a(n+1) = A073846(a(n)) = A018252(ceiling(a(n)/2))*A000035(a(n)) + A000040(ceiling(a(n)/2))*A059841(a(n)).

a(n-1) = A073898(a(n)) = 2*A010051(a(n))*A000720(a(n)) + (1-A010051(a(n)))*(2*A018252(a(n))-1).

A260315 Decimal expansion of the generating constant c of A139244.

Original entry on oeis.org

1, 9, 6, 6, 8, 9, 1, 7, 6, 1, 7, 9, 0, 1, 7, 6, 3, 6, 5, 3, 3, 3, 5, 0, 5, 7, 2, 0, 2, 2, 6, 6, 9, 8, 2, 7, 6, 1, 3, 2, 2, 1, 2, 1, 1, 8, 6, 8, 6, 3, 3, 5, 3, 4, 2, 2, 0, 1, 8, 5, 8, 2, 8, 8, 3, 1, 2, 6, 2, 5, 6, 4, 4, 2

Offset: 1

Chayim Lowen, Jul 22 2015

A139244(n-1) = ceiling(c^(2^n)).

The digit 4 appears for the first time only at position n=53, making it the last one to appear in the sequence.

			1.96689176179017636533350572... .

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Cf. A139244.

A256625 Product of the first n lucky numbers.

Original entry on oeis.org

1, 3, 21, 189, 2457, 36855, 773955, 19348875, 599815125, 19793899125, 732374267625, 31492093507875, 1543112581885875, 78698741676179625, 4958020725599316375, 332187388615154197125, 22920929814445639601625, 1673227876454531690918625, 125492090734089876818896875, 9913875167993100268692853125

Offset: 1

Chayim Lowen, Jul 11 2015

For all n A057616(n)=a(m) for some m, that is, A256625 is a superset of A057616.

			For n=4, a(4) = A000959(1)*A000959(2)*A000959(3)*A000959(4) = 1*3*7*9 = 189.

Robert Israel, Table of n, a(n) for n = 1..315

Cf. A000959 (lucky numbers).
Cf. A000142 (factorial), A002110 (product of the first n primes).
Cf. A057616 (subset).

N:= 100:  # to use all lucky numbers <= N
L:= [seq(2*i+1, i=0..N)]:
for n from 2 while n < nops(L) do
  r:= L[n];
  L:= subsop(seq(r*i=NULL, i=1..nops(L)/r), L);
od:
seq(convert(L[1..n],`*`),n=1..nops(L)) ;  # Robert Israel, Jul 14 2015

luckies = 2*Range@ 200 - 1; f[n_] := Block[{k = luckies[[n]]}, luckies = Delete[luckies, Table[{k}, {k, k, Length@ luckies, k}]]]; Do[f@ n, {n, 2, 30}]; Table[Times @@ Take[luckies, n], {n, 20}] (* Michael De Vlieger, Jul 12 2015, based on code from Robert G. Wilson v at A000959 *)

a(n+1) = a(n)*A000959(n+1).
a(n) > A002110(n) > A000142(n) for all n>1 (conjectured).
A000312(n) = o(a(n)) where o(a(n)) is "little o" notation (conjectured).
Sum_{k >=1} 1/a(n) = 1.386678865477327..., the sum converges because the inequality a(n) > A000142(n) holds for large enough n.

a(12)-a(20) from Michael De Vlieger, Jul 12 2015

cp's OEIS Frontend

User: Chayim Lowen

A262149 Rocket sequence 50: a(0)=50, a(n)=A073846(a(n-1)).

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A261621 Rocket Sequence 42: a(0) = 42, a(n) = A073846(a(n-1)).

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A263570 Smallest positive integer such that n iterations of A073846 are required to reach an even number.

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A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).

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A261314 Rocket Sequence 34: a(0)=34, a(n) = A073846(a(n-1)).

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A260315 Decimal expansion of the generating constant c of A139244.

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A256625 Product of the first n lucky numbers.

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