Bill McEachen - cp's OEIS frontend

A379757 a(n) = a(n-1) + 1 with two exceptions: if a(n-1) is prime, a(n) = a(n-2) + a(n-1), or if a(n-1) is a power, a(n) = a(n-1) / (root factor), with initial three terms are 0, 1, 2.

0, 1, 2, 3, 5, 8, 4, 2, 6, 7, 13, 20, 21, 22, 23, 45, 46, 47, 93, 94, 95, 96, 97, 193, 290, 291, 292, 293, 585, 586, 587, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 2361, 2362, 2363, 2364, 2365, 2366, 2367, 2368, 2369, 2370, 2371, 4741, 4742, 4743

Offset: 1

Bill McEachen, Jan 02 2025

The construction rules are very basic, but lead to somewhat surprising results. Terms that are perfect powers are extremely rare (only n=6,7 so far). Additionally, the sequence is nearly all composites. Comparing to A000045, eight early distinct terms are in common, but it is unclear when another intersection is seen.

			We know a(1)=0, a(2)=1, a(3)=2. Since a(3) is prime, a(4)=a(2)+a(3)=3. Since a(4) is prime, a(5)=a(3)+a(4)=5. Similarly, a(6)=a(4)+a(5)=8. Since a(6) is a perfect power, a(7) = a(6)/2 since 8=2^3. Since a(7)=4 is another perfect power, a(8)=4/2=2. Since a(8) is prime, a(9)=a(7)+a(8)=6. For clarity, if a(n-1) = r^k, then a(n) = a(n-1)/r.

Cf. A000045, A001597, A089580.

a[n_] := a[n] = If[n < 4, n-1, If[PrimeQ[a[n-1]], a[n-1] + a[n-2], If[(g = GCD @@ FactorInteger[a[n-1]][[;; , 2]]) > 1, a[n-1]^(1 - 1/g), a[n-1] + 1]]]; Array[a, 54] (* Amiram Eldar, Apr 10 2025 *)

Conjecture: log(a(n)) ~ k*sqrt(n).

A381329 Number of steps for n to reach 1 under the map x -> x/2 if x is even, x -> 2*x+1 if x is prime or a perfect power, otherwise x -> gpf(x)-1 where gpf(x) = A006530(x).

Original entry on oeis.org

0, 1, 5, 2, 16, 6, 4, 3, 10, 17, 15, 7, 20, 5, 3, 4, 8, 11, 9, 18, 7, 16, 14, 8, 6, 21, 19, 6, 7, 4, 8, 5, 18, 9, 7, 12, 4, 10, 8, 19, 23, 8, 8, 17, 3, 15, 13, 9, 19, 7, 5, 22, 11, 20, 18, 7, 12, 8, 6, 5, 21, 9, 7, 6, 8, 19, 4, 10, 17, 8, 9, 13, 8, 5, 3, 11, 18

Offset: 1

Bill McEachen, Feb 20 2025

nonn

Does every n reach 1?

			n=7 reaches 1 by 7 -> 15 -> 4 -> 2 -> 1 which is a(7)=4 steps.
Starting from 47993 yields 4362, 2181, 726, 363, 10, 5, 11, 23, 47, 95, 18, 9, 19, 39, 12, 6, 3, 7, 15, 4, 2, 1. Thus a(47993)=22.

Cf. A006530.

s[n_] := Which[n == 1, 1, EvenQ[n], n/2, PrimeQ[n] || GCD @@ ((f = FactorInteger[n])[[;; , 2]]) > 1, 2*n + 1, True, f[[-1, 1]] - 1]; a[n_] := -1 + Length@ NestWhileList[s, n, # > 1 &]; Array[a, 100] (* Amiram Eldar, Feb 20 2025 *)

A377371 a(n) = k*(a(n-1)+n), k=-1 for prime n, otherwise k=1 (a(1)=1).

Original entry on oeis.org

1, -3, 0, 4, -9, -3, -4, 4, 13, 23, -34, -22, 9, 23, 38, 54, -71, -53, 34, 54, 75, 97, -120, -96, -71, -45, -18, 10, -39, -9, -22, 10, 43, 77, 112, 148, -185, -147, -108, -68, 27, 69, -112, -68, -23, 23, -70, -22, 27, 77, 128, 180, -233, -179, -124, -68, -11

Offset: 1

Bill McEachen, Dec 27 2024

From the infinitude of the primes, k<0 will always be seen, and the +/- halves of the graph are essentially mirror images. The number of sign changes through n should be on the order of primepi(n).

			a(1)=1. For n=2 (prime), a(2) = -(1+2) = -3. For n=3 (prime), a(3) = -(-3+3) = 0.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

j = 1; {j}~Join~Reap[Do[k = (1 - 2 Boole[PrimeQ[n]])*(j + n); j = Sow[k], {n, 2, 57}] ][[-1, 1]] (* Michael De Vlieger, Dec 28 2024 *)

a(n) = a(n-1) + n for composite n, else -(a(n-1) + n).

A378814 a(n) = round(n/(A000005(A071904(n))-2)).

Original entry on oeis.org

1, 1, 2, 4, 3, 3, 4, 4, 2, 10, 6, 6, 7, 4, 8, 8, 4, 9, 6, 10, 11, 11, 12, 12, 6, 4, 14, 14, 7, 15, 31, 16, 17, 17, 18, 6, 19, 19, 20, 10, 10, 21, 22, 22, 8, 46, 12, 12, 25, 25, 26, 26, 9, 9, 28, 28, 29, 15, 30, 30, 31, 31, 32, 32, 9, 11, 34, 34, 17, 18, 36

Offset: 1

Bill McEachen, Dec 08 2024

nonn

Nearly all of the data falls on lines discussed below. There are a few "outliers" visible on the graph. There are <120 such outliers in the first 20000 terms (about 0.6%). Many of the outlier indices belong to A037040. The lines are n, (n+0)/2, (n+2)/4, (n+4)/6, (n+6)/8,....

			Let n=14, A071904(n)=63, tau(63)=6 so a(14)=round(14/(6-2))=4.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Bill McEachen, Plot 20K terms
Bill McEachen, A037040 overlap (TEMP)

Cf. A000005, A037040, A071904.

MapIndexed[Floor[#2[[1]]/# + 1/2] &, DivisorSigma[0, Select[Range[9, 500, 2], CompositeQ]] - 2] (* Paolo Xausa, Dec 16 2024 *)

A378677 a(n)=a(n-1) + prime(n) for n prime, and a(n)=-a(n-1) otherwise, with a(0)=0, with duplicates removed afterwards.

Original entry on oeis.org

0, 3, 8, -8, -3, 14, -14, 17, -17, 24, -24, 35, -35, 32, -32, 51, -51, 58, -58, 69, -69, 88, -88, 91, -91, 100, -100, 111, -111, 130, -130, 147, -147, 136, -136, 195, -195, 158, -158, 209, -209, 192, -192, 239, -239, 222, -222, 287, -287, 260, -260, 303, -303

Offset: 0

Bill McEachen, Dec 03 2024

sign

Let b = subset of positive terms for n>4. We have A073131= b(m+2)-b(m) , A006450= b(m+2)+b(m) and A299644= b(m+2)+2*b(m+1)+b(m).

			n=1 is not prime, so a(1)= -a(0)= 0. n=2 is prime, so a(2)=a(1)+prime(2)=0+3=3. n=5 is prime, so a(5)=3, but note that it duplicates a(2). n=6 is not prime, so a(6)= -a(5)=-3. After terms are computed, duplicates are only then removed, which will alter indices accordingly.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A006450, A073131, A299644.

Module[{n = 0}, DeleteDuplicates[NestList[If[PrimeQ[++n], # + Prime[n], -#] &, 0, 200]]] (* Paolo Xausa, Dec 06 2024 *)

a(n) = a(n-1) + a prime for n odd >4.

a(n) = -a(n-1) for a(n-1)>0, n>1.

A378500 a(1) = 2, then a(n) = a(n-1) - 2 for n even, a(n) = a(n-1) + 3 for n an odd prime or odd prime power, and a(n) = a(n-1) + 2 otherwise.

Original entry on oeis.org

2, 0, 3, 1, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6, 8, 6, 9, 7, 10, 8, 10, 8, 11, 9, 12, 10, 13, 11, 14, 12, 15, 13, 15, 13, 15, 13, 16, 14, 16, 14, 17, 15, 18, 16, 18, 16, 19, 17, 20, 18, 20, 18, 21, 19, 21, 19, 21, 19, 22, 20, 23, 21, 23, 21, 23, 21, 24, 22, 24, 22, 25

Offset: 1

Bill McEachen, Nov 30 2024

nonn

Asymptotically the sequence should reach long periods of stasis, with rare increases with each new prime so as to yield a strict permutation of the positive integers if duplicates are removed.

			3 is prime so a(3) = a(2) + 3 = 0 + 3 = 3. 4 is even so a(4) = a(3) - 2 = 3 - 2 = 1.

Bill McEachen, Table of n, a(n) for n = 1..10000

Module[{n = 1}, NestList[# + Which[EvenQ[++n], -2, PrimePowerQ[n], 3, True, 2] &, 2, 100]] (* Paolo Xausa, Dec 06 2024 *)

first(n)=my(v=vector(n)); v[1]=2; for(k=2,n, v[k]=v[k-1]+if(k%2==0, -2, isprimepower(k), 3, 2)); v \\ Charles R Greathouse IV, Dec 01 2024

a(n) = a(n-1) - 2 for n even.
a(n) = a(n-1) + 3 for n an odd prime or odd prime power.
a(n) = a(n-1) + 2 for n an odd composite not a prime power.
a(n) ~ n/log n. - Charles R Greathouse IV, Dec 01 2024

Corrected by Charles R Greathouse IV, Dec 01 2024

A375205 PrimePi(greatest prime < sqrt(Q)) - PrimePi(greatest prime factor(Q) < sqrt(Q)), with Q = A082686(n).

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 3, 2, 3, 1, 2, 3, 0, 2, 1, 3, 2, 3, 1, 0, 4, 2, 4, 4, 3, 1, 2, 0, 4, 2, 3, 4, 1, 4, 3, 2, 4, 0, 1, 3, 4, 4, 2, 0, 4, 1, 3, 2, 4, 3, 4, 0, 1, 4, 3, 2, 5, 4, 2, 1, 3, 5, 4, 5, 3

Offset: 1

Bill McEachen, Oct 15 2024

nonn

New records appear to be in consecutive numerical order, suggesting every integer should be seen in this infinite sequence. Considering a(n)=0, empirically a power fit Y=k*x^c correlates well with the "x-th" occurrence. For example, the 491st 0 value is at n=99808.

			A082686(8)=51, with square root = 7.14... so the greatest prime < 7.14 is 7, while the greatest prime factor of 51 < 7.14 is 3. The prime count from 3 to 7 is 2, so a(8)=2.
A082686(999)=2883 with square root = 53.69... so the greatest prime < 53.69 is 53, while the greatest prime factor of 2883 < 53.69 is 31. The prime count from 31 to 53 is 5, so a(999)=5.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A082686.

nmax=85;q={};m=15;Until[Length[q]==nmax,If[ !PrimeQ[m]&&EvenQ[DivisorSigma[0, m]],AppendTo[q,m]];m=m+2];Table[PrimePi[NextPrime[Sqrt[q[[n]]],-1]]-PrimePi[Select[First/@FactorInteger[q[[n]]],#James C. McMahon, Dec 06 2024 *)

A375235 Records of A112591.

Original entry on oeis.org

1, 6, 12, 28, 58, 126, 252, 506, 1012, 2042, 4082, 8190, 16366, 32742, 65518, 131056, 262114, 524280, 1048554, 2097146, 4194278, 8388594, 16777208, 33554390, 67108858, 134217716, 268435396, 536870852, 1073741814, 2147483614, 4294967284, 8589934580, 17179869158

Offset: 1

Bill McEachen, Aug 06 2024

nonn

Sequence closely parallel to A000295.

			The first term of A112591 = 1 is a record and is a(1). The next A112591 value > 1 is 6 which is a(2).

Cf. A000295, A014210 (primes where records occur), A014234, A112591.

a[n_] := BitXor @@ NextPrime[2^n, {-1, 1}]; a[1] = 1; Array[a, 33] (* Amiram Eldar, Aug 08 2024 *)

a(n)= if(n==1,1,bitxor(precprime(2^n), nextprime(2^n) ))

a(n) = previous_prime(2^n) XOR next_prime(2^n) = A112591(A014234(n)) for n > 1.

More terms from Amiram Eldar, Aug 06 2024

A375010 a(n) = prime(n-1) - floor((prime(n-2) + prime(n-1) + prime(n)) / 3).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 2, -1, 1, 1, 0, 0, 0, 2, -1, 1, 1, -1, 1, 0, 0, 2, 1, 0, 1, 0, -3, 4, 0, 2, -2, 3, -1, 0, 1, 0, 0, 2, -2, 3, 0, 1, -3, 0, 3, 1, 0, 0, 2, -2, 2, 0, 0, 2, -1, 1, 1, -2, -1, 4, 1, 0, -3, 3, -1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 2, -1, 0, 3, -2

Offset: 3

Bill McEachen, Jul 27 2024

			For n=500001 a(n) = 7368787 - floor((7368743 + 7368787 + 7368791)/3) = 14.

Bill McEachen, Table of n, a(n) for n = 3..10002

seq[len_] := (#[[2]] - Floor[Total[#]/3]) & /@ Partition[Prime[Range[len]], 3, 1]; seq[100] (* Amiram Eldar, Jul 27 2024 *)

A374965 a(n) = 2*a(n-1) + 1 for a(n-1) not prime, otherwise a(n) = prime(n) - 1; with a(1)=1.

Original entry on oeis.org

1, 3, 4, 9, 19, 12, 25, 51, 103, 28, 57, 115, 231, 463, 46, 93, 187, 375, 751, 70, 141, 283, 82, 165, 331, 100, 201, 403, 807, 1615, 3231, 6463, 12927, 25855, 51711, 103423, 156, 313, 166, 333, 667, 1335, 2671, 192, 385, 771, 1543, 222, 445, 891, 1783, 238, 477

Offset: 1

Bill McEachen, Jul 25 2024

Sequence is clearly infinite and not monotonic. Primes are sparse.
When is the next prime after n=10016 ? [Answer from N. J. A. Sloane, Aug 01 2024: The point of Bill's question is that a(10016) is the prime 838951, which is in fact the 289th prime in this sequence, as can be seen from A375028 and A373799. Thanks to the work of Lucas A. Brown (see A050412), we now know that the answer to Bill's question is that the 290th prime is the 102410-digit prime 104917*2^340181 - 1 = 5079...8783, which is a(350198). It was a very good question!]
It appears that the trajectories for different initial conditions a(1) converge to a few attractors. For all prime values and most nonprime values of a(1), the trajectories converge to the same attractor with prime 838951 at n=10016. For a(1) = 147, 295, 591, 1183, ... the trajectories converge to prime 85796863 at n=4390. For a(1) = 658, the trajectory reaches a prime with 240983 digits after 800516 steps. For a(1) = 509202, the trajectory never reaches a prime (see A050412, A052333). - Chai Wah Wu, Jul 29 2024

			a(1) = 1 is not a prime, so a(2) = 2*1+1 = 3. a(2) is a prime, so a(3) = prime(3)-1 = 4. a(4) = 2*4+1 = 9.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

The primes are listed in A375028 (see also A373798 and A373804).

Cf. A050412 and A052333.

a[n_] := a[n] = If[!PrimeQ[a[n-1]], 2*a[n-1] + 1, Prime[n]-1]; a[1] = 1; Array[a, 60] (* Amiram Eldar, Jul 25 2024 *)
nxt[{n_,a_}]:={n+1,If[!PrimeQ[a],2a+1,Prime[n+1]-1]}; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Jul 28 2024 *)

from itertools import islice
from sympy import isprime, nextprime
def A374965_gen(): # generator of terms
    a, p = 1, 3
    while True:
        yield a
        a, p = p-1 if isprime(a) else (a<<1)+1, nextprime(p)
A374965_list = list(islice(A374965_gen(),30)) # Chai Wah Wu, Jul 29 2024

cp's OEIS Frontend

User: Bill McEachen

A379757 a(n) = a(n-1) + 1 with two exceptions: if a(n-1) is prime, a(n) = a(n-2) + a(n-1), or if a(n-1) is a power, a(n) = a(n-1) / (root factor), with initial three terms are 0, 1, 2.

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A381329 Number of steps for n to reach 1 under the map x -> x/2 if x is even, x -> 2*x+1 if x is prime or a perfect power, otherwise x -> gpf(x)-1 where gpf(x) = A006530(x).

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A377371 a(n) = k*(a(n-1)+n), k=-1 for prime n, otherwise k=1 (a(1)=1).

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A378814 a(n) = round(n/(A000005(A071904(n))-2)).

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A378677 a(n)=a(n-1) + prime(n) for n prime, and a(n)=-a(n-1) otherwise, with a(0)=0, with duplicates removed afterwards.

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A378500 a(1) = 2, then a(n) = a(n-1) - 2 for n even, a(n) = a(n-1) + 3 for n an odd prime or odd prime power, and a(n) = a(n-1) + 2 otherwise.

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A375205 PrimePi(greatest prime < sqrt(Q)) - PrimePi(greatest prime factor(Q) < sqrt(Q)), with Q = A082686(n).

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A375235 Records of A112591.

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