A090252 -id:A090252 - cp's OEIS frontend

A354148 Index of prime(n) in A090252.

2, 3, 4, 6, 8, 9, 10, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 100

Offset: 1

N. J. A. Sloane, May 22 2022

nonn

All the primes appear in A090252 and in their correct order.

			A090252 begins 1, 2, 3, 5, 4, 7, 9, ..., so the indices of the primes are 2, 3, 4, 6, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..9800

Cf. A090252, A354146.

from math import gcd, prod
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink = [1], {1}, 2
    for n in count(2):
        k, s = mink, n - n//2
        prodall = prod(alst[n-n//2-1:n-1])
        while k in aset or gcd(prodall, k) != 1: k += 1
        alst.append(k); aset.add(k)
        if isprime(k): yield n
        while mink in aset: mink += 1
print(list(islice(agen(), 83))) # Michael S. Branicky, May 23 2022

A354159 Terms 2*p (p prime) in A090252, divided by 2, in order of appearance.

Original entry on oeis.org

2, 13, 103, 239, 499, 1567, 3257, 6971, 14447, 30259, 63317, 130699

Offset: 1

N. J. A. Sloane, May 30 2022

By definition, 2*a(n) is a term in A354255.

Conjecture: the indices of the terms 2*p in A090252 are terms in A083329.

			The indices in A090252 of the initial terms 2*p, the terms 2*p themselves, and the primes p are [5, 4, 2], [47, 26, 13], [767, 206, 103], [3071, 478, 239].

Cf. A083329, A090252, A354150, A354151, A354255.

a(6)-a(8) from Michael S. Branicky, Jun 04 2022 using A354255
a(9)-a(11) from Hugo van der Sanden, Jun 14 2022
a(12) from Jinyuan Wang, Jul 15 2022

A355057 a(n) = product of prohibited prime factors of A090252(n).

Original entry on oeis.org

1, 1, 2, 6, 15, 30, 70, 210, 462, 6006, 51051, 102102, 277134, 6374082, 10623470, 223092870, 588153930, 18232771830, 51893273670, 2127624220470, 5381637734130, 252936973504110, 6702829797858915, 13405659595717830, 41628100849860630, 2539314151841498430, 7397132529277408470

Offset: 1

Michael De Vlieger, Jun 16 2022

nonn

Let s(n) = A090252(n) and let K(n) = A007947(n) = squarefree kernel of n.
Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n.
Therefore a(n) is the product of primes p that cannot divide s(n).
a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. - N. J. A. Sloane, Jun 17 2022

			a(1) = 1;
a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;
a(3) = a(2) * K(s(2)) / K(s((3-1)/2)) = 1 * 2 / 1 = 2;
a(4) = a(3) * K(s(3)) = 2 * 3 = 6;
a(5) = a(4) * K(s(4)) / K(s((5-1)/2)) = 6 * 5 / 2 = 15;
a(6) = a(5) * K(s(5)) = 15 * 2 = 30;
a(7) = a(6) * K(s(6)) / K(s((7-1)/2)) = 30 * 7 / 3 = 70;
etc.

N. J. A. Sloane, Table of n, a(n) for n = 1..500
Michael De Vlieger, Annotated chart of prime factors of a(n), n = 1..64, plotting prime p | a(n) at (n, pi(p)) in black, and prime q | s(n) at (n, pi(q)) in red.
Michael De Vlieger, Plot of prime p | a(n), n = 1..2^12, plotting p | a(n) at (n, pi(p)) in black.

See A354758 for another version.
A354765 is a binary encoding.
Cf. A007947, A090252, A354764.

# To get first M terms, from N. J. A. Sloane, Jun 18 2022
with(numtheory);
M:=20; ans:=[1,1,2];
for i from 4 to M do
S:={}; j1:=floor((i+1)/2); j2:=i-1;
  for j from j1 to j2 do S:={op(S), op(factorset(b252[j]))} od:
t2 := product(S[k], k = 1..nops(S));
ans:=[op(ans),t2];
od:
ans;

Block[{s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 120, -1]], m = 1}, Reap[Do[m *= Times @@ FactorInteger[s[[If[# == 0, 1, #] &[i - 1]]]][[All, 1]]; If[IntegerQ[#] && # > 0, m /= Times @@ FactorInteger[s[[#]]][[All, 1]]] &[(i - 1)/2]; Sow[m], {i, Length[s]}] ][[-1, -1]] ]

from math import prod, lcm, gcd
from itertools import count, islice
from collections import deque
from sympy import primefactors
def A355057_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1:
                yield prod(primefactors(b))
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A355057_list = list(islice(A355057_gen(),20)) # Chai Wah Wu, Jun 18 2022

a(n) = a(n-1) * K(s(n-1)) / K(s((n-1)/2)), where the last operation is only carried out iff (n-1)/2 is an integer.

A354164 Indices of nonprimes in A090252.

Original entry on oeis.org

1, 5, 7, 11, 14, 15, 23, 29, 31, 32, 47, 59, 60, 63, 65, 95, 96, 119, 120, 121, 127, 128, 131, 132, 191, 193, 239, 241, 243, 244, 255, 257, 263, 264, 265, 383, 387, 388, 479, 480, 483, 484, 487, 488, 489, 511, 512, 515, 527, 529, 531, 571, 767, 775, 776, 777, 959, 960, 961, 967, 968, 969, 975, 976, 977, 979, 980, 1023, 1031, 1055, 1056, 1059, 1063

Offset: 1

N. J. A. Sloane, May 31 2022

nonn

Russ Cox, Table of n, a(n) for n = 1..3527 (first 374 terms from N. J. A. Sloane)
Russ Cox, Table of nonprime entries in A090252: n, A090252(n), # of prime factors, n = 1..3527.
Hugo van der Sanden, Table of nonprime entries in the first 10^9 terms of A090252. [See file header for description]

Cf. A090252, A354148, A354165, A354166.

A354758 a(n) = Product_{k = ceiling(n/2)..n-1} A090252(k).

Original entry on oeis.org

1, 1, 2, 6, 15, 60, 140, 1260, 2772, 36036, 153153, 1225224, 3325608, 76488984, 212469400, 4461857400, 11763078600, 364655436600, 1037865473400, 42552484409400, 107632754682600, 5058739470082200, 33514148989294575, 536226383828713200, 1665124033994425200

Offset: 1

Rémy Sigrist, Jun 06 2022

nonn

The prime divisors of a(n) are forbidden in A090252(n).

			a(7) = A090252(4) * A090252(5) * A090252(6) = 5 * 4 * 7 = 140.

Michael De Vlieger, Table of n, a(n) for n = 1..587
Rémy Sigrist, Representation of the prime divisors of a(n) (blue pixels) and A090252(n) (red pixels)
Rémy Sigrist, PARI program

A355057 is another version.

Cf. A090252, A354757.

With[{s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 25, -1]]}, Table[Product[s[[k]], {k, Ceiling[n/2], n - 1}], {n, Length[s]}]] (* Michael De Vlieger, Aug 28 2022 *)

PARI
```
See Links section.
```

gcd(a(n), A090252(n)) = 1.

A354160 Products of exactly two distinct primes in A090252, in order of appearance.

Original entry on oeis.org

21, 55, 26, 85, 57, 161, 319, 217, 481, 205, 731, 517, 159, 1121, 1403, 871, 355, 1241, 869, 2407, 1691, 413, 3007, 2323, 206, 1391, 4033, 565, 5207, 2227, 5891, 6533, 4321, 453, 1007, 623, 4867, 2231, 6161, 2119, 11189, 6401, 12709, 7421, 2159, 9563, 8213, 1507, 15247, 9259, 4031, 12367, 597, 2869, 11183, 1561, 13393, 7099, 3611, 14213, 478, 24823

Offset: 1

N. J. A. Sloane, May 30 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..2517
Michael De Vlieger, Annotated log-log scatterplot of A090252(n), n = 1..2^12, labeling primes in red, highlighting composite prime powers in gold, squarefree semiprimes in green and labeling them in boldface.

Cf. A090252, A354159, A354161, A354162, A354163.

Select[Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 2000, -1]], PrimeNu[#] == PrimeOmega[#] == 2 &] (* Michael De Vlieger, Jun 16 2022 *)

from itertools import count, islice
from collections import deque
from math import gcd, lcm
from sympy import factorint
def A354160_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1:
                if len(fm := factorint(m)) == sum(fm.values()) == 2:
                    yield m
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354160_list = list(islice(A354160_gen(),25)) # Chai Wah Wu, May 31 2022

A354161 Index of A354160(n) in A090252.

Original entry on oeis.org

15, 29, 47, 59, 63, 65, 121, 131, 193, 239, 241, 243, 255, 257, 265, 387, 479, 483, 487, 489, 515, 527, 529, 531, 767, 775, 777, 959, 961, 967, 969, 977, 979, 1023, 1031, 1055, 1059, 1063, 1143, 1551, 1553, 1555, 1921, 1923, 1935, 1937, 1939, 1951, 1953, 1955, 1959, 1961, 2047, 2063, 2064, 2111, 2113, 2119, 2127, 2288, 3071, 3073, 3105, 3107

Offset: 1

N. J. A. Sloane, May 30 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..2517

Cf. A090252, A354159, A354160, A354162, A354163.

from itertools import count, islice
from collections import deque
from math import gcd, lcm
from sympy import factorint
def A354161_gen(): # generator of terms
    aset, aqueue, c, b, f, i = {1}, deque([1]), 2, 1, True, 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1:
                i += 1
                if len(fm := factorint(m)) == sum(fm.values()) == 2:
                    yield i
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354161_list = list(islice(A354161_gen(),25)) # Chai Wah Wu, May 31 2022

A354165 Nonprimes in A090252 in order of appearance.

Original entry on oeis.org

1, 4, 9, 8, 25, 21, 16, 55, 27, 49, 26, 85, 121, 57, 161, 32, 169, 125, 289, 319, 81, 361, 217, 529, 64, 481, 205, 731, 517, 841, 159, 1121, 343, 961, 1403, 128, 871, 1369, 355, 1681, 1241, 1849, 869, 2209, 2407, 243, 2809, 1691, 413, 3007, 2323, 3721, 206, 1391, 4489, 4033, 565, 5041, 5207, 2227, 5329, 5891, 1331, 6241, 6533, 4321, 6889, 453

Offset: 1

N. J. A. Sloane, May 31 2022

nonn

Russ Cox, Table of n, a(n) for n = 1..3527 (first 374 terms from N. J. A. Sloane)
Russ Cox, Table of nonprime entries in A090252: n, A090252(n), # of prime factors, n = 1..3527.

Cf. A090252, A354164, A354166.

A354166 Number of nonprimes among first n terms of A090252.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17

Offset: 1

N. J. A. Sloane, May 31 2022

nonn

It appears that this sequence controls the growth of A090252. That is, almost all the terms of A090252 lie on the line A090252(n) = k-th prime, where k = n - a(n).

N. J. A. Sloane, Table of n, a(n) for n = 1..34886

Cf. A090252, A354164, A354165.

A354255 Even numbers in A090252 in order of appearance.

Original entry on oeis.org

2, 4, 8, 16, 26, 32, 64, 128, 206, 256, 478, 512, 998, 1024, 2048, 3134, 4096, 6514, 8192, 13942, 16384, 28894, 32768, 60518, 65536, 126634, 131072, 261398, 262144

Offset: 1

Michael S. Branicky, May 21 2022

The n-th even term in A090252 appears at index k <= A083329(n).
Conjecture: The indices of even numbers in A090252 are precisely the numbers {A083329(n), n >= 1}. See A090252 for discussion. - N. J. A. Sloane, May 22 2022
Taking logs to base 2 of these terms produces 1., 2., 3., 4., 4.700439718, 5., 6., 7., 7.686500527, 8., 8.900866807, 9., 9.962896004, 10., 11., 11.61378946, 12., 12.66932800, 13., 13.76714991, 14. - N. J. A. Sloane, Jun 01 2022

Cf. A083329, A090252, A247665, A248379, A353730, A354146, A354159.

from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink = [1], {1}, 2
    for n in count(2):
        k, s = mink, n - n//2
        prodall = prod(alst[n-n//2-1:n-1])
        while k in aset or gcd(prodall, k) != 1: k += 1
        alst.append(k); aset.add(k)
        if k%2 == 0: yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 9))) # Michael S. Branicky, May 23 2022

a(14) from Michael S. Branicky, May 26 2022
a(15)-a(21) from Michael S. Branicky, Jun 01 2022 using gzipped b-file in A090252
a(22)-a(26) from Hugo van der Sanden, Jun 14 2022
a(27)-a(29) from Jinyuan Wang, Jul 15 2022

cp's OEIS Frontend

A354148 Index of prime(n) in A090252.

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A354159 Terms 2*p (p prime) in A090252, divided by 2, in order of appearance.

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A355057 a(n) = product of prohibited prime factors of A090252(n).

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A354164 Indices of nonprimes in A090252.

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A354758 a(n) = Product_{k = ceiling(n/2)..n-1} A090252(k).

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A354160 Products of exactly two distinct primes in A090252, in order of appearance.

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A354161 Index of A354160(n) in A090252.

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A354165 Nonprimes in A090252 in order of appearance.

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A354166 Number of nonprimes among first n terms of A090252.

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