A090252 -id:A090252 - cp's OEIS frontend

A355893 Let A090252(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A090252(n); a(1)=0 by convention.

0, 1, 10, 100, 2, 1000, 20, 10000, 100000, 1000000, 3, 10000000, 100000000, 200, 1010, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 4, 10000000000000000

Offset: 1

N. J. A. Sloane, Aug 23 2022

nonn

A090252 and A354169 are similar in many ways. This sequence and A355892 illustrate this.
This compressed format only make sense if all e_i are less than 10, that is, for n <= 24574.
It is believed that 6 does not appear in A090252, so 11 is missing from the present sequence.

			The initial terms of A090252 are:
1 -> 0
2 = 2^1 ->1
3 = 2^0 3^1 -> 10
5 = 2^0 3^0 5^1 -> 100
4 = 2^2 -> 2
7 = 2^0 3^0 5^0 7^1 -> 1000
9 = 2^0 3^2 -> 20
...
The terms, right-justified, for comparison with A355892, are:
.1 ...................................0
.2 ...................................1
.3 ..................................10
.4 .................................100
.5 ...................................2
.6 ................................1000
.7 ..................................20
.8 ...............................10000
.9 ..............................100000
10 .............................1000000
11 ...................................3
12 ............................10000000
13 ...........................100000000
14 .................................200
15 ................................1010
16 ..........................1000000000
17 .........................10000000000
18 ........................100000000000
19 .......................1000000000000
20 ......................10000000000000
21 .....................100000000000000
22 ....................1000000000000000
23 ...................................4
24 ...................10000000000000000
...

Michael De Vlieger, Table of n, a(n) for n = 1..1073

Cf. A054841, A090252, A354169.

See A354150 for indices of powers of 2 in A090252.

nn = 24, s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; nn, -1]]; f[n_] := If[n == 1, 0, Function[g, FromDigits@ Reverse@ ReplacePart[Table[0, {PrimePi[g[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ n]; Array[f[s[[#]]] &, nn] (* Michael De Vlieger, Aug 24 2022 *)

a(n) = A054841(A090252(n)). - Stefano Spezia, Aug 24 2022

A354151 Lengths of successive runs of primes in A090252.

Original entry on oeis.org

3, 1, 3, 2, 7, 5, 1, 14, 11, 2, 1, 29, 22, 5, 2, 58, 1, 45, 1, 1, 10, 1, 5, 117, 3, 90, 2, 2, 21, 2, 11, 1, 1, 39, 195, 7, 181, 5, 5, 1, 42, 7, 23, 2, 3, 79, 391, 14, 1, 362, 1, 11, 1, 1, 11, 1, 1, 3, 1, 85, 15, 46, 5, 7, 158, 782, 1, 29, 1, 3, 1, 725, 1, 1, 2, 22, 2, 2, 23, 2, 3, 7, 2, 170, 31, 93, 1, 1, 11, 15, 317, 1, 1089, 475, 2, 58, 1, 1, 3, 7, 2, 1450

Offset: 1

N. J. A. Sloane, May 27 2022

nonn

			A090252 begins 1, 2, 3, 5, 4, 7, 9, 11, 13, 17, 8, 19, 23, 25, 21, 29, 31, ... which, writing N for a nonprime and P for a prime, is NPPPNPNPPPNPPNPP...  The runs of primes have lengths 3, 1, 3, 2, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..2148 [Based on Russ Cox's 5 million term data file for A090252]

Cf. A090252, A354148, A249033.

from math import gcd, prod
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink, run = [1], {1}, 2, 0
    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): run += 1
        elif run > 0: yield run; run = 0
        while mink in aset: mink += 1
print(list(islice(agen(), 45))) # Michael S. Branicky, May 28 2022

A354162 Products of exactly two distinct odd primes in A090252, in order of appearance.

Original entry on oeis.org

21, 55, 85, 57, 161, 319, 217, 481, 205, 731, 517, 159, 1121, 1403, 871, 355, 1241, 869, 2407, 1691, 413, 3007, 2323, 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, 24823

Offset: 1

N. J. A. Sloane, May 30 2022

nonn

Odd terms in A354160. - Chai Wah Wu, May 31 2022

Cf. A090252, A354159, A354160, A354161, A354163.

from itertools import count, islice
from collections import deque
from math import gcd, lcm
from sympy import factorint
def A354162_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 m % 2 and 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
A354162_list = list(islice(A354162_gen(),25)) # Chai Wah Wu, May 31 2022

A354163 Index of A354162(n) in A090252.

Original entry on oeis.org

15, 29, 59, 63, 65, 121, 131, 193, 239, 241, 243, 255, 257, 265, 387, 479, 483, 487, 489, 515, 527, 529, 531, 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, 3073, 3105, 3107, 3111, 3113, 3839

Offset: 1

N. J. A. Sloane, May 30 2022

nonn

Cf. A090252, A354159, A354160, A354161, A354162.

from itertools import count, islice
from collections import deque
from math import gcd, lcm
from sympy import factorint
def A354163_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 m % 2 and 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
A354163_list = list(islice(A354163_gen(),25)) # Chai Wah Wu, May 31 2022

A355012 Prime powers (p^k, k >= 2) in A090252 in order of appearance.

Original entry on oeis.org

4, 9, 8, 25, 16, 27, 49, 121, 32, 169, 125, 289, 81, 361, 529, 64, 841, 343, 961, 128, 1369, 1681, 1849, 2209, 243, 2809, 3721, 4489, 5041, 5329, 1331, 6241, 6889, 3481, 256, 10609, 11449, 11881, 625, 12769, 7921, 9409, 10201, 2197, 26569, 32041, 32761, 16129

Offset: 1

Michael De Vlieger, Jun 15 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1009
Michael De Vlieger, Annotated log-log scatterplot of A090252(n), n = 1..2^12, labeling primes in red, squarefree semiprimes in green, highlighting composite prime powers in gold and labeling them in boldface.

Cf. A090252, A246547, A354165, A355013, A355014.

Select[Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 4000, -1]], CompositeQ[#] && PrimePowerQ[#] &]

A355013 Index of A355012(n) in A090252.

Original entry on oeis.org

5, 7, 11, 14, 23, 31, 32, 60, 95, 96, 119, 120, 127, 128, 132, 191, 244, 263, 264, 383, 388, 480, 484, 488, 511, 512, 571, 776, 960, 968, 975, 976, 980, 1056, 1535, 1536, 1552, 1556, 1919, 1920, 2112, 2128, 2287, 3103, 3104, 3844, 3848, 3872, 3875, 3904, 3920

Offset: 1

Michael De Vlieger, Jun 15 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1009

Cf. A090252, A246547, A354164, A355012, A355014.

Position[Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 4000, -1]], _?(CompositeQ[#] && PrimePowerQ[#] &)][[All, 1]]

A354149 Odd numbers in A090252 in order of appearance.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 21, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 55, 79, 27, 49, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 85, 121, 223, 227, 57, 229, 161, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

N. J. A. Sloane, May 22 2022

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..9988

Cf. A090252, A354146, A354148.

from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink = [1], {1}, 2; yield 1
    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 == 1: yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 77))) # Michael S. Branicky, May 23 2022

A354150 Index of 2^n in A090252, or -1 if 2^n does not appear.

Original entry on oeis.org

1, 2, 5, 11, 23, 95, 191, 383, 1535, 6143, 24575, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367

Offset: 0

N. J. A. Sloane, May 26 2022

It is conjectured (see A090252) that the indices of even terms in A090252 are {3*2^i-1, i >= 0} (A083329), so the positive terms of the present sequence should be a subsequence of A083329.

For n = 1,...,18, the terms are 3*2^k - 1 for k = 0,1,2,3,5,6,7,9,11,13,14,16,18,20,22,24,26,28. Will the formula a(n) = 3*2^(2*n-8)-1 hold for all n >= 11? This appears to be yet another example of the influence of A029744 on A090252 and A354691.- N. J. A. Sloane, Aug 24 2022

Cf. A090252, A083329, A354148, A354149, A354255 (even terms in A090252).

A354154 a(1) = 0; for n>1, a(n) = prime(n-1) - A090252(n).

Original entry on oeis.org

0, 0, 0, 0, 3, 4, 4, 6, 6, 6, 21, 12, 14, 16, 22, 18, 22, 22, 20, 24, 24, 20, 63, 24, 28, 30, 30, 30, 52, 30, 86, 78, 48, 48, 42, 48, 48, 50, 54, 54, 46, 48, 44, 52, 44, 46, 173, 54, 60, 60, 56, 54, 58, 50, 58, 60, 64, 58, 186, 156, 58, 56, 236, 78, 150, 80, 78, 90, 86, 90, 86, 84, 88, 90, 92, 96, 90, 82, 86, 88, 92, 88, 84, 84, 84, 86, 84, 82, 84

Offset: 1

N. J. A. Sloane, May 28 2022

nonn

Theorem: a(n) >= 0 for all n.

			A090252 begins
  1, 2, 3, 5, 4,  7,  9, 11, 13, ...
and we subtract these numbers from
  1, 2, 3, 5, 7, 11, 13, 17, 19, ...
to get
  0, 0, 0, 0, 3,  4,  4,  6,  6, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A090252.

from math import gcd, prod
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    alst, aset, mink, p = [1], {1}, 2, 1
    yield 0
    for n in count(2):
        k, s, p = mink, n - n//2, nextprime(p)
        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); yield p - k
        while mink in aset: mink += 1
print(list(islice(agen(), 89))) # Michael S. Branicky, May 28 2022

A354764 Squarefree kernel of A090252(n) (cf. A007947).

Original entry on oeis.org

1, 2, 3, 5, 2, 7, 3, 11, 13, 17, 2, 19, 23, 5, 21, 29, 31, 37, 41, 43, 47, 53, 2, 59, 61, 67, 71, 73, 55, 79, 3, 7, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 26, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 85, 11, 223, 227, 57, 229, 161, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281

Offset: 1

N. J. A. Sloane, Jun 18 2022

nonn

Cf. A007947, A090252.

from itertools import count, islice
from math import prod, gcd, lcm
from collections import deque
from sympy import primefactors
def A354764_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(m))
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354764_list = list(islice(A354764_gen(),20)) # Chai Wah Wu, Jun 18 2022

cp's OEIS Frontend

A355893 Let A090252(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A090252(n); a(1)=0 by convention.

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Formula

A354151 Lengths of successive runs of primes in A090252.

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A354162 Products of exactly two distinct odd primes in A090252, in order of appearance.

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A354163 Index of A354162(n) in A090252.

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A355012 Prime powers (p^k, k >= 2) in A090252 in order of appearance.

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Mathematica

A355013 Index of A355012(n) in A090252.

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A354149 Odd numbers in A090252 in order of appearance.

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A354150 Index of 2^n in A090252, or -1 if 2^n does not appear.

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A354154 a(1) = 0; for n>1, a(n) = prime(n-1) - A090252(n).

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A354764 Squarefree kernel of A090252(n) (cf. A007947).

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