A143207 -id:A143207 - cp's OEIS frontend

A288162 Numbers whose prime factors are 2 and 13.

26, 52, 104, 208, 338, 416, 676, 832, 1352, 1664, 2704, 3328, 4394, 5408, 6656, 8788, 10816, 13312, 17576, 21632, 26624, 35152, 43264, 53248, 57122, 70304, 86528, 106496, 114244, 140608, 173056, 212992, 228488, 281216, 346112, 425984, 456976, 562432, 692224, 742586, 851968, 913952

Offset: 1

Bernard Schott, Jun 06 2017

nonn

Numbers k such that phi(k)/k = 6/13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A033845, A033846, A033847, A033848, A033849, A033850, A033851, A086780, A107326, A143207.

[n:n in [1..100000] | Set(PrimeDivisors(n)) eq {2,13}];  // Marius A. Burtea, May 10 2019

Select[Range[920000],FactorInteger[#][[All,1]]=={2,13}&] (* Harvey P. Dale, Jun 18 2021 *)

is(n) = factor(n)[, 1]~==[2, 13] \\ Felix Fröhlich, Jun 06 2017

list(lim)=my(v=List(),t); for(n=1,logint(lim\2,13), t=13^n; while((t<<=1)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 11 2017

a(n) = 26 * A107326(n). - David A. Corneth, Jun 06 2017

Sum_{n>=1} 1/a(n) = 1/12. - Amiram Eldar, Dec 22 2020

A386252 Numbers m of the form 2^i * 3^j * 5^k such that i, j, k > 0 and m+1 and m-1 are both prime numbers.

Original entry on oeis.org

30, 60, 150, 180, 240, 270, 600, 810, 1620, 3000, 4050, 4800, 9000, 9720, 15360, 21600, 23040, 33750, 138240, 180000, 281250, 345600, 737280, 3456000, 6144000, 6561000, 10125000, 13668750, 15552000, 17496000, 20995200, 22118400, 24000000, 30000000, 54675000

Offset: 1

Ken Clements, Jul 16 2025

nonn

			a(1) = 2^1 * 3^1 * 5^1 = 30 where 29 and 31 are prime numbers.
a(2) = 2^2 * 3^1 * 5^1 = 60 where 59 and 61 are prime numbers.
a(3) = 2^1 * 3^1 * 5^2 = 150 where 149 and 151 are prime numbers.
a(4) = 2^2 * 3^2 * 5^1 = 180 where 179 and 181 are prime numbers.

Ken Clements, Table of n, a(n) for n = 1..926

Subsequence of A143207.

Cf. A027856, A384530, A080185.

seq[max_] := Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &]; seq[10^8] (* Amiram Eldar, Jul 17 2025 *)

from math import log10
from gmpy2 import is_prime
l2, l3, l5 = log10(2), log10(3), log10(5)
upto_digits = 20
sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = []
    for exponent_sum in range(3, limit+1):
        for i in range(1, exponent_sum -1):
             for j in range(1, exponent_sum - i):
                k = exponent_sum - i - j
                log_N = i*l2 + j*l3 + k*l5
                if log_N <= upto_digits:
                    N = 2**i * 3**j * 5**k
                    if is_prime(N-1) and is_prime(N+1):
                        unsorted_result.append((N, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([n for n, _ in TP_pi_3_upto_sum(sum_limit) ])

A069819 Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.

Original entry on oeis.org

30, 60, 90, 120, 150, 180, 240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810, 858, 900, 960, 1080, 1200, 1350, 1440, 1500, 1620, 1716, 1722, 1800, 1920, 2160, 2250, 2400, 2430, 2574, 2700, 2880, 3000, 3240, 3432, 3444, 3600, 3750, 3840, 4050, 4320, 4500

Offset: 1

Benoit Cloitre, Apr 28 2002

Sequence is generated by A007850(n). For example: 30, 858, 1722 (30 = 2*3*5, 858 = 2*3*11*13, 1722 = 2*3*11*13) generate numbers of the form 2^a*3^b*5^c (A143207), 2^a*3^b*7^c*41^d, 2^a*3^b*11^c*13^d, (a,b,c,d => 1), which are in the sequence.

Equivalently, numbers k such that Sum_{p|k} 1/p - Product_{p|k} 1/p, where p are the prime divisors of k, is a positive integer. All these terms have at least 3 prime factors. When k is a term and p is a prime divisor of k, then p*k is another term (see Diophante link). - Bernard Schott, Dec 19 2021

			For k = 30 = 2*3*5, 1/(Sum_{p|n} (1/p) - 1) = 1/(1/2 + 1/3 + 1/5 - 1) = 30 hence 30 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..4233 (terms below 10^10)
Diophante, A1862 - Inversons les facteurs (in French).

Cf. A007850.

A143207 is a subsequence.

Select[Range[4320], (sum = Plus @@ (1/FactorInteger[#][[;;,1]])) > 1 && IntegerQ[1/(sum - 1)] &] (* Amiram Eldar, Feb 03 2020 *)

isok(k) = my(f=factor(k), x=1/(sum(i=1, #f~, 1/f[i,1]) -1)); (x>1) && (denominator(x)==1); \\ Michel Marcus, Dec 19 2021

from sympy import factorint
from fractions import Fraction
def ok(n):
    s = sum(Fraction(1, p) for p in factorint(n))
    return s > 1 and (s - 1).numerator == 1
print([k for k in range(1, 4501) if ok(k)]) # Michael S. Branicky, Dec 19 2021

A372400 Position of 30^n among 5-smooth numbers A051037.

Original entry on oeis.org

1, 18, 83, 228, 486, 888, 1466, 2255, 3283, 4583, 6189, 8134, 10445, 13158, 16305, 19916, 24027, 28667, 33870, 39665, 46086, 53166, 60937, 69429, 78675, 88709, 99561, 111263, 123849, 137347, 151793, 167219, 183658, 201139, 219695, 239359, 260165, 282141, 305320

Offset: 0

Michael De Vlieger, Jun 03 2024

nonn

Also position of 30^(n+1) in A143207.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A002110, A051037, A143207, A202821, A372401, A372402.

Table[Sum[Floor@ Log[5, 30^n/(2^i*3^j)] + 1, {i, 0, Log[2, 30^n]}, {j, 0, Log[3, 30^n/2^i]}], {n, 0, 38}]

a(n)=my(t=30^n,u=5*t); sum(a=0,logint(t,5), u\=5; sum(b=0,logint(u,3), logint(u\3^b,2)+1)) \\ Charles R Greathouse IV, Sep 18 2024

# uses imports/function in A372401
print(list(islice(A372401gen(p=5), 40))) # Michael S. Branicky, Jun 05 2024

from sympy import integer_log
def A372400(n):
    c, x = 0, 30**n
    for i in range(integer_log(x,5)[0]+1):
        for j in range(integer_log(y:=x//5**i,3)[0]+1):
            c += (y//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = k*n^3 + (3k/2)*n^2 + O(n) where k = (log 30)^3/(6 log 2 log 3 log 5) = 5.35057081984.... - Charles R Greathouse IV, Sep 19 2024

A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

Original entry on oeis.org

36, 144, 216, 324, 576, 900, 1296, 1728, 2304, 2916, 3600, 5184, 5832, 7776, 8100, 9216, 11664, 13824, 14400, 20736, 22500, 26244, 27000, 32400, 36864, 44100, 46656, 57600, 72900, 82944, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 186624, 202500, 216000

Offset: 1

Michael De Vlieger, Jul 25 2025

nonn

Perfect powers k^m, m > 1, for k in A055932.
Union of {k^m : rad(k) | P(i), m >= 2}, rad = A007947, P = A002110. Therefore perfect powers in A033845, A143207, A147571, A147572, etc. are proper subsets.
Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading. Terms that also appear in A368682 are marked by "#":
                         Exponents
 n      a(n)             2.3.5.7.11
-----------------------------------
 1       36 =    6^2  #  2.2
 2      144 =   12^2  #  4.2
 3      216 =    6^3  #  3.3
 4      324 =   18^2     2.4
 5      576 =   24^2  #  6.2
 6      900 =   30^2  #  2.2.2
 7     1296 =    6^4  #  4.4
 8     1728 =   12^3  #  6.3
 9     2304 =   48^2  #  8.2
10     2916 =   54^2     2.6
11     3600 =   60^2  #  4.2.2
12     5184 =   72^2  #  6.4
26    44100 =  210^2  #  2.2.2.2
90  5336100 = 2310^2  #  2.2.2.2.2

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fast Mathematica algorithm for A055932.

Cf. A001597, A002110, A006530, A007947, A033845, A053669, A055932, A126706, A131605, A143207, A147571, A147572, A246547, A304250, A365308 (subset), A368682 (subset), A369374 (superset).

(* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Intersection of A131605 and A055932 = A304250 \ A246547.

A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

Original entry on oeis.org

4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241

Offset: 1

Michel Marcus, Dec 17 2020

nonn

Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - David A. Corneth, Dec 19 2020
From Bernard Schott, Jan 19 2021: (Start)
Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)

			   n  a(n) prime factor set
   1    4  [2]           A000079
   2    9  [3]           A000244
   3   25  [5]           A000351
   4   18  [2, 3]        A033845
   5   49  [7]           A000420
   6   80  [2, 5]        A033846
   7  121  [11]          A001020
   8  169  [13]          A001022
   9  112  [2, 7]        A033847
  10  135  [3, 5]        A033849
  11  289  [17]          A001026
  12  361  [19]          A001029
  13  441  [3, 7]        A033850
  14  352  [2, 11]       A033848
  15  529  [23]          A009967
  16  416  [2, 13]       A288162
  17  841  [29]          A009973
  18  360  [2, 3, 5]     A143207

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Cf. A000203 (sigma), A007947 (rad).
Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
Subsequence: A001248 (squares of primes).

u(n) = {my(fn=factor(n)[,1]); for (k = n, n^2, my(fk = factor(k)); if (fk[,1] == fn, if (factorback(fk[,1])^2 < sigma(fk), return (k));););}
lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", ");););}

a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020

A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 20, 16, 18, 40, 28, 32, 24, 80, 56, 30, 64, 36, 100, 112, 60, 42, 128, 48, 160, 196, 90, 84, 66, 256, 54, 200, 224, 120, 126, 132, 78, 512, 72, 320, 392, 150, 168, 198, 156, 88, 1024, 96, 400, 448, 180, 252, 264, 234, 176, 104, 2048, 108, 500, 784, 240, 294, 396, 312, 352, 208, 140, 4096, 144, 640, 896, 270, 336, 528, 468, 704, 416, 280, 204, 8192

Offset: 1

Frank M Jackson, Dec 30 2024

A permutation of the practical numbers.
This sequence is presented as an array of rows. The first row contains a single term of value 1. Subsequent rows are infinite sequences and are presented as a square array by listing the antidiagonals downwards that is 1: 2; 4,6; 8,12,20; etc.
The first column contains the primitive practical numbers A267124; each row lists all practical numbers (A005153) having the same primitive practicle progenitor and which is the first term in each row. See A379325 comments for further details. If T[1,m] is squarefree then the row is identical to the same squarefree row in A284457.
Every primitive practical number A267124(n) is the progenitor of a disjoint subsequence of the practical numbers. If the PP column represents the sequence of primitive practical numbers A267124, the table below give the 7 initial terms of the disjoint sequences of practical numbers A005153 generated by the initial 7 terms of the sequence of primitive practical numbers.
PP: Disjoint subsequence of A005153
--  -------------------------------
1:   1
2:   2,  4,  8, 16, 32, 64,128, . . .- A000079 with offset 1,1
6:   6, 12, 18, 24, 36, 48, 54, . . .- A033845
20: 20, 40, 80,100,160,200,320, . . .
28: 28, 56,112,196,224,392,448, . . .
30: 30, 60, 90,120,150,180,240, . . .- A143207
42: 42, 84,126,168,252,294,336
...
Row 1 is T[1,1] = 1 and only has one term in the subsequence.
Row 7 is T[1,7] = 2*3*7; T[2,7] = 2^2*3*7; T[3,7] = 2*3^2*7; T[4,7] = 2^3*3*7; T[5,7] = 2^2*3^2*7, etc.

			a(14) = 80 and it is T[3,4] = 2^4*5. Its primitive progenitor is 20 = 2^2*5 and its equivalence class are the terms of row 4.

Frank M Jackson, Mathematica program

Cf. A000079, A005117, A005153, A007947, A033845, A143207, A267124, A284457, A379325.

Mathematica
```
(* See link above *)
```

A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 48, 50, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 210, 240, 252, 270, 300, 330, 360, 390

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system (RRS) does not intersect A126706 (i.e., the sequence of numbers that are neither squarefree nor prime powers). Alternatively, numbers k whose RRS is a subset of A303554 (i.e., the union of powers of primes and squarefree numbers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q = A382248(k). Then this sequence is that of k such that k < m.
There are 72 terms in this sequence.
Sequences A048597 and A051250 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = 2^2 * 3 = 12.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 16, 32}.
  11 is in the sequence since 11 < m, m = 2^2 * 3 = 12, but 13 is not, since 13 > 12.
  9 is in the sequence since 9 < m, m = 2^2 * 5 = 20.
  25 is not a term since 25 > 12, and 27 is not a term since 27 > 20.
For omega = 2, we have the subset {6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 34, 36, 38, 40, 44, 48, 50, 54, 72, 96, 108, 144, 162}.
  38 = 2*19 is a term since 38 < 45, 45 = 3^2 * 5, but 46 = 2*23 is not, since 46 > 45.
  15 = 3*5 is a term since 15 < 20, but 21 is not, since 21 > 20 and 35 is not, since 35 > 12.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162}, since m = 5^2 * 7 = 175.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, 50}, since m = 3^2 * 7 = 63.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28}, since m = 3^2 * 5 = 45, etc.
For omega = 3, we have the subset {30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 240, 252, 270, 300, 360, 450, 480}, of which {30, 42, 66, 70, 78, 102, 114, 138, 174} are squarefree.
  Intersection with A143207 = {k : rad(k) = 30} is {30, 60, 90, .., 480} because m = 7^2 * 11 = 539.
  Intersection with 42*A108319 = {k : rad(k) = 42} is {42, 84, 126, 168}, since m = 5^2 * 11 = 275, etc.
For omega = 4, we have the subset {210, 330, 390, 420, 510, 630, 840, 1050, 1260, 1470}, of which {210, 330, 390, 510} are squarefree.
  Intersection with A147571 = {k : rad(k) = 210} is {210, 420, 630, 840, 1050, 1260, 1470} since m = 11^2 * 13 = 1573, etc.
For omega = 5, we have 2310 = 2*3*5*7*11, a term since 2310 < 13*17 = 2873; 2730 = 2*3*5*7*13 is not a term.
There are no terms larger than 2310, since the intersection with A147572 = {2310}, 2730 is not a term, and k = Product_{i=1..j} prime(i), k > prime(j+1)^2 * prime(j+2) for j > 5. Therefore the sequence is finite like A051250.

Michael De Vlieger, Table of n, a(n) for n = 1..72

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A126706, A303554, A380539, A382248, A382960.

Select[Range[30030], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A010869 Constant sequence: a(n) = 30.

Original entry on oeis.org

30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30

Offset: 0

N. J. A. Sloane

Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

Table[30, {60}] (* Vincenzo Librandi, Jan 19 2012 *)

a(n) = A007947(A143207(n)). - Reinhard Zumkeller, Sep 13 2011

G.f.: 30/(1-x). - Vincenzo Librandi, Jan 19 2012

A114991 Primes of the form 2^a * 3^b * 5^c + 1 for positive a, b, c.

Original entry on oeis.org

31, 61, 151, 181, 241, 271, 541, 601, 751, 811, 1201, 1621, 1801, 2161, 2251, 3001, 4051, 4801, 4861, 6481, 7681, 8101, 8641, 9001, 9601, 9721, 11251, 14401, 15361, 19441, 21601, 21871, 22501, 23041, 24001, 32401, 33751, 36451, 37501, 43201, 54001, 57601, 58321

Offset: 1

Jonathan Vos Post, Feb 22 2006

			a(1) = 31 = 2^1 * 3^1 * 5^1 + 1.
a(2) = 61 = 2^2 * 3^1 * 5^1 + 1.
a(3) = 151 = 2^1 * 3^1 * 5^2 + 1.
a(4) = 181 = 2^2 * 3^2 * 5^1 + 1.
Values include 30000001 = 2^7 * 3^1 * 5^7 + 1, 90000000001 = 2^9 * 3^2 * 5^9 + 1.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000040, A005109, A030430.

Cf. A143207.

cp's OEIS Frontend

A288162 Numbers whose prime factors are 2 and 13.

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A386252 Numbers m of the form 2^i * 3^j * 5^k such that i, j, k > 0 and m+1 and m-1 are both prime numbers.

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A069819 Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.

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A372400 Position of 30^n among 5-smooth numbers A051037.

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A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

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A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

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A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

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