A046022 -id:A046022 - cp's OEIS frontend

A129906 n!/(n^2) when an integer.

1, 20, 630, 4480, 36288, 3326400, 444787200, 5811886080, 81729648000, 19760412672000, 6082255020441600, 115852476579840000, 2322315553259520000, 1077167364120207360000, 24817936069329577574400, 596585001666576384000000, 14936720782466875392000000

Offset: 1

Jonathan Vos Post, Jun 09 2007

Also see comments in A046022 = primes together with 1 and 4.

Alois P. Heinz, Table of n, a(n) for n = 1..150

Cf. A000142, A000290, A046022, A056653.

f[n_]:=If[PrimeQ[n]||n == 4,0,n!/n^2];Select[Table[f[n],{n,1,40}],#>0&]  (* Geoffrey Critzer, Oct 26 2012 *)
Select[#!/#^2&/@Range[30],IntegerQ] (* Harvey P. Dale, Jul 17 2024 *)

a(n) = A000142(m)/A000290(m) where m = A056653(n). - Jason Yuen, Sep 20 2024

More terms from Alois P. Heinz, Oct 26 2012

A141295 Largest m<=n such that all k with 1<=k<=m are divisors of n or coprime to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 5, 5, 3, 11, 7, 13, 3, 5, 5, 17, 3, 19, 5, 5, 3, 23, 8, 9, 3, 5, 5, 29, 3, 31, 5, 5, 3, 9, 7, 37, 3, 5, 5, 41, 3, 43, 5, 5, 3, 47, 8, 13, 3, 5, 5, 53, 3, 9, 5, 5, 3, 59, 7, 61, 3, 5, 5, 9, 3, 67, 5, 5, 3, 71, 9, 73, 3, 5, 5, 13, 3, 79, 5, 5, 3, 83, 7, 9, 3, 5, 5, 89, 3, 13, 5, 5, 3

Offset: 1

Reinhard Zumkeller, Jun 23 2008

nonn

n mod a(n) = 0 or GCD(n,a(n)) = 1;
a(n) = n iff n=1 or n=4 or n is prime; a(A046022(n))=A046022(n);
a(p^2) = 2*p - 1 for odd primes p.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027750, A038566, A076274.

A268281 Numbers n such that n-tau(n), phi(n) and n form a Heronian triangle, where tau=A000005 is the number of divisors and phi=A000010 the totient.

Original entry on oeis.org

5, 34, 53, 90, 120, 440, 780, 1954, 120994, 140453, 28813276834

Offset: 1

Frank M Jackson, Jan 29 2016

For all n, n > tau(n) and n > phi(n) and if n is prime then n-tau(n) = n-2 and phi(n) = n-1. So n = 5 gives the triangle {3, 4, 5} which is a primitive Pythagorean triangle and this is the only one. Other Pythagorean triangles are {30, 16, 34} and {756, 192, 780}, the remainder are only Heronian.
It is not known if this sequence is infinite. Prime numbers in the sequence are 5, 53 and 140453 and generate triangles {3, 4, 5}, {51, 52, 53} and {140451, 140452, 140453}.
If n = 2p where p is prime then n-tau(n) = n-4 and phi(n) = n/2-1. So n = 34 gives the triangle {16, 30, 34}. Similar numbers in this sequence are a(8), a(9) and a(11). See A272365 for generating Heronian triangles with sides n, n-4, n/2-1.
a(12) > 2*10^12. - Giovanni Resta, Apr 14 2016
Next prime value of a(n) after 140453 is > 2*10^5719. See A003500 for generating Heronian triangles with consecutive sides. - Frank M Jackson, Apr 19 2016
A003500(n)+1 is a member of this sequence iff it is prime. Also A272365(n) is a member of this sequence iff A272365(n)/2 is prime. - Frank M Jackson, Apr 29 2016

			a(2) = 34 because the triangle so formed has sides 30, 16, 34. It is Heronian with integer area 240 and is also Pythagorean. It is the second Heronian triangle.
The triangle corresponding to a(11) has sides n = 28813276834, n-tau(n) = 28813276830, phi(n) = 14406638416, and area 200960614753814018640.

Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A000005, A000010, A003500, A046022, A049820, A084820, A272365.

triples[n_] := ({a, b, c}={n-DivisorSigma[0, n], EulerPhi[n], n}; s=(a+b+c)/2; If[a+b>c&&IntegerQ[Sqrt[s(s-a)(s-b)(s-c)]], {a, b, c}, {}]); lst={}; Do[If[triples[n]!={}, AppendTo[lst, Last[triples[n]]]], {n, 1, 200000}]; lst

a(11) from Giovanni Resta, Apr 14 2016

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Rémy Sigrist, Jan 27 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298882.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * s_p(k)) = p * r_p(k),
- for example: a(11 * A051038(k)) = 11 * A008364(k).

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
Rémy Sigrist, PARI program for A298268
Index entries for sequences that are permutations of the natural numbers

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

PARI
```
See Links section.
```

a(1) = 1.
a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2^k) = 2 * k for any k > 0,
- a(2 * p) = p^2 for any prime p,
- a(3 * p) = p * A151800(p) for any odd prime p.

A335402 Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 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, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Gus Wiseman, Jun 06 2020

An integer partition is normal if it covers an initial interval of positive integers.
Conjecture: The sequence consists of 0, 1, 4, and all primes except 3.
From Chai Wah Wu, Jun 22 2020: (Start)
The above conjecture is true.
Proof: The cases of 0, 1, 4 can be checked by inspection. Next we show that if n is prime and not equal to 3, then n is a term. Let n be prime and consider a palindromic normal partition of n covering the integers 1,...,k with k > 1. Then the multiplicity of 1 and k are the same and the multiplicities of 2 and k-1 are the same, etc.
If k is even, then n is of the form (k+1)r. Since n is prime, this implies that n = k+1. Since n >= k(k+1)/2. this means that k = 2 and n = 3.
If k is odd, then n is of the form (k+1)r + w(k+1)/2. Let m = (k+1)/2, then n = m(2r+w). Since n is prime and r,w > 0, this means that m = 1, k = 1, a contradiction.
Next we show that if n is composite and not equal to 4, then n is not a term.
Suppose n = pq for 1 < p <= q. If p is odd, let k = p-1 > 1.
Consider the partition covering 1,..,k where the multiplicity is 1 except for 1 and k where the multiplicity is q-k/2 + 1 > 0. This is a normal palindromic partition summing up to pq = n.
If p is even, without loss of generality we can choose p = 2. Since n != 4, q >= 3. In this case, choosing k = 3 with 1 and 3 having multiplicity 1 and 2 having multiplicity q-2 > 0 results in a normal palindromic partition of 2q = n. QED
It is clear that if n is not a term, then any multiple of n is also not a term.
(End)

			There are 4 normal integer partitions of 10 whose sequence of multiplicities is a palindrome, namely (4321), (33211), (32221), (1111111111), so 10 does not belong to the sequence. The normal integer partitions of 7 are (3211), (2221), (22111), (211111), (1111111), none of which has palindromic multiplicities except the last, so 7 belongs to the sequence.

Positions of 1's in A317086.
Palindromic-multiplicity partitions are counted by A317085.
Normal integer partitions are counted by A000009.
Heinz numbers of normal palindromic-multiplicity partitions are A317087.
Cf. A000041, A000837, A025065, A046022, A124010, A242414.

Select[Range[0,30],Length[Select[IntegerPartitions[#],And[Or[#=={},Union[#]==Range[First[#]]],Length/@Split[#]==Reverse[Length/@Split[#]]]&]]==1&]

# from definition
from sympy.utilities.iterables import partitions
from sympy import integer_nthroot
A335402_list = []
for m in range(0,101):
    for d in partitions(m,k=integer_nthroot(2*m,2)[0]):
        l = len(d)
        if l > 0 and not(l == 1 and 1 in d):
            k = max(d)
            if l == k:
                for i in range(k//2):
                    if d[i+1] != d[k-i]:
                        break
                else:
                    break
    else:
        A335402_list.append(m) # Chai Wah Wu, Jun 22 2020

# from formula
from sympy import prime
A335402_list = [0,1,2,4] + [prime(i) for i in range(3,100)] # Chai Wah Wu, Jun 22 2020

n is a term if and only if n = 0, 1, 2, 4 or a prime > 3. - Chai Wah Wu, Jun 22 2020

a(22)-a(59) from Chai Wah Wu, Jun 22 2020

A338554 Number of non-constant integer partitions of n whose parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 0, 9, 0, 13, 6, 18, 0, 33, 0, 40, 14, 54, 0, 87, 5, 99, 27, 133, 0, 211, 0, 226, 55, 295, 18, 443, 0, 488, 100, 637, 0, 912, 0, 1000, 198, 1253, 0, 1775, 13, 1988, 296, 2434, 0, 3358, 59, 3728, 489, 4563, 0, 6241, 0, 6840, 814

Offset: 0

Gus Wiseman, Nov 07 2020

nonn

			The a(6) = 2 through a(15) = 6 partitions (empty columns indicated by dots, A = 10, B = 11, C = 12):
  (42)  .  (62)   (63)  (64)    .  (84)     .  (86)      (96)
           (422)        (82)       (93)        (A4)      (A5)
                        (442)      (A2)        (C2)      (C3)
                        (622)      (633)       (644)     (663)
                        (4222)     (642)       (662)     (933)
                                   (822)       (842)     (6333)
                                   (4422)      (A22)
                                   (6222)      (4442)
                                   (42222)     (6422)
                                               (8222)
                                               (44222)
                                               (62222)
                                               (422222)

A046022 lists positions of zeros.
A082023(n) - A059841(n) is the 2-part version, n > 2.
A303280(n) - 1 is the strict case (n > 1).
A338552 lists the Heinz numbers of these partitions.
A338553 counts the complement, with Heinz numbers A338555.
A000005 counts constant partitions, with Heinz numbers A000961.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A018783 counts non-relatively prime partitions (ordered: A178472), with Heinz numbers A318978.
A282750 counts relatively prime partitions by sum and length.
Cf. A000010, A008284, A051424, A082024, A289508, A302698, A304712.

Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&GCD@@#>1&]],{n,0,30}]

For n > 0, a(n) = A018783(n) - A000005(n) + 1.

A382492 a(n) is the least number that has exactly n 3-smooth divisors.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 72, 4096, 192, 144, 216, 65536, 288, 262144, 432, 576, 3072, 4194304, 864, 1296, 12288, 2304, 1728, 268435456, 2592, 1073741824, 3456, 9216, 196608, 5184, 6912, 68719476736, 786432, 36864, 10368, 1099511627776, 15552, 4398046511104

Offset: 1

Amiram Eldar, Mar 29 2025

The record values occur at A046022.

All the terms are in A003586 and A025487.

Amiram Eldar, Table of n, a(n) for n = 1..3322

Cf. A003586, A005179, A025487, A037143, A046022, A072078, A382493.

a[n_] := Min[Table[2^(d-1)*3^(n/d-1), {d, Divisors[n]}]]; Array[a, 60]

a(n) = vecmin(apply(d -> 2^(d-1)*3^(n/d-1), divisors(n)));

a(n) = Min_{d|n} (2^(d-1)*3^(n/d-1)).
a(n) = 2^A382493(n) * 3^(n/(A382493(n)+1)-1).
a(p) = 2^(p-1) for prime p.
a(n) = A005179(n) if n is in A037143.

A029716 Partial sums of Kempner numbers A002034.

Original entry on oeis.org

1, 3, 6, 10, 15, 18, 25, 29, 35, 40, 51, 55, 68, 75, 80, 86, 103, 109, 128, 133, 140, 151, 174, 178, 188, 201, 210, 217, 246, 251, 282, 290, 301, 318, 325, 331, 368, 387, 400, 405, 446, 453, 496, 507, 513, 536, 583, 589, 603, 613, 630, 643, 696, 705, 716, 723, 742

Offset: 1

N. J. A. Sloane

nonn

Comment from Jonathan Vos Post, May 18 2010 (Start):
The subsequence of primes begins: 3, 29, 103, 109, 151, 251, 331, 613, 643, 1033, 1151, 1277, 1307, 1399.
The subsequence of perfect powers begins: 1, 25, 128, 400, 1296. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Li Hailong and Zhao Xiaopeng, On the Smarandache function and the k-th roots of a positive integer, 2004, p. 119.

Cf. A000142, A002034, A007672 - A002034(n)!/n, A064759, A094371, A094372, A046022, A094404. Cf. also A006530, A057109, A001113, A122378, A122379, A122416, A122417. [Jonathan Vos Post, May 18 2010]

Accumulate[Table[found = 0; m = 1; While[found == 0, If[IntegerQ[m!/n], found = 1, m++]]; m, {n, 1, 100}]] (* Vaclav Kotesovec, Jul 29 2021 *)

a(n) ~ Pi^2 * n^2 / (12 * log(n)) [Li Hailong and Zhao Xiaopeng, 2004]. - Vaclav Kotesovec, Jul 29 2021

More terms from Vaclav Kotesovec, Jul 29 2021

A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 16, 6, 32, 11, 64, 13, 128, 9, 256, 17, 512, 19, 1024, 12, 2048, 23, 4096, 10, 8192, 18, 16384, 29, 32768, 31, 65536, 24, 131072, 15, 262144, 37, 524288, 27, 1048576, 41, 2097152, 43, 4194304, 36, 8388608, 47, 16777216, 14, 33554432, 48

Offset: 1

Rémy Sigrist, Jan 28 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298268.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * r_p(k)) = p * s_p(k),
- for example: a(11 * A008364(k)) = 11 * A051038(k).

			The first terms, alongside A020639(n), are:
  n     a(n)    lpf(n)
  --    ----    ------
   1       1      1
   2       2      2
   3       3      3
   4       4      2
   5       5      5
   6       8      2
   7       7      7
   8      16      2
   9       6      3
  10      32      2
  11      11     11
  12      64      2
  13      13     13
  14     128      2
  15       9      3
  16     256      2
  17      17     17
  18     512      2
  19      19     19
  20    1024      2

Index entries for sequences that are permutations of the natural numbers

Cf. A008364, A020639, A046022, A051038, A055396, A078898, A083140, A125624, A298268 (inverse).

a(1) = 1.
a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0.
a(n) = A125624(A055396(n), A078898(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2 * k) = 2^k for any k > 0,
- a(p^2) = 2 * p for any prime p,
- a(p * q) = 3 * p for any pair of consecutive odd primes (p, q).

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

Original entry on oeis.org

1, 2, 12, 56, 180, 304, 336, 936, 1696, 1824, 2484, 5040, 5328, 6664, 8384, 8512, 9900, 10176, 13176, 14040, 25632, 26208, 27360, 33372, 33712, 37260, 39808, 39984, 47488, 50304, 51072, 52200, 65232, 69552, 79920, 126900, 128448, 142272, 149184, 152640, 162648, 167776, 184064, 193752, 197640

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

Numbers k such that A007947(k)*A156061(k) = k or A156061(k) = A003557(k).

			9900 is a term because 9900 = 2^2 * 3^2 * 5^2 * 11 = prime(1)^2 * prime(2)^2 * prime(3)^2 * prime(5) = 1*prime(1) * 2*prime(2) * 3*prime(3) * 5*prime(5).

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003557, A005117, A007947, A008478, A033286, A046022, A048768, A078779, A109298, A156061.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); a[1] = 1; Select[Range[200000], a[#] == # &]

isok(n) = {my(f=factor(n)); prod(k=1, #f~, primepi(f[k,1])*f[k,1]) == n;} \\ Michel Marcus, May 08 2018

cp's OEIS Frontend

A129906 n!/(n^2) when an integer.

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A141295 Largest m<=n such that all k with 1<=k<=m are divisors of n or coprime to n.

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A268281 Numbers n such that n-tau(n), phi(n) and n form a Heronian triangle, where tau=A000005 is the number of divisors and phi=A000010 the totient.

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A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

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A335402 Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.

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A338554 Number of non-constant integer partitions of n whose parts have a common divisor > 1.

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A382492 a(n) is the least number that has exactly n 3-smooth divisors.

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A029716 Partial sums of Kempner numbers A002034.

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