A061742 -id:A061742 - cp's OEIS frontend

A316974 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

1, 1, 4, 14, 49, 173, 652, 2494

Offset: 0

Gus Wiseman, Jul 17 2018

Also the number of unlabeled multigraphs with n edges, allowing loops, spanning an initial interval of positive integers with no equivalent vertices (two vertices are equivalent if in every edge the multiplicity of the first is equal to the multiplicity of the second). For example, non-isomorphic representatives of the a(2) = 4 multigraphs are {(1,2),(1,3)}, {(1,1),(1,2)}, {(1,1),(2,2)}, {(1,1),(1,1)}.

			Non-isomorphic representatives of the a(3) = 14 strict multiset partitions:
  (112233),
  (1)(12233), (11)(2233), (12)(1233), (112)(233),
  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (2)(11)(233), (11)(22)(33), (12)(13)(23),
  (1)(2)(3)(123), (1)(2)(12)(33), (1)(2)(13)(23).

Cf. A001055, A007716, A007717, A007719, A020554, A020555, A045778, A050535, A053419, A061742, A094574, A162247, A316892, A316972.

a(7) from Andrew Howroyd, Feb 07 2020

A181555 a(n) = A002110(n)^n.

Original entry on oeis.org

1, 2, 36, 27000, 1944810000, 65774855015100000, 733384949590939374729000000, 9037114296609938214167920266348510000000, 78354300210436852307898467208663359164858967744100000000

Offset: 0

Matthew Vandermast, Oct 31 2010

For n>0, a(n)= first counting number whose prime signature consists of n repeated n times (cf. A002024). Subsequence of A025487.

			a(4) = 1944810000 = 210^4 = 2^4 * 3^4 * 5^4 * 7^4.

Amiram Eldar, Table of n, a(n) for n = 0..26

A061742(n) = A002110(n)^2. See also A006939, A066120, A166475, A167448.
A000005(a(n)) = A000169(n). The divisors of a(n) appear as the first A000169(n) terms of A178479, with A178479(A000169(n)) = a(n).
A071207(n, k) gives the number of divisors of n with (n-k) distinct prime factors, A181567(n, k) gives the number of divisors of n with k prime factors counted with multiplicity.
Cf. A001221, A001222, A079474, A146290, A146292.

a[0] = 1; a[n_] := Product[Prime[i], {i, 1, n}]^n; Array[a, 9, 0] (* Amiram Eldar, Aug 08 2019 *)

a(n) = A079474(2n,n). - Alois P. Heinz, Aug 22 2019

A070284 Smallest of 4 consecutive numbers each divisible by a square.

Original entry on oeis.org

242, 844, 845, 1680, 1681, 2888, 2889, 3174, 3624, 3625, 3750, 5046, 5047, 8475, 8523, 8954, 10050, 10827, 10924, 10925, 11322, 13374, 14748, 14749, 15775, 15848, 15849, 16575, 17404, 17405, 19647, 19940, 19941, 20574, 21462

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

This sequence has positive density in N; the density is around 0.0025.

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of quadruples of terms = {44100k+29349, 44100k+29350, 44100k+29351, 44100k+29352} = {9(4900k+3261), 25(1764k+1174), 49(900k+599), 4(11025k+7338)}; starting terms in this sequence = {29349, 73449, 117549, ...}; difference = A002110(4)^2 = 210^2. - Labos Elemer, Nov 25 2002

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

Cf. A070258, A068781, A045882, A049535, A077647, A078143.

f[n_] := Union[Transpose[FactorInteger[n]][[2]]][[ -1]]; a = 0; b = 1; c = 0; Do[d = f[n]; If[a > 1 && b > 1 && c > 1 && d > 1, Print[n - 3]]; a = b; b = c; c = d, {n, 4, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[60000],4,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)

is(n)=for(i=n,n+3, if(!issquarefree(n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A070284 = { A070258[k] | A070258[k+1] = A070258[k]+1 }. - M. F. Hasler, Feb 01 2016

More terms from Robert G. Wilson v, May 09 2002

b-file from Charles R Greathouse IV, Jul 23 2010

A115964 Denominator of Sum_{i=1..n} 1/prime(i)^3.

Original entry on oeis.org

8, 216, 27000, 9261000, 12326391000, 27081081027000, 133049351085651000, 912585499096480209000, 11103427767506874702903000, 270801499821725167129101267000, 8067447481189014453943055845197000

Offset: 1

Jonathan Vos Post, Mar 14 2006

Numerators are in A115963.

Also the primorials cubed. - Reikku Kulon, Sep 18 2008

			1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.

Cf. A115963 (numerators).
Cf. A024451 (numerator of sum_{i=1..n} 1/prime(i)), A002110 (primorial, also denominator of sum_{i=1..n} 1/prime(i)), A061015 (numerator of sum_{i=1..n} 1/prime(i)^2).
Cf. A000040, A075986/A075987, A106830/A034386.
Cf. A061742, A100778. - Reikku Kulon, Sep 18 2008

a[n_]:=Product[Prime[i]^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

a(n) = denominator of Sum_{i=1..n} 1/A000040(i)^3.

a(n) = A002110(n)^3. - Reikku Kulon, Sep 18 2008

A078143 Smallest term of a run of at least 9 consecutive integers which are not squarefree.

Original entry on oeis.org

8870024, 33908368, 49250144, 69147868, 70918820, 111500620, 112931372, 164786748, 167854344, 200997948, 203356712, 207543320, 211014920, 216785256, 221167422, 221167423, 221167424, 236645624, 240574368, 262315467, 262315468

Offset: 1

Labos Elemer, Nov 22 2002

nonn

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like squares of primorials, A061742(7)]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: arithmetic progression subsequences of starting 9-chains is {mk+69147868+j} where j=0..8, m=510510^2 because square prime factors of a(4)+j=68147868+j are 4, 49, 121, 169, 4, 9, 289, 25, 4 resp. for j=0..8; k goes to infinity; 7th primorial is sufficient, 9th is not necessary. Construction is provable for arbitrary long [>9] chains. - Labos Elemer, Nov 25 2002

More precisely, if in one run {a(n)+j, j=0..8} the maximum smallest square factor is p^2, then an infinite subsequence is given by {a(n)+(p#)^2*k, k=0..oo}, where p# = A034386(p). One may get a smaller step taking the least L^2 which has a square factor in common with each of the 9 consecutive terms. - M. F. Hasler, Feb 03 2016

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A013929, A045882 (first of the k-chains), A051681.

Cf. A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077640 (7-chains), A077647 (8-chains), A078143 (9-chains), A268313 (10-chains), A268314 (11-chains).

s9[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 8}]]; Do[If[Equal[s9[n], 0], Print[n]], {n, 8000000, 1000000000}]

is(n)=for(i=n,n+8, if(!issquarefree(i), return(0))); 1 \\ Charles R Greathouse IV, Nov 05 2017

A078143 = { A077647[k] | A077647[k+1] = A077647[k]+1 } = { A077640[k] | A077640[k+2] = A077640[k]+2 } = { A078144[k] | A078144[k+4] = A078144[k]+4 } etc. Note that A049535 is defined differently. - M. F. Hasler, Feb 01 2016

a(n) < 4666864390*n. With more work this bound can be decreased significantly. - Charles R Greathouse IV, Nov 05 2017

a(6)-a(21) from Donovan Johnson, Nov 26 2008

A078144 Starts for strings of at least five consecutive nonsquarefree numbers.

Original entry on oeis.org

844, 1680, 2888, 3624, 5046, 10924, 14748, 15848, 17404, 19940, 22020, 22021, 22624, 23272, 24647, 24648, 25772, 29348, 30248, 30923, 30924, 33172, 36700, 37248, 38724, 39444, 40472, 45372, 47672, 47673, 47724, 47824, 48372, 49488

Offset: 1

Labos Elemer, Nov 25 2002

nonn

			Squares dividing 5-string=844+j, j=0,..,4 are as follows:4,169,9,121,16 resp. Each term initiates an arithmetic progression with suitable large difference. Such progressions are constructible by solving suitable linear Diophantine equations. E.g., quintet = {m*k+3689649, m*k+3689650, m*k+3689651, m*k+3689652, m*k+3689653} = {9*(592900*k+409961), 25*(213444*k+147586), 49*(108900*k+75299), 4*(1334025*k+922413), 121*(44100*k+30493)}; m=2310*2310=A002110(5)^2=A061742(5)=5336100.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A045882 (min terms), A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077647 (8-chains), A078143 (9-chains), A188296.

s5[x_] := Total[Table[Abs[MoebiusMu[x + j]], {j, 0, 4}]] == 0; Select[Range[50000], s5]
Flatten[Position[Partition[SquareFreeQ/@Range[60000],5,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
SequencePosition[Table[If[SquareFreeQ[n],0,1],{n,50000}],{1,1,1,1,1}][[All,1]] (* Harvey P. Dale, Oct 16 2022 *)

Equals { A070284[k] | A070284[k+1] = A070284[k]+1 }. - M. F. Hasler, Feb 01 2016

a(n) = A188296(n) - 2. - Amiram Eldar, Feb 09 2021

A369427 The number of unitary divisors of n that are squares of primes.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 23 2024

The number of exponents in the prime factorization of n that are equal to 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001248, A038109, A056169, A056170, A061742, A085541, A085548.

Similar sequences: A125070, A293439, A366074, A367512, A369426, A369428.

f[p_, e_] := If[e == 2, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x == 2, 1, 0), factor(n)[, 2]));

Additive with a(p^e) = 1 if e = 2, and 0 otherwise.
a(n) > 0 if and only if n is in A038109.
a(A061742(n)) = n, and a(k) < n for all k < A061742(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 - 1/p^3) = A085548 - A085541 = 0.27748478074162196208... .

A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 6, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 12, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 10, 3, 4, 53, 18, 5, 8, 3, 2, 59, 4, 61, 2, 9, 64, 5, 2, 67, 4, 3, 2, 71, 24, 73, 2, 15, 4, 7

Offset: 1

Gus Wiseman, Aug 17 2024

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An anti-run is a sequence with no adjacent equal parts. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 540 are (1,1,2,2,2,3), with maximal anti-runs ((1),(1,2),(2),(2,3)), with minima (1,1,2,2), with Heinz number 36, so a(540) = 36.
The prime indices of 990 are (1,2,2,3,5), with maximal anti-runs ((1,2),(2,3,5)), with minima (1,2), with Heinz number 6, so a(990) = 6.

bigomega is A001222(a(n)) = A375136(n).
Least prime factor is A020639(a(n)) = A020639(n).
Least prime index is A055396(a(n)) = A055396(n).
Heinz weights are A056239(a(n)) = A374706(n).
The greatest prime index A061395(a(n)) is the maximum of row n of A375128.
Firsts for omega (except first term) are half A061742.
Prime indices A112798(a(n)) are row n of A375128.
Positions of prime-powers are A375396, counted by A115029.
Positions of squarefree numbers are A375398, counted by A375134.
A000041 counts integer partitions, strict A000009.
A027748 lists distinct prime factors, sum A008472.
A304038 lists distinct prime indices, sum A066328.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A000040, A046660, A124768, A320324, A358836, A374683, A374700, A375133.

Table[Times@@Prime/@If[n==1,{},Min /@ Split[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],UnsameQ]],{n,100}]

A101436 Number of exponents in prime factorization of n which are primes.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0

Offset: 1

Leroy Quet, Jan 18 2005

First occurrence of k: 1,4,36,900,44100 (A061742). - Robert G. Wilson v, Jan 25 2005

			36 = 2^2 *3^2. Since 2 is a prime and occurs twice as an exponent in the prime factorization of 36, a(36) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A010051, A061742, A123391, A125029, A125070, A125072.

f[n_] := Length[ Select[ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]], PrimeQ[ # ] &]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jan 25 2005 *)
Table[Count[Transpose[FactorInteger[n]][[2]],?PrimeQ],{n,120}] (* _Harvey P. Dale, Mar 21 2016 *)

A101436(n) = vecsum(apply(e -> isprime(e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 19 2017

Additive with a(p^e) = A010051(e). - Antti Karttunen, Jul 19 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (P(p)-P(p+1)) = 0.39847584805803104040..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 29 2023

More terms from Robert G. Wilson v, Jan 25 2005

A136370 Numerator of 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2.

Original entry on oeis.org

3, 31, 739, 37111, 4446331, 756766039, 217803584371, 78887714418031, 41637516941042299, 35066922176061410359, 33657455280704707522099, 46117280789485930425170431, 77468081652660425646977758411, 143331051198625503752852285686039

Offset: 1

Alexander Adamchuk, Dec 27 2007

It seems that the denominator of 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2 is A061742(n), which is the square of the product of the first n primes, but this is not immediately obvious. - Petros Hadjicostas, May 14 2020

			The first few fractions are 3/4, 31/36, 739/900, 37111/44100, 4446331/5336100, 756766039/901800900, ... = A136370/A061742. - _Petros Hadjicostas_, May 14 2020

Michael S. Branicky, Table of n, a(n) for n = 1..195

Possible denominators are A061742.

Cf. A024530, A136365, A136366, A136367, A136368, A136369, A136371.

Table[Numerator[1 - Sum[(-1)^(k+1)/Prime[k]^2, {k, 1, n}]], {n, 1, 20}]

a(n) = numerator(1 - sum(k=1, n, (-1)^(k+1)/prime(k)^2)); \\ Michel Marcus, May 14 2020

from sympy import prime
from fractions import Fraction
from itertools import accumulate, count, islice
def A136370gen(): yield from map(lambda x: (1-x).numerator, accumulate(Fraction((-1)**(k+1), prime(k)**2) for k in count(1)))
print(list(islice(A136370gen(), 14))) # Michael S. Branicky, Jun 26 2022

A136370/A061742 tends to 1 - A242301 = 0.83718375333639858423166... - Vaclav Kotesovec, May 14 2020

Definition corrected by Alexander Adamchuk, Sep 15 2010

a(14) and beyond from Michael S. Branicky, Jun 26 2022

cp's OEIS Frontend

A316974 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

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A181555 a(n) = A002110(n)^n.

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A070284 Smallest of 4 consecutive numbers each divisible by a square.

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A115964 Denominator of Sum_{i=1..n} 1/prime(i)^3.

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A078143 Smallest term of a run of at least 9 consecutive integers which are not squarefree.

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A078144 Starts for strings of at least five consecutive nonsquarefree numbers.

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A369427 The number of unitary divisors of n that are squares of primes.

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A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

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