A046387 -id:A046387 - cp's OEIS frontend

A380530 Positions of records in A380528.

1, 4, 10, 42, 366, 3246, 37266, 631266, 11563926, 271591926

Offset: 1

Antti Karttunen, Feb 09 2025

Questions: Are all terms squarefree after 4, and do all terms end (in base ten) with digit 6 from the fifth term onward? Is the sequence infinite? Are there any terms with more than four prime factors? See also the conjecture given in A380475, and A380476.
For n = 1..10, A380528(a(n)) = A008578(n). If a(11) exists, it is > 2^30.
Note that for squarefree n with exactly 5 prime factors (A046387), it might be possible that primes obtained as A380528(A046387(.)) has no upper limit. First k in A046387 such that A380528(k) is a prime 2 .. 17 are: 15015, 2730, 2310, 83886, 1551066, 71559186, 1245223986. See also the last two examples in A380470 for two similar cases with exactly 6 prime factors. - Antti Karttunen, May 08 2025

Index entries for sequences related to primorial base.

Cf. A008578 (conjectured to give the record values), A046387, A380528, A380459, A380468, A380470, A380475, A380476.

A078329 Primes p such that mu(p+1)=-1, where mu denotes the moebius function.

Original entry on oeis.org

2, 29, 41, 101, 109, 113, 137, 173, 181, 229, 257, 281, 317, 353, 373, 401, 409, 433, 601, 617, 641, 653, 677, 709, 761, 821, 829, 853, 937, 941, 977, 1009, 1021, 1033, 1069, 1117, 1129, 1181, 1193, 1297, 1361, 1373, 1433, 1489, 1597, 1613, 1669, 1697

Offset: 1

Shyam Sunder Gupta, Nov 21 2002

Contains primes followed by primes (i.e., the 2), primes followed by sphenic numbers (A007304), or followed by elements of A046387, A123321, A115343 etc. - R. J. Mathar, Aug 14 2019

Primes followed by numbers that are the product of an odd number of distinct primes (A030059). - Joerg Arndt, Aug 14 2019

			29 is in the sequence because 29 is prime and mu(30)=-1.

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

Cf. A008683, A030059.

Cf. A007304, A046387, A123321, A115343.

Select[Prime[Range[300]],MoebiusMu[#+1]==-1&] (* Harvey P. Dale, Feb 28 2013 *)

A340467 a(n) is the n-th squarefree number having n prime factors.

Original entry on oeis.org

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330

Offset: 1

Alois P. Heinz, Jan 08 2021

nonn

a(n) is the n-th product of n distinct primes.
All terms are even.
This sequence differs from A073329 which has also nonsquarefree terms.

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.

Alois P. Heinz, Table of n, a(n) for n = 1..350

Main diagonal of A340316.
Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.
Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

a(n) = A340316(n,n).
a(n) = A005117(m) <=> A072047(m) = n = A340313(m).
A001221(a(n)) = A001222(a(n)) = n.
a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.

A375264 Products of prime 5-tuples (p, p+4, p+6, p+10, p+12) where p = A022007(n).

Original entry on oeis.org

323323, 11769028333, 23075277662228863, 498323779650735373, 5774946339890457283, 964079483344036314403, 1069516017850492496773, 2764561257197073595813, 160896800504464381875853, 3214487754103795836790843, 5525765859191774942946523, 10569235342244010823461223, 1140114641548139662904736973

Offset: 1

Michael De Vlieger, Aug 16 2024

nonn

Subsequence of A046387.

All terms are congruent to 13 (mod 30), since they are products m of the following residues (mod 30): {7, 11, 13, 17, 19}, and m mod 30 is congruent to 13.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022007, A046387, A375264.

Map[Times @@ NextPrime[#, Range[0, 4]] &, Select[Prime@ Range[2^16], AllTrue[{# + 4, # + 6, # + 10, # + 12}, PrimeQ] &]]

A077438 Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.

Original entry on oeis.org

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 900, 961, 1369, 1681, 1764, 1849, 2209, 2809, 3481, 3721, 4356, 4489, 4900, 5041, 5329, 6084, 6241, 6889, 7921, 9409, 10201, 10404, 10609, 11025, 11449, 11881, 12100, 12769, 12996, 16129, 16900

Offset: 1

Benoit Cloitre, Nov 30 2002

nonn

From Robert G. Wilson v, Dec 28 2016: (Start)
Union of {A000040, A007304, A046387, A123321, A115343, etc}^2 = Union of {A001248, A162143, etc} = A030059(n)^2.
Number of terms < 10^k: 2, 4, 12, 30, 98, 303, 957, ..., . (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 800 terms from G. C. Greubel)

Cf. A000040, A001248, A007304, A030059, A046387, A088245, A115343, A123321, A162143.

fQ[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[#] MoebiusMu[n/#]^2 & /@ d) == -1]; Select[Range@17000, fQ] (* Robert G. Wilson v, Dec 28 2016 *)

isok(n) = sumdiv(n, d, moebius(d)*moebius(n/d)^2) == -1; \\ Michel Marcus, Nov 08 2013

is(n)=if(!issquare(n,&n), return(0)); my(f=factor(n)[,2]); #f%2 && vecmax(f)==1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = A030059(n)^2.
From Amiram Eldar, Jun 16 2020: (Start)
Sum_{k>=1} 1/a(k) = 9/(2*Pi^2) = A088245.
Sum_{k>=1} 1/a(k)^2 = 15/(2*Pi^4). (End)

A375263 Products of prime 5-tuples (p, p+2, p+6, p+8, p+12) where p = A022006(n).

Original entry on oeis.org

85085, 1062347, 13710311357, 7260418981910177, 1070581737243007155797, 2766839311485334046837, 4100268039496291731347, 5485857138434357935487, 160955606471252225494877, 518874528458711923617827, 62276368576118330117536697, 124939289421198925662649757, 129296510584439756429395217

Offset: 1

Michael De Vlieger, Aug 16 2024

nonn

Subsequence of A046387.

Aside from a(1) = 85085 = 5 (mod 30), all terms are congruent to 17 (mod 30), since a(k), k > 2 is a product m of the following residues (mod 30): {11, 13, 17, 19, 23}, and m mod 30 is congruent to 17.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022006, A046387, A375264.

Map[Times @@ NextPrime[#, Range[0, 4]] &, Select[Prime@ Range[2^16], AllTrue[{# + 2, # + 6, # + 8, # + 12}, PrimeQ] &]]

A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

Original entry on oeis.org

3, 5, 27, 7, 10, 11, 9, 25, 14, 13, 20, 17, 15, 21, 7625597484987, 19, 24, 23, 28, 22, 26, 29, 50, 32, 33, 3125, 44, 31, 42, 37, 49, 34, 35, 38, 100, 41, 39, 46, 45, 43, 66, 47, 52, 56, 51, 53, 80, 121, 98, 55, 54, 59, 68, 57, 63, 58, 62, 61, 84, 67, 65, 75

Offset: 2

Rémy Sigrist, Feb 07 2017

nonn

The prime tower factorization of a number is defined in A182318.
The prime tower factorization equivalence classes are described in A279686.
For any n>1, a(n)=least k>n such that A279690(n)=A279690(k).
This sequence is a permutation of the complement of A279686.
This sequence is to prime tower factorization what A081761 is to prime signature.

Rémy Sigrist, Table of n, a(n) for n = 2..2500
Rémy Sigrist, Illustration of the first terms

Cf. A000040, A006881, A007304, A046386, A046387, A051674, A067885, A081761, A182318, A279686, A279690

a(n) = my (c=a279690(n)); my (k=n+1); while (c!=a279690(k), k++); k

a(A000040(n)) = A000040(n+1) for any n>0.
a(A006881(n)) = A006881(n+1) for any n>0.
a(A051674(n)) = A051674(n+1) for any n>0.
a(A007304(n)) = A007304(n+1) for any n>0.
a(A046386(n)) = A046386(n+1) for any n>0.
a(A046387(n)) = A046387(n+1) for any n>0.
a(A067885(n)) = A067885(n+1) for any n>0.

A327829 Squarefree numbers with a prime number of prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Sebastian F. Orellana, Sep 26 2019

nonn

210 is the first integer in A120944 but not here: it has 4 prime factors.

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

Subsequence of A120944.
A006881, A007304, A046387 are subsequences.
A046386, A067885 are not subsequences.

Select[Range@ 161, And[SquareFreeQ@ #, PrimeQ@ PrimeNu@ #] &] (* Michael De Vlieger, Sep 29 2019 *)

isok(n) = issquarefree(n) && isprime(omega(n)); \\ Michel Marcus, Sep 27 2019

A006881 UNION A007304 UNION A046387 UNION A123321 UNION .... - R. J. Mathar, Oct 13 2019

Corrected and extended by Michel Marcus, Sep 27 2019

A353019 Heptagonal numbers which are products of five distinct primes.

Original entry on oeis.org

32890, 48790, 102718, 167314, 236698, 239785, 260338, 330694, 360430, 389470, 455182, 749938, 884170, 932386, 960070, 1007110, 1104565, 1334806, 1397638, 1423930, 1488802, 1515934, 1610818, 1679770, 1721005, 1741810, 1952314, 2046205, 2312167, 2365363, 2473570, 2503501, 2513518, 2558842

Offset: 1

Massimo Kofler, Apr 17 2022

nonn

A squarefree subsequence of heptagonal numbers.

			32890 = 2*5*11*13*23 = 115(5*115-3)/2.
48790 = 2*5*7*17*41 = 140(5*140-3)/2.
102718 = 2*7*11*23*29 = 203(5*203-3)/2.
167314 = 2*7*17*19*37 = 259(5*259-3)/2.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A000566 and A046387.

f:= proc(n) local k, F;
  k:= n*(5*n-3)/2;
  F:= ifactors(k)[2];
  if F[..,2] = [1,1,1,1,1] then k fi
end proc:
map(f, [$1..2000]); # Robert Israel, Jul 29 2025

Select[Table[n*(5*n - 3)/2, {n, 1, 1000}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Apr 17 2022 *)

from sympy import factorint
from itertools import count, islice
def agen():
    for h in (n*(5*n-3)//2 for n in count(1)):
        f = factorint(h, multiple=True)
        if len(f) == len(set(f)) == 5: yield h
print(list(islice(agen(), 34))) # Michael S. Branicky, May 28 2022

A354976 Squarefree tetrahedral numbers which are products of five distinct primes.

Original entry on oeis.org

7770, 14190, 98770, 121485, 129766, 273819, 383306, 457310, 632710, 735130, 848046, 971970, 1072445, 1456935, 1543465, 2027795, 2135445, 2190670, 2731135, 3136805, 3817670, 4775385, 4869634, 5159805, 5564321, 5989445, 6099006, 6209895, 8579746, 9145270, 9735114, 9886435

Offset: 1

Massimo Kofler, Jun 15 2022

nonn

A squarefree subsequence of tetrahedral numbers.

			7770 = 35*36*37/6 = 2*3*5*7*37
14190 = 43*44*45/6 = 2*3*5*11*43
98770 = 83*84*85/6 = 2*5*7*17*83
121485 = 89*90*91/6 = 3*5*7*13*89

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A000292 and A046387.

Subsequence of A070755.

q:= n-> is(map(x-> x[2], ifactors(n)[2])=[1$5]):
select(q, [n*(n+1)*(n+2)/6$n=1..500])[];  # Alois P. Heinz, Jun 15 2022

Select[Table[n*(n+1)*(n+2)/6, {n, 1, 400}], FactorInteger[#][[;;, 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Jun 15 2022 *)
Select[Binomial[Range[500]+2,3],PrimeNu[#]==PrimeOmega[#]==5&] (* Harvey P. Dale, Jul 05 2025 *)

Definition clarified by Harvey P. Dale, Jul 05 2025

cp's OEIS Frontend

A380530 Positions of records in A380528.

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A078329 Primes p such that mu(p+1)=-1, where mu denotes the moebius function.

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A340467 a(n) is the n-th squarefree number having n prime factors.

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A375264 Products of prime 5-tuples (p, p+4, p+6, p+10, p+12) where p = A022007(n).

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A077438 Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.

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A375263 Products of prime 5-tuples (p, p+2, p+6, p+8, p+12) where p = A022006(n).

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A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

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A327829 Squarefree numbers with a prime number of prime factors.

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