A046387 -id:A046387 - cp's OEIS frontend

A355095 Taxicab numbers (sums of 2 cubes in more than 1 way) which are products of five distinct primes.

16387189, 16776487, 17045567, 24767171, 38253878, 39639691, 40183262, 41892515, 44409995, 51278929, 60271939, 73842713, 106496767, 122810129, 129380329, 145908847, 154245637, 156234169, 176427433, 197842337, 243578881, 271688534, 272264167, 292940137, 300694303, 373333697, 389675503, 401947273

Offset: 1

Massimo Kofler, Jun 19 2022

nonn

A squarefree subsequence of taxicab numbers.

			16387189 = 5^3 + 254^3 = 197^3 + 206^3 = 7*13*31*37*157.
38253878 = 87^3 + 335^3 = 173^3 + 321^3 = 2*13*19*211*367.
44409995 = 138^3 + 347^3 = 176^3 + 339^3 = 5*7*97*103*127.

David A. Corneth, Table of n, a(n) for n = 1..3057 (terms <= 10^13)
Eric Weisstein's World of Mathematics, Taxicab Number

Intersection of A001235 and A046387.

A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

Original entry on oeis.org

2, 39, 1279786, 8377774397163159586

Offset: 1

Jon E. Schoenfield, Nov 22 2022

			The first two squarefree numbers are 1 and 2; 1 has 0 prime factors and 2 has 1 prime factor, so a(1)=2.
At k=39, in the interval [1..k], there are 12 squarefree numbers with 1 prime factor (i.e., 12 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), and 12 squarefree numbers with 2 prime factors (i.e., 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39). k=39 is the smallest such positive number for which these two counts are the same (and are positive), so a(2)=39.
At k=1279786, the interval [1..k] includes 265549 squarefree numbers with 2 prime factors and the same number of squarefree numbers with 3 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(3)=1279786. There are 75 numbers with this property, the last one being 1281378.
At k=8377774397163159586, the interval [1..k] includes 1356557942402075858 squarefree numbers with 3 prime factors and the same number of squarefree numbers with 4 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(4)=8377774397163159586. There are 14 numbers with this property, the last one being 8377774397163162544. - _Henri Lifchitz_, Jan 31 2025

Cf. A005117, A072047.
Cf. 1 to 5 distinct primes: A000040, A006881, A007304, A046386, A046387.
Cf. 6 to 10 distinct primes: A067885, A123321, A123322, A115343, A281222.
Cf. A340316.

a(n) = my(nbm = 0, nbn = 0); for (k=1, oo, if (issquarefree(k), my(o=omega(k)); if (o==n, nbn++); if (o==n-1, nbm++); if (nbm && (nbn==nbm), return(k)))); \\ Michel Marcus, Nov 25 2022

from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A356683(n):
    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(k,n): return sum(primepi(k//prod(c[1] for c in a))-a[-1][0] for a in g(k,0,1,1,n)) if n>1 else primepi(k)
    return 2 if n==1 else next(k for k in count(1) if (x:=f(k,n-1))>0 and x==f(k,n)) # Chai Wah Wu, Aug 31 2024

a(4) from Henri Lifchitz, Jan 31 2025

A359304 Oblong numbers which are products of five distinct primes.

Original entry on oeis.org

4290, 4830, 6006, 11130, 12210, 13110, 16770, 23870, 27390, 33306, 34410, 34782, 37830, 44310, 49062, 56406, 60762, 64770, 66822, 70490, 71022, 74802, 82082, 84390, 95790, 101442, 103362, 104006, 109230, 119370, 125670, 127806, 133590, 137270, 148610, 151710, 158802

Offset: 1

Massimo Kofler, Dec 25 2022

nonn

			4290 = 65*66 = 2*3*5*11*13.
4830 = 69*70 = 2*3*5*7*23.
6006 = 77*78 = 2*3*7*11*13.
11130 = 105*106 = 2*3*5*7*53.

Intersection of A002378 and A046387.

Select[(#*(# + 1)) & /@ Range[400], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Dec 25 2022 *)

select(x->((omega(x)==5) && (bigomega(x)==5)), vector(500, k, k*(k+1))) \\ Michel Marcus, Dec 25 2022

A359644 Number of numbers <= 10^n that are products of 5 distinct primes.

Original entry on oeis.org

0, 0, 0, 24, 910, 18387, 286758, 3884936, 48396263, 571221133, 6499261245, 72047682376, 783561421371, 8399470576016, 89038389261794, 935562667202846, 9761003371437806, 101253973351371824, 1045354835981786609

Offset: 1

Peter Dolland, Jan 09 2023

			a(4) = 24 = #{2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, 9870}

Cf. A006880, A036351, A046387, A215218, A359642.

a(n) = my(N=10^n); (f(m,p,k,j=1)=my(s=sqrtnint(N\m, k), count=0); if(k==2, forprime(q=p, s, count += primepi(N\(m*q)) - j; j+=1); return(count)); forprime(q=p, s, count += f(m*q, q+1, k-1, j+1); j+=1); count); f(1, 2, 5); \\ Daniel Suteu, Jan 10 2023

a(14) from Daniel Suteu, Jan 10 2023

a(15)-a(19) from Henri Lifchitz, Feb 01 2025

A376929 Products of 5 distinct primes that are sandwiched between sphenic numbers.

Original entry on oeis.org

50610, 52206, 63546, 65190, 71890, 73830, 77406, 84930, 89310, 89870, 90390, 92598, 98210, 116754, 119210, 120990, 123410, 125994, 131054, 132430, 132870, 137410, 140998, 141702, 144430, 148190, 150306, 151810, 159942, 160854, 162470, 164406, 165110, 167314, 170562, 172938, 174306, 176946, 185658

Offset: 1

Massimo Kofler, Oct 11 2024

nonn

All terms are even.

Dickson's conjecture implies that there are infinitely many terms, e.g. there should be infinitely many k such that p = 241 + 104533*k, q = 229 + 99330*k, and r = 107 + 46410*k are all prime, and then 210*p is a term (with 210*p = 2*3*5*7*p, 210*p-1 = 13*17*q, and 210*p+1 = 11*43*r). - Robert Israel, Nov 12 2024

			50610 is a term because 50610=2*3*5*7*241 is the product of five distinct primes and 50609=13*17*229, 50611=11*43*107 are sphenic numbers.
52206 is a term because 52206=2*3*7*11*113 is the product of five distinct primes and 52205=5*53*197, 52207=17*37*83 are sphenic numbers.

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

Intersection of A046387 and 2*A376734.

Cf. A376380, A351382, A007304.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  if F[..,2] <> [1$5] then return false fi;
  F:= ifactors(n-1)[2];
  if F[..,2] <> [1$3] then return false fi;
  F:= ifactors(n+1)[2];
  F[..,2] = [1$3]
end proc:
select(filter, [seq(i,i=2..2*10^5,4)]); # Robert Israel, Nov 12 2024

SequencePosition[Map[#[[;; , 2]] &, FactorInteger[Range[200000]]], {{1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Oct 11 2024 *)

A068316 Run lengths of the Moebius function applied to A051270 (numbers with 5 distinct prime factors).

Original entry on oeis.org

5, 1, 1, 1, 6, 2, 4, 3, 4, 1, 2, 1, 6, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 1, 4, 1, 2, 2, 3, 1, 2, 5, 2, 2, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2

Offset: 1

Jani Melik, Feb 26 2002

nonn

			If we consider A051270 and apply the Moebius function mu(n) to it we get a sequence of values: (-1,-1,-1,-1,-1),0,(-1),0,(-1,-1,-1,-1,-1,-1),0,0,(-1,-1,-1,-1),0,0,0,(-1,-1,-1,-1),0,(-1,-1),0,(-1, ... If we then look at the lengths of runs of equal terms, we get the sequence.
If we consider the values of A051270 which are not in A046387 we get numbers which are not squarefree, so mu(A051270(.)) is zero: 4620, 5460, 6930, ...

Cf. A046387, A051270.

runl := 1 :
for n from 2 to 1000 do
    if numtheory[mobius](A051270(n)) = numtheory[mobius](A051270(n-1)) then
        runl := runl+1 ;
    else
        printf("%d,",runl) ;
        runl := 1;
    end if;
end do: # R. J. Mathar, Oct 13 2019

Corrected and extended by R. J. Mathar, Oct 13 2019

A356355 9-gonal numbers which are products of five distinct primes.

Original entry on oeis.org

24486, 71214, 90321, 116754, 123234, 156774, 181374, 265926, 287574, 445179, 450186, 483414, 488631, 595959, 688866, 698214, 781869, 791826, 845994, 912646, 937839, 970734, 1030614, 1042041, 1100121, 1266909, 1463514, 1659801, 2014386, 2041026, 2171334, 2187906

Offset: 1

Massimo Kofler, Oct 15 2022

nonn

A squarefree subsequence of 9-gonal numbers a(n) = n*(7*n-5)/2.

			24486 = 2*3*7*11*53 = 84*(7*84-5)/2.
445179 = 3*7*17*29*43 = 357*(7*357-5)/2.
781869 = 3*11*19*29*43 = 473*(7*473-5)/2.
1030614 = 2*3*13*73*181 = 543*(7*543-5)/2.

Intersection of A001106 and A046387.

Select[Table[n*(7*n - 5)/2, {n, 1, 1000}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Oct 15 2022 *)

select(x->omega(x)==5&&bigomega(x)==5, vector(800, k, k*(7*k-5)/2)) \\ Michel Marcus, Nov 26 2022

A357590 Triangular numbers which are products of five distinct primes.

Original entry on oeis.org

3570, 8778, 9870, 12090, 13530, 20706, 20910, 21945, 24090, 24310, 26565, 33670, 40470, 40755, 47586, 54285, 57630, 57970, 63546, 66430, 69006, 72390, 76245, 87990, 88410, 91806, 92235, 94395, 94830, 98790, 121278, 130305, 132870, 133386, 141778, 148785, 154290, 159330, 163878, 167910

Offset: 1

Massimo Kofler, Oct 05 2022

nonn

A squarefree subsequence of triangular numbers (T(n) = n*(n+1)/2).

			3570 = 2*3*5*7*17 = 84*85/2.
21945 = 3*5*7*11*19 = 209*210/2.
121278 = 2*3*17*29*41 = 492*493/2.
154290 = 2*3*5*37*139 = 555*556/2.

Intersection of A000217 and A046387.

Cf. A068443, A128896, A333771.

q:= n-> map(i-> i[2], ifactors(n)[2])=[1$5]:
select(q, [seq(n*(n+1)/2, n=0..1000)])[];  # Alois P. Heinz, Oct 05 2022

Select[Accumulate @ Range[600], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Oct 05 2022 *)

A376949 Products of 5 distinct primes that are sandwiched between squarefree semiprime numbers.

Original entry on oeis.org

5610, 6510, 6630, 7770, 9570, 10230, 12558, 13530, 14190, 14586, 14790, 15834, 16590, 16770, 17710, 19734, 20670, 21630, 22610, 22890, 23478, 23730, 24882, 25662, 26970, 28490, 28938, 29370, 31110, 31710, 33306, 33558, 33990, 34410, 34710, 35178, 35742, 36330, 37230, 37758, 38346, 39130, 40890

Offset: 1

Massimo Kofler, Oct 10 2024

nonn

All terms are even.

			5610 is a term because 5610=2*3*5*11*17 is the product of five distinct primes and 5609=71*79, 5611=31*181 are semiprime numbers.
6510 is a term because 6510=2*3*5*7*31 is the product of five distinct primes and 6509=23*283, 6511=17*383 are semiprime numbers.

Intersection of A046387 and 2*A350101.

Cf. A046387, A376380, A006881.

SequencePosition[Map[#[[;; , 2]] &, FactorInteger[Range[41000]]], {{1, 1}, {1, 1, 1, 1, 1}, {1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Oct 10 2024 *)

A384507 Initial term of first run of exactly n consecutive numbers with exactly 5 distinct prime factors.

Original entry on oeis.org

2310, 254540, 1042404, 21871365, 129963314, 830692265, 4617927894, 18297409143, 41268813542, 287980277114, 1182325618032, 6455097761454, 14207465691240, 54049709480208, 90987640183352, 1164838922284960, 546525829796442

Offset: 1

Toshitaka Suzuki, Jun 23 2025

See A087978 for further details and an explanation of why this sequence is finite.
First differs from A087978 at n=16.
a(18) > 2 * 10^15.

			a(16) > a(17) because the first run of 16 consecutive integers i with A001221(i)=5 is not a maximal run.

Cf. A001221, A046387, A080569, A087977, A087978, A185032, A185042.

cp's OEIS Frontend

A355095 Taxicab numbers (sums of 2 cubes in more than 1 way) which are products of five distinct primes.

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A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

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A359304 Oblong numbers which are products of five distinct primes.

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A359644 Number of numbers <= 10^n that are products of 5 distinct primes.

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A376929 Products of 5 distinct primes that are sandwiched between sphenic numbers.

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A068316 Run lengths of the Moebius function applied to A051270 (numbers with 5 distinct prime factors).

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A356355 9-gonal numbers which are products of five distinct primes.

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A357590 Triangular numbers which are products of five distinct primes.

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A376949 Products of 5 distinct primes that are sandwiched between squarefree semiprime numbers.

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