A046389 -id:A046389 - cp's OEIS frontend

A338469 Products of three odd prime numbers of odd index.

125, 275, 425, 575, 605, 775, 935, 1025, 1175, 1265, 1331, 1445, 1475, 1675, 1705, 1825, 1955, 2057, 2075, 2255, 2425, 2575, 2585, 2635, 2645, 2725, 2783, 3175, 3179, 3245, 3425, 3485, 3565, 3685, 3725, 3751, 3925, 3995, 4015, 4175, 4301, 4475, 4565, 4715

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd and > 1. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     125: {3,3,3}     1825: {3,3,21}    3425: {3,3,33}
     275: {3,3,5}     1955: {3,7,9}     3485: {3,7,13}
     425: {3,3,7}     2057: {5,5,7}     3565: {3,9,11}
     575: {3,3,9}     2075: {3,3,23}    3685: {3,5,19}
     605: {3,5,5}     2255: {3,5,13}    3725: {3,3,35}
     775: {3,3,11}    2425: {3,3,25}    3751: {5,5,11}
     935: {3,5,7}     2575: {3,3,27}    3925: {3,3,37}
    1025: {3,3,13}    2585: {3,5,15}    3995: {3,7,15}
    1175: {3,3,15}    2635: {3,7,11}    4015: {3,5,21}
    1265: {3,5,9}     2645: {3,9,9}     4175: {3,3,39}
    1331: {5,5,5}     2725: {3,3,29}    4301: {5,7,9}
    1445: {3,7,7}     2783: {5,5,9}     4475: {3,3,41}
    1475: {3,3,17}    3175: {3,3,31}    4565: {3,5,23}
    1675: {3,3,19}    3179: {5,7,7}     4715: {3,9,13}
    1705: {3,5,11}    3245: {3,5,17}    4775: {3,3,43}

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

A046316 allows all primes (strict: A046389).
A338471 allows all odd primes (strict: A307534).
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005408 lists odds (strict: A056911).
A008284 counts partitions by sum and length.
A014311 is a ranking of 3-part compositions (strict: A337453).
A014612 lists Heinz numbers of 3-part partitions (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A066207 lists numbers with all even prime indices (strict: A258117).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict 3-part partitions.
Cf. A001221, A001222, A002620, A005117, A037144, A056239, A112798.

N:= 10000: # for terms <= N
P0:= [seq(ithprime(i),i=3..numtheory:-pi(floor(N/25)),2)]:
sort(select(`<=`,[seq(seq(seq(P0[i]*P0[j]*P0[k],k=1..j),j=1..i),i=1..nops(P0))], N)); # Robert Israel, Nov 12 2020

Select[Range[1,1000,2],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (m%2) && (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338469(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
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(5,integer_nthroot(x,3)[0]+1),3)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A075808 Palindromic odd composite numbers that are the products of an odd number of distinct primes.

Original entry on oeis.org

555, 595, 777, 969, 1001, 1221, 1551, 1771, 3333, 3553, 5335, 5555, 5665, 5885, 5995, 7337, 7557, 7667, 7777, 7887, 9339, 9669, 9779, 9889, 11211, 11811, 12121, 12621, 12921, 13731, 14241, 14541, 15051, 15951, 16261, 16761, 17171, 18381

Offset: 1

Jani Melik, Oct 13 2002

			555 = 3*5*37, 595 = 5*7*17 and 777 = 3*7*37 are palindromic, odd, composite and products of an odd number of distinct primes.
50505 = 3 * 5 * 7 * 13 * 37 is the first term with five factors.
125 = 5^3 and 5445 = 3^2 * 5 * 11^2 are not terms since they are not the products of distinct primes.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A046389, A356750.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)=-1 and not isprime(n); end; a := []; for n from 1 to 30000 by 2 do if test(n) then a := [op(a),n]; end; od; a;

Select[Range[2,20000], ! PrimeQ[#] && OddQ[#] && PalindromeQ[#] &&
   OddQ[Length[Transpose[FactorInteger[#]][[2]]]] &&
Max[Transpose[FactorInteger[#]][[2]]] == 1 &] (* Tanya Khovanova, Aug 26 2022 *)

from sympy import isprime, factorint
from itertools import count, islice, product
def cond(n):
    if n%2 == 0 or isprime(n): return False
    f = factorint(n)
    return len(f) == sum(f.values()) and len(f)&1
def oddpals(): # generator of odd palindromes
    yield from [1, 3, 5, 7, 9]
    for d in count(2):
        for first in "13579":
            for p in product("0123456789", repeat=(d-2)//2):
                left = "".join(p); right = left[::-1]
                for mid in [[""], "0123456789"][d%2]:
                    yield int(first + left + mid + right + first)
def agen(): yield from filter(cond, oddpals())
print(list(islice(agen(), 38))) # Michael S. Branicky, Aug 25 2022

Edited by Dean Hickerson, Oct 21 2002

Name edited by Tanya Khovanova, Aug 26 2022

A180257 a(n) = 2^(p*r) mod q for the n-th odd number with exactly three distinct prime factors p < q < r.

Original entry on oeis.org

2, 2, 3, 1, 3, 1, 2, 2, 1, 2, 1, 6, 3, 4, 2, 1, 3, 2, 2, 1, 3, 7, 2, 1, 8, 4, 2, 10, 5, 6, 1, 3, 2, 1, 2, 5, 1, 3, 10, 7, 2, 1, 2, 2, 2, 8, 10, 2, 4, 3, 2, 1, 8, 15, 2, 5, 2, 7, 1, 2, 10, 1, 4, 6, 12, 3, 8, 1, 6, 2, 8, 3, 8, 9, 1, 11, 4, 3, 1, 2, 7, 2, 15, 10, 8

Offset: 1

Juri-Stepan Gerasimov, Jan 17 2011

nonn

			a(1) = 2^(3*7) mod 5 = 2 because A046389(1) = 105 = 3*5*7.

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

Cf. A046389, A180197.

More terms and corrected a(10) and a(13) from Nathaniel Johnston, Jan 18 2011

A378209 Antiderivatives of 334406399, numbers k for which A003415(k) = A024451(9) = A003415(A002110(9)).

Original entry on oeis.org

223092870, 975351895, 1527890095, 1885679383, 2189118743, 2329696457, 2338611863, 3485765789, 4586671213, 5453593183, 5472849253, 5674340053, 8071055747, 8931775397, 9332889127, 9453996491, 9601098443, 10293819917, 12717530039, 17343441881, 18636581773, 19498393573, 20167656703, 23244839627, 23515890737, 23556538969

Offset: 1

Antti Karttunen, Nov 20 2024

Apart from the initial term A002110(9), all other terms are products of three distinct odd primes, A046389. Compare to the comments in A369239.
Note that A024451(9) = 334406399 = 43 * 163 * 47711 == -1 (mod 12). Compare the sequences A369450, A369451 and A369452 to see why there is such a sudden peak in A377993 at n=9, when compared to the nearby terms before and after.
For all n=1..330, A327969(a(n)) <= 7 = A099307(a(n)), because, when we apply A003415 successively, we get: A003415(334406399) -> 9835475 [= A369651(9)] -> 4893565 -> 978718 -> 564671 (which is a prime) -> 1 -> 0.

Antti Karttunen, Table of n, a(n) for n = 1..330
Index entries for sequences related to primorial numbers

Row 9 of irregular triangle A377992.
Subsequence of A099308, and after the initial term, subsequence of A046389.
Cf. A002110, A003415, A024451, A099307, A327969, A369239, A369651.
Cf. A369450, A369451, A369452.

A075815 Palindromic odd composite numbers with an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

99, 171, 333, 343, 363, 555, 575, 595, 747, 777, 909, 969, 1001, 1221, 1331, 1551, 1771, 3333, 3553, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 7337, 7557, 7667, 7777, 7887, 9009, 9339, 9559, 9669, 9779, 9889, 11211, 11511, 11711, 11811, 12121

Offset: 1

Jani Melik, Oct 13 2002

			5445=3^2*5*11^2 and 5555=5*11*101 are palindromic, odd, composite and have an odd number of prime factors.

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

Cf. A046389.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and type(numtheory[bigomega](n),odd) and not isprime(n); end; a := []; for n from 1 to 13000 by 2 do if test(n) then a := [op(a),n]; end; od; a;

Select[Range[9,13000,2],PalindromeQ[#]&&CompositeQ[#]&&OddQ[ PrimeOmega[ #]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 28 2021 *)

Edited by Dean Hickerson, Oct 21 2002

A146961 Numbers k = pqr, with odd primes p < q < r, such that Sister Beiter's cyclotomic coefficient conjecture is false.

Original entry on oeis.org

20213, 125609, 136477, 141317, 150271, 198493, 199177, 212971, 239039, 273229, 282367, 291343, 311201, 332777, 373901, 393313, 398563, 412357, 442091, 449527, 449647, 450131, 456569, 461263, 469249, 470741, 475057, 522461, 524837, 532363

Offset: 1

T. D. Noe, Nov 03 2008

nonn

In 1968, Sister Beiter conjectured that for k = p*q*r, with odd primes p < q < r, the maximum coefficient (in absolute value) of the cyclotomic polynomial Phi(k,x) is <= (p+1)/2. Up to 10^6, all counterexamples have p > 7. Gallot and Moree prove the conjecture is false for p > 7.

Robin Visser, Table of n, a(n) for n = 1..200
A. S. Bang, Om Ligningen phi_n(x) = 0, Nyt tidsskrift for matematik, Vol. 6, Afdeling B (1895), pp. 6-12 (7 pages).
Yves Gallot and Pieter Moree, Counter-examples to Sister Beiter's cyclotomic coefficient conjecture, MPIM Preprint Series 2007 (141).
Nathan Kaplan, Flat cyclotomic polynomials of order three, Journal of Number Theory, Volume 127, Issue 1, November 2007, Pages 118-126.
G. S. Kazandzidis, On the cyclotomic polynomial: Coefficients, Bull. Soc. Math. Gr`ece (N.S.) 4 (1963), no. 1, 1-11.
Carlo Sanna, A Survey on Coefficients of Cyclotomic Polynomials, arXiv:2111.04034 [math.NT], 2021.
Wikipedia, Marion Beiter.

Subsequence of A046389.

isok(m) = if ((m%2) && (bigomega(m)==3) && (omega(m)==3), my(p=vecmin(factor(m)[,1])); vecmax(apply(abs, Vec(polcyclo(m)))) > (p+1)/2;); \\ Michel Marcus, Jan 16 2023

from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs
for n in range(3, 100000, 2):
    pqr = Integer(n).prime_factors()
    if (len(pqr) == 3) and (product(pqr) == n):
        coeffs = cyclotomic_coeffs(n, sparse=False)
        max_coeff = max(abs(c) for c in coeffs)
        if (max_coeff > (pqr[0]+1)//2): print(n)  # Robin Visser, Aug 17 2023

A180258 a(n) = 2^(q*r) mod p for the n-th odd number with exactly 3 distinct prime factors p < q < r.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jan 17 2011

nonn

			a(1) = 2^(5*7) mod 3 = 2, because A046389(1) = 105 = 3*5*7 and 3 < 5 < 7.

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

Cf. A046389, A180197, A180257.

More terms and corrected a(28) from Nathaniel Johnston, Jan 18 2011

A361075 Products of exactly 7 distinct odd primes.

Original entry on oeis.org

4849845, 5870865, 6561555, 7402395, 7912905, 8273265, 8580495, 8843835, 9444435, 10015005, 10140585, 10465455, 10555545, 10705695, 10818885, 10975965, 11565015, 11696685, 11996985, 12267255, 12777765, 12785955, 13096545, 13408395, 13498485, 13528515, 13667745, 13803405

Offset: 1

Karl-Heinz Hofmann, Mar 01 2023

nonn

			a(1)     =   4849845 = 3*5*7*11*13*17*19
a(9663)  = 253808555 = 5*7*11*13*17*19*157
a(9961)  = 258573315 = 3*5*7*11*13*17*1013
a(10000) = 259173915 = 3*5*7*11*13*41*421

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A065091, A046388 (2 distinct odd primes).
Cf. A046389 (3 distinct odd primes), A046390 (4 distinct odd primes).
Cf. A046391 (5 distinct odd primes), A168352 (6 distinct odd primes).

import numpy
from sympy import nextprime, sieve, primepi
k_upto = 14 * 10**6
array = numpy.zeros(k_upto,dtype="i4")
sieve_max_number = primepi(nextprime(k_upto // 255255))
for s in range(2,sieve_max_number):
    array[sieve[s]:k_upto][::sieve[s]] += 1
for s in range(2,sieve_max_number):
    array[sieve[s]**2:k_upto][::sieve[s]**2] = 0
print([k for k in range(1,k_upto,2) if array[k] == 7])

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A361075(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(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,2,1,7)))
    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,n,n) # Chai Wah Wu, Sep 10 2024

A379762 Products of 4 distinct prime numbers (or tetraprimes) that are abundant.

Original entry on oeis.org

210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1190, 1218, 1230, 1254, 1290, 1302, 1326, 1330, 1410, 1430, 1482, 1518, 1554, 1590, 1610, 1722, 1770, 1794, 1806, 1830, 1870, 1914, 1938, 1974, 2002, 2010, 2030, 2046, 2090, 2130, 2170, 2190, 2210, 2226, 2262, 2346, 2370, 2418, 2442, 2470, 2478, 2490, 2530, 2562, 2590, 2622

Offset: 1

Massimo Kofler, Jan 09 2025

nonn

This sequence is not 2*{A046389}. 2618  = 2*1309 is not in this sequence, while 1309 is in A046389.
Contains 6*p*q if p and q are distinct primes > 3. The first term not of this form is 770. - Robert Israel, Jan 09 2025
a(43) = 2002 is the only term coprime to 15. - Charles R Greathouse IV, Jan 13 2025

			210 is a term because 210=2*3*5*7 is the product of four distinct primes and it is smaller than the sum of its proper divisors 366.
1155 is not a term because 1155=3*5*7*11 is the product of four distinct primes and it is larger than the sum of its proper divisors 1149.

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

Intersection of A005101 and A046386.

Cf. A001065, A378717, A379917.

filter:= proc(n) local F,t;
   F:= ifactors(n)[2];
   F[..,2] = [1,1,1,1] and mul(t[1]+1, t = F) > 2*n
end proc:
select(filter, [seq(i,i=2..3000, 4)]); # Robert Israel, Jan 09 2025

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1, 1} && Times @@ (1 + 1/f[[;; , 1]]) > 2]; Select[Range[3000], q] (* Amiram Eldar, Jan 09 2025 *)

list(lim)=my(v=List(select(k->k<=lim, [1430, 1870, 2002, 2090, 2210, 2470, 2530, 2990, 3190, 3230, 3410, 3770, 4030, 4070, 4510, 4730, 5170, 5830]))); forprime(p=5,sqrtint(lim\6), my(t=6*p); forprime(q=p+2,lim\t, listput(v,t*q))); forprime(p=11,lim\70,listput(v,70*p)); Set(v) \\ Charles R Greathouse IV, Jan 13 2025

a(n) == 2 (mod 4).

a(n) ~ (1/6)*n log n/log log n. - Charles R Greathouse IV, Jan 13 2025

cp's OEIS Frontend

A338469 Products of three odd prime numbers of odd index.

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A075808 Palindromic odd composite numbers that are the products of an odd number of distinct primes.

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A180257 a(n) = 2^(p*r) mod q for the n-th odd number with exactly three distinct prime factors p < q < r.

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A378209 Antiderivatives of 334406399, numbers k for which A003415(k) = A024451(9) = A003415(A002110(9)).

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A075815 Palindromic odd composite numbers with an odd number of prime factors (counted with multiplicity).

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A146961 Numbers k = p*q*r, with odd primes p < q < r, such that Sister Beiter's cyclotomic coefficient conjecture is false.

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A180258 a(n) = 2^(q*r) mod p for the n-th odd number with exactly 3 distinct prime factors p < q < r.

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A361075 Products of exactly 7 distinct odd primes.

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A146961 Numbers k = pqr, with odd primes p < q < r, such that Sister Beiter's cyclotomic coefficient conjecture is false.