A108166 -id:A108166 - cp's OEIS frontend

A108164 Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).

49, 91, 133, 169, 217, 247, 259, 301, 361, 403, 427, 469, 481, 511, 553, 559, 589, 679, 703, 721, 763, 793, 817, 871, 889, 949, 961, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1261, 1267, 1273, 1333, 1339, 1351, 1369, 1387, 1393, 1417, 1477, 1501, 1561, 1591

Offset: 1

Jonathan Vos Post, Jun 13 2005

These are the products of terms from A107890 excluding multiples of 3.
Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 = the product of two primes of the form 6n+1 (A002476),
A108166 = the product of two primes of the form 6n-1 (A007528),
A108172 = the product of a prime of the form 6n+1 and a prime of the form 6n-1.
The product of two primes of the form 6n+1 is a semiprime of the form 6n+1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Robert Israel, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.

Cf. A001358, A002476, A107890.

N:= 2000: # To get all terms <= N
P:= select(isprime, [seq(i,i=7..N/7, 6)]):
sort(select(`<=`,[seq(seq(P[i]*P[j],j=1..i),i=1..nops(P))],N)); # Robert Israel, Dec 27 2018

With[{nn=50},Take[Times@@@Tuples[Select[6*Range[nn]+1,PrimeQ],2]// Union,nn]] (* Harvey P. Dale, May 20 2021 *)

{a(n)} = {p*q where both p and q are in A002476}.

Edited and extended by Ray Chandler, Oct 15 2005

A112771 Semiprimes of the form 6n + 1.

Original entry on oeis.org

25, 49, 55, 85, 91, 115, 121, 133, 145, 169, 187, 205, 217, 235, 247, 253, 259, 265, 289, 295, 301, 319, 355, 361, 391, 403, 415, 427, 445, 451, 469, 481, 493, 505, 511, 517, 529, 535, 553, 559, 565, 583, 589, 649, 655, 667, 679, 685, 697, 703, 721, 745

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

Union of A108164 and A108166.

Subsequence of A091300. - Zak Seidov, Dec 28 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

IsSemiprime:=func; [s: n in [2..150] | IsSemiprime(s) where s is 6*n + 1]; // Vincenzo Librandi, Sep 22 2012

Select[6 Range[0, 200] + 1, PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

a(n) = 6 * A112775(n) +1.

A108172 Semiprimes p*q where p is a prime of the form 6n+1 (A002476) and q is a prime of the form 6n-1 (A007528).

Original entry on oeis.org

35, 65, 77, 95, 119, 143, 155, 161, 185, 203, 209, 215, 221, 287, 299, 305, 323, 329, 335, 341, 365, 371, 377, 395, 407, 413, 437, 473, 485, 497, 515, 527, 533, 545, 551, 581, 611, 623, 629, 635, 671, 689, 695, 707, 713, 731, 737, 749, 755, 767, 779, 785

Offset: 1

Jonathan Vos Post, Jun 13 2005

Also semiprimes of the form 6n-1 (or 6n+5).
Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 - the product of two primes of the form 6n+1 (A002476),
A108166 - the product of two primes of the form 6n-1 (A007528),
A108172 - the product of a prime of the form 6n+1 and a prime of the form 6n-1.
The product of a prime of the form 6n+1 and a prime of the form 6n-1 is a semiprime of the form 6n-1.
There are 740 of these numbers below 10,000.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001358, A002476, A007528, A190299.

Select[Range[15,1000,2], Last/@FactorInteger[#]=={1,1} && IntegerQ[(#-2)/3]&] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)

list(lim)=my(v=List(),t); forprime(p=5, lim\7, if(p%6<5, next); forprime(q=7, lim\5, if(q%6>1, next); t=p*q; if(t>lim, break); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = 6 * A112776(n) + 5.

Edited by Ray Chandler, Oct 15 2005

A352274 Numbers whose squarefree part is congruent to 1 modulo 6 or 3 modulo 18.

Original entry on oeis.org

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 55, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 85, 91, 93, 97, 100, 103, 108, 109, 111, 112, 115, 117, 121, 124, 127, 129, 133, 139, 144, 145, 147, 148, 151, 156, 157, 163, 165, 169, 171, 172

Offset: 1

Peter Munn, Mar 10 2022

Numbers of the form 4^i * 3^j * (6k+1), i, j, k >= 0. Numbers whose prime factorization has an even number of factors of 2 and an even number of factors of the form 6k+5 (therefore also an even number of factors of the form 3k+2).
Closed under multiplication.
Includes the nonzero Loeschian numbers (A003136). The two sequences have few early differences (the first extra number here is a(22) = 55, followed by 85, 115, 145, ...), but their densities diverge progressively, driven by the presence here - and absence from A003136 - of the nonsquare terms of A108166. Asymptotic densities are 1/3 and 0 respectively.
Term by term, the sequence is one half of its complement within A225837.

			4 = 2^2 has square part 2^2, therefore squarefree part 4/2^2 = 1, which is congruent to 1 mod 6, so 4 is in the sequence.
63 = 3^2 * 7 has square part 3^2, therefore squarefree part 63/3^2 = 7, which is congruent to 1 mod 6, so 63 is in the sequence.
21 = 3*7 has square part 1^2 and squarefree part 21, which is congruent to 3 mod 18, so 21 is in the sequence.
72 = 2^3 * 3^2 has square part 2^2 * 3^2 = 6^2, therefore squarefree part 72/6^2 = 2, which is congruent to 2 mod 6 and to 2 mod 18, so 72 is not in the sequence.

Eric Weisstein's World of Mathematics, Squarefree Part.

Intersection of any two of A003159, A026225 and A225837.
Closure of A084089 under multiplication by 3.
Cf. A007913.
Subsequences: A003136\{0}, A108166, A352272.

isok(m) = core(m) % 6 == 1 || core(m) % 18 == 3;

from itertools import count
def A352274(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in count(0,2):
            i2 = 1<x:
                break
            for j in count(0):
                k = i2*3**j
                if k>x:
                    break
                c -= (x//k-1)//6+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

Formula

{a(n): n >= 1} = {m >= 1 : A007913(m) == 1 (mod 6)} U {m >= 1 : A007913(m) == 3 (mod 18)} = {A352272(m): m >= 1} U {3*A352272(m): m >= 1}.

{A225837(n): n >= 1} = {a(m): m >= 1} U {2*a(m): m >= 1}.

A176275 Numbers of the form 6k+1 with the least prime divisor of the form 6m-1.

Original entry on oeis.org

25, 55, 85, 115, 121, 145, 175, 187, 205, 235, 253, 265, 289, 295, 319, 325, 355, 385, 391, 415, 445, 451, 475, 493, 505, 517, 529, 535, 565, 583, 595, 625, 649, 655, 667, 685, 697, 715, 745, 775, 781, 799, 805, 835, 841, 865, 895, 901, 913, 925, 943, 955, 979, 985, 1003, 1015, 1045, 1075

Offset: 1

Vladimir Shevelev, Apr 14 2010

nonn

All terms of A108166 are in the sequence.

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

Cf. A176274, A176262, A176256, A176257, A176258, A176255, A108166, A016969, A007528.

Select[6*Range[200] + 1, IntegerQ[(FactorInteger[#][[1, 1]] + 1)/6] &] (* Harvey P. Dale, Sep 19 2018 *)

isok(n) = ((n % 6) == 1) && (n != 1) && ((vecmin(factor(n)[,1]) % 6) == 5); \\ Michel Marcus, Feb 08 2016

Corrected by R. J. Mathar, May 03 2013

A167857 Numbers whose divisors are represented by an integer polynomial.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 19, 22, 23, 25, 29, 31, 34, 37, 41, 43, 46, 47, 49, 53, 55, 58, 59, 61, 67, 71, 73, 79, 82, 83, 85, 89, 91, 94, 97, 101, 103, 106, 107, 109, 113, 115, 118, 121, 127, 131, 133, 137, 139, 142, 145, 149, 151, 157, 163, 166, 167, 169, 171

Offset: 1

T. D. Noe, Nov 13 2009

nonn

That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.

Every prime has this property, as do 1 and 9, the squares of primes of the form 6k+1, and semiprimes p*q with p and q both primes of the form 3k-1 or 3k+1. Terms of the form p^2*q also appear. We can find terms of the form p^m for any m. For example, 2311^13 is the smallest 13th power that appears. For any m, it seems that p^m appears for p a prime of the form k*m#+1, where m# is the product of the primes up to m. Are there terms with three distinct prime divisors?

			The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A108164, A108166, A112774 (forms of semiprimes)
Cf. A002476 (primes of the form 6k+1)
Cf. A132230 (primes of the form 30k+1)
Cf. A073103 (primes of the form 210k+1)
Cf. A073917 (least prime of the form k*prime(n)#+1)

Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]

is(n)=my(d=divisors(n));denominator(content(polinterpolate([0..#d-1],d))) == 1 \\ Charles R Greathouse IV, Jan 29 2016

A176278 Numbers of the form 6k-1 with the least prime divisor of the form 6m+1.

Original entry on oeis.org

77, 119, 161, 203, 221, 287, 299, 329, 371, 377, 413, 437, 497, 533, 539, 551, 581, 611, 623, 689, 707, 749, 767, 779, 791, 833, 893, 917, 923, 959, 1001, 1007, 1043, 1079, 1121, 1127, 1157, 1169, 1211, 1253, 1271, 1313, 1337, 1349, 1379, 1391, 1421, 1457

Offset: 1

Vladimir Shevelev, Apr 14 2010

nonn

By definition, all terms are composite numbers.

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

Cf. A176275, A176274, A176262, A176258, A176257, A176256, A176255, A108166, A016969, A007528.

Select[Range[11,2581,6],1==Mod[FactorInteger[ # ][[1,1]],6]&] (* Zak Seidov, Apr 14 2010 *)

isok(n) = ((n % 6) == 5) && ((vecmin(factor(n)[,1]) % 6) == 1); \\ Michel Marcus, Feb 08 2016

Corrected (erroneous 341 deleted) and extended by Zak Seidov, Apr 14 2010

cp's OEIS Frontend

A108164 Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).

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A112771 Semiprimes of the form 6n + 1.

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A108172 Semiprimes p*q where p is a prime of the form 6n+1 (A002476) and q is a prime of the form 6n-1 (A007528).

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A352274 Numbers whose squarefree part is congruent to 1 modulo 6 or 3 modulo 18.

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A176275 Numbers of the form 6k+1 with the least prime divisor of the form 6m-1.

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A167857 Numbers whose divisors are represented by an integer polynomial.

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A176278 Numbers of the form 6k-1 with the least prime divisor of the form 6m+1.

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