A002476 -id:A002476 - cp's OEIS frontend

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).

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

A154728 Products of three consecutive primes of the form 6n+1 (see A002476).

Original entry on oeis.org

1729, 7657, 21793, 49321, 97051, 175741, 298351, 386389, 559399, 789289, 1089019, 1425829, 1924177, 2665603, 3295273, 3864241, 4631971, 5694079, 6951667, 8103877, 9363547, 10775137, 12307147, 14956219, 18091147, 21243961, 24066037

Offset: 1

Omar E. Pol, Jan 18 2009, Jan 21 2009

Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

			13, 19, 31 are three consecutive primes of the form 6n+1 and 13*19*31 = 7657. - _Emeric Deutsch_, Jan 21 2009

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

Cf. A001235, A002476, A011541, A154716, A154717, A154729.

a := proc (n) if `mod`(ithprime(n), 6) = 1 then ithprime(n) else end if end proc: A := [seq(a(n), n = 1 .. 100)]: seq(A[j]*A[j+1]*A[j+2], j = 1 .. 30); # Emeric Deutsch, Jan 21 2009

Times@@@Partition[Select[Prime[Range[100]],IntegerQ[(#-1)/6]&],3,1] (* Harvey P. Dale, Jan 13 2019 *)

Extended by Emeric Deutsch, Jan 21 2009

A272200 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the first case.

Original entry on oeis.org

13, 19, 43, 61, 97, 103, 109, 127, 157, 163, 181, 193, 241, 277, 283, 331, 349, 373, 379, 409, 433, 463, 487, 499, 523, 601, 607, 619, 631, 661, 673, 691, 727, 733, 757, 769, 787, 811, 859, 883, 937, 967, 991

Offset: 1

Wolfdieter Lang, Apr 28 2016

The other primes congruent to 1 modulo 3 are given in A272201.
Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1 (see also A001479). The present sequence gives such primes corresponding to A(m+1) == 1 (mod 3). The ones corresponding to A(m+1) not == 1 (mod 3) (the complement) are given in A272201.
This bisection of the primes from A002476 is needed in the formula for the coefficients of the q-expansion (q = exp(2*Pi*i*z), Im(z) > 0) of the modular cusp form (eta(6*z))^4|A000727%20which%20gives%20the%20coefficients%20of%20the%20q-expansion%20of%20F(q)%20=%20Eta64(q%5E(1/6))/q%5E(1/6)%20=%20(Product">{z=z(q)} = Eta64(q) with Dedekind's eta function. See A000727 which gives the coefficients of the q-expansion of F(q) = Eta64(q^(1/6))/q^(1/6) = (Product{m>=0} (1 - q^m))^4. The coefficients F(q) = Sum_{n>=0} f(6*n+1)*q^n are given in the Finch link on p. 5, using multiplicativity. For primes congruent to 1 modulo 6 the formula involves x_p and y_p which are the present A and B for prime p == 1 (mod 3).
See also the p-defects of the elliptic curve y^2 = x^3 + 1, related to (eta(6*z))^4, given in A272198 with another (equivalent) way to find the coefficients of the Eta64(q) expansion, hence those of F(q).

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Cf. A000727, A001479, A002476, A001480, A272198, A272201 (complement relative to A002476).

filter:= proc(n) local S,x,y;
    if not isprime(n) then return false fi;
    S:= remove(hastype,[isolve(x^2+3*y^2=n)],negative);
    subs(S[1],x) mod 3 = 1
end proc:
select(filter, [seq(i,i=7..1000,6)]); # Robert Israel, Apr 29 2019

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] == 1];
Select[Range[7, 1000, 6], filterQ] (* Jean-François Alcover, Apr 21 2020, after Robert Israel *)

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 1 (mod 3), for m >= 1. A(m) = A001479(m+1).

A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case.

Original entry on oeis.org

7, 31, 37, 67, 73, 79, 139, 151, 199, 211, 223, 229, 271, 307, 313, 337, 367, 397, 421, 439, 457, 541, 547, 571, 577, 613, 643, 709, 739, 751, 823, 829, 853, 877, 907, 919, 997

Offset: 1

Wolfdieter Lang, Apr 28 2016

The other primes congruent to 1 modulo 3 are given in A272200, where also more details are given.

Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1 (see also A001479). The present sequence gives these primes corresponding to A(m+1) not congruent 1 modulo 3. The ones corresponding to A(m+1) == 1 (mod 3) (the complement) are given in A272200.

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

Cf. A000727, A001479, A002476, A001480, A272198, A272200 (complement relative to A002476).

filter:= proc(n) local S,x,y;
    if not isprime(n) then return false fi;
    S:= remove(hastype,[isolve(x^2+3*y^2=n)],negative);
    subs(S[1],x) mod 3 <> 1
end proc:
select(filter, [seq(i,i=7..1000,6)]); # Robert Israel, Apr 29 2019

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] != 1];
Select[Range[7, 1000, 6], filterQ] (* Jean-François Alcover, Apr 21 2020, after Robert Israel *)

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) not == 1 (mod 3), for m >= 1. A(m) = A001479(m+1).

A334481 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^2).

Original entry on oeis.org

1, 0, 3, 3, 5, 3, 7, 8, 8, 8, 4, 6, 1, 3, 5, 2, 8, 4, 3, 0, 8, 2, 8, 4, 6, 1, 8, 4, 9, 7, 6, 2, 1, 8, 3, 3, 9, 4, 7, 5, 1, 7, 6, 7, 7, 4, 8, 1, 4, 9, 1, 6, 3, 0, 1, 2, 3, 2, 4, 8, 9, 2, 5, 1, 0, 3, 2, 7, 7, 7, 7, 4, 2, 3, 9, 4, 0, 7, 0, 3, 6, 1, 5, 8, 7, 5, 3, 2, 0, 5, 9, 1, 7, 2, 4, 0, 8, 1, 4, 0, 1, 1, 7, 3, 9

Offset: 1

Vaclav Kotesovec, May 02 2020

Product_{k>=1} (1 - 1/A002476(k)^2) = 1/A175646 = 0.9671040753637981066150556834173635260473412207450...

Let Zeta_{6,1}(4) = 1/ Product_{k>=1}(1-1/A002476(k)^4) = 1.0004615089.. and Zeta_{6,1}(2)= A175646 as tabulated in arXiv:1008.2547. Then this constant equals Zeta_{6,1}(2)/Zeta_{6,1}(4). - R. J. Mathar, Jan 12 2021

			1.03353788846135284308284618497621833947517677481...

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{6,1}(4) and Zeta_{6,1}(2) in Section 3.2.

Cf. A002476, A175646, A334477, A334482.

A334481 * A334482 = 54/(5*Pi^2).

More digits from Vaclav Kotesovec, Jun 27 2020

A344473 Numbers of the form (q1^b1)(q2^b2)(q3^b3)(q4^b4)(q5^b5)... where q1=7, q2=13, q3=19, q4=31, q5=37, ... (A002476) and b1>=b2>=b3>=b4>=b5...

Original entry on oeis.org

1, 7, 49, 91, 343, 637, 1729, 2401, 4459, 8281, 12103, 16807, 31213, 53599, 57967, 84721, 117649, 157339, 218491, 375193, 405769, 593047, 753571, 823543, 1101373, 1529437, 1983163, 2626351, 2840383, 2989441, 4151329, 4877509, 5274997, 5764801, 7709611

Offset: 1

Jianing Song, May 20 2021

A343771 is a subsequence.

			12103 is a term since 12103 = 7^2 * 13 * 19.
22477 is not a term since 22477 = 7 * 13^2 * 19, the exponents are not nonincreasing.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A002476, A343771, A054994.

\\ following program for A054994
list_A344473(lim) =
{
    my(u = [1], v = List(), w = v, pr, t = 1);
    forprime(p = 7, ,
        if (p % 3 > 1, next);
        t *= p;
        if (t > lim,
            break);
        listput(w, t)
    );
    for (i = 1, #w,
        pr = 1;
        for (e = 1, logint(lim\ = 1, w[i]),
            pr *= w[i];
            for (j = 1, #u,
                t = pr * u[j];
                if (t > lim,
                    break);
                listput(v, t)
            )
        );
        if (w[i] ^ 2 < lim, u = Set(concat(Vec(v), u)); v = List());
    );
    Set(concat(Vec(v), u));
}
list_A344473(100000)

A373590 Numbers whose number of prime factors (with multiplicity) is a multiple of 3, and all of them are of the type 3m+1 (in A002476).

Original entry on oeis.org

1, 343, 637, 931, 1183, 1519, 1729, 1813, 2107, 2197, 2527, 2821, 2989, 3211, 3283, 3367, 3577, 3871, 3913, 4123, 4693, 4753, 4921, 5047, 5239, 5341, 5551, 5719, 6097, 6223, 6253, 6643, 6727, 6811, 6859, 7189, 7267, 7399, 7657, 7693, 7987, 8029, 8113, 8827, 8869, 8911, 9139, 9331, 9373, 9457, 9583, 9709, 9751, 9919

Offset: 1

Antti Karttunen, Jun 10 2024

nonn

A multiplicative semigroup: if m and n are in the sequence, then so is m*n.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Intersection of A004611 and A145784.
Subsequence of A373597 (see also A373589).
Cf. A001222, A002476.

isA373590(n) = if(bigomega(n)%3, 0, my(f = factor(n)); for(i = 1, #f~, if((f[i, 1]%3) != 1, return(0))); (1));

A272205 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 4 or 5 modulo 6.

Original entry on oeis.org

19, 37, 43, 73, 103, 127, 163, 229, 283, 313, 331, 337, 379, 397, 421, 457, 463, 487, 499, 523, 541, 577, 607, 613, 619, 631, 691, 709, 727, 787, 811, 829, 853, 859, 877, 883, 967, 991, 997

Offset: 1

Wolfdieter Lang, May 05 2016

The other part of this bisection appears in A272204.
Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. The present sequence gives such primes corresponding to A(m) == 4, 5 (mod 6). The ones corresponding to A(m) == 1, 2 (mod 6) (the complement) are given in A272205.
The corresponding A001479 entries are 4, 5, 4, 5, 10, 10, 4, 11, 16, 11, 16, 17, 4, 17, 11, 5, 10, 22, 16, 4, 23, 23, 10, 5, 16, 22, 4, 11, 22, 28, 28, 23, 29, 28, 17, 4, 10, 22, 5, ...
See  A272204 for a comment on the relevance of this bisection in connection with the signs of the q-expansion coefficients of the modular cusp form eta^{12}(12*z) / (eta^4(6*z)*eta^4(24*z)).

Cf. A001479, A001480, A002476, A047239, A187076, A272203, A272204 (complement relative to A002476).

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 4, 5 (mod 6), for m >= 1. A(m) = A001479(m+1).

A095015 Number of 6k+1 primes (A002476) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 11, 20, 37, 69, 126, 228, 434, 806, 1514, 2845, 5361, 10212, 19308, 36747, 70135, 134065, 256824, 492871, 946880, 1822913, 3513737, 6780428, 13103565, 25348226, 49090715, 95167496, 184660541, 358633265, 697094862, 1356047971, 2639884795, 5142811282

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A002476, A036378, A095016.

a(n) = A036378(n) - A095016(n) for n >= 2.

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A272204 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 1 or 2 modulo 6.

Original entry on oeis.org

7, 13, 31, 61, 67, 79, 97, 109, 139, 151, 157, 181, 193, 199, 211, 223, 241, 271, 277, 307, 349, 367, 373, 409, 433, 439, 547, 571, 601, 643, 661, 673, 733, 739, 751, 757, 769, 823, 907, 919, 937

Offset: 1

Wolfdieter Lang, May 05 2016

The other part of this bisection appears in A272205.
Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. The present sequence gives all such primes corresponding to A(m) == 1, 2 (mod 6). The ones corresponding to A(m) not == 1, 2 (mod 6) (the complement), that is == 4, 5 (mod 6), are given in A272205.
The corresponding A001479 entries are 2, 1, 2, 7, 8, 2, 7, 1, 8, 2, 7, 13, 1, 14, 8, 14, 7, 14, 13, 8, 7, 2, 19, 19, 1, 14, 20, 8, 13, 20, 19, 25, 25, 8, 26, 13, 1, 26, 20, 26, 13, ...
This bisection of the 1 (mod 3) primes A002476 is needed to determine the sign in the formula for the coefficients of the q-expansion (q = exp(2*Pi*i*z), Im(z) > 0) of the modular weight 2 cusp form
  eta^{12}(12*z) / (eta^4(6*z)*eta^4(24*z)) |A187076%20which%20gives%20the%20coefficients%20of%20the%20q-expansion%20of%20F(q)%20=%20Eta(q%5E%7B1/6%7D)%20/%20q%5E%7B1/6%7D%20=%20Product">{z=z(q)} =: Eta(q) with Dedekind's eta function. See A187076 which gives the coefficients of the q-expansion of F(q) = Eta(q^{1/6}) / q^{1/6} = Product{m>=0} (1 - q^(2*m))^{12} / ((1 - q^m)*(1 - q^(4*m)))^4. The q-expansion coefficients (called b(n)) of the modular cusp form are given there using multiplicativity. Note that there x can also be negative, whereas here A is positive.

Cf. A001479, A001480, A002476, A047239, A187076, A272203, A272205 (complement relative to A002476).

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 1, 2 (mod 6), for m >= 1. A(m) = A001479(m+1).

cp's OEIS Frontend

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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A154728 Products of three consecutive primes of the form 6n+1 (see A002476).

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A272200 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the first case.

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A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case.

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A334481 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^2).

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A344473 Numbers of the form (q1^b1)(q2^b2)(q3^b3)(q4^b4)(q5^b5)... where q1=7, q2=13, q3=19, q4=31, q5=37, ... (A002476) and b1>=b2>=b3>=b4>=b5...

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Examples

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A373590 Numbers whose number of prime factors (with multiplicity) is a multiple of 3, and all of them are of the type 3m+1 (in A002476).

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PARI

A272205 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 4 or 5 modulo 6.

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