A007528 -id:A007528 - cp's OEIS frontend

A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

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

Offset: 1

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) * (1 + 1/A007528(k)^s) = 6^s * zeta(s) / ((2^s + 1) * (3^s + 1) * zeta(2*s)).
For s > 0, Product_{k>=1} ((A007528(k)^(2*s+1) - 1) / (A007528(k)^(2*s+1) + 1)) * ((A002476(k)^(2*s+1) + 1) / (A002476(k)^(2*s+1) - 1)) = 6 * A002114(s)^2 * (4*s + 2)! / ((2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2) = Bernoulli(2*s)^2 * (4*s + 2)! * (zeta(2*s + 1, 1/6) - zeta(2*s + 1, 5/6))^2 / (8*Pi^2 * (2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2 * zeta(2*s)^2).

			1.0036025402212598967043239333321878591705394771...

Cf. A002476, A175646, A334424, A334426, A334478, A334481.

A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

A334477 * A334479 = 810*zeta(3)/Pi^6.

More digits from Vaclav Kotesovec, Jun 27 2020

A334478 Decimal expansion of Product_{k>=1} (1 - 1/A002476(k)^3).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).
For s > 1, Product_{k>=1} (1 - 1/A002476(k)^s) * (1 - 1/A007528(k)^s) = 6^s / ((2^s - 1)*(3^s - 1)*zeta(s)).

			0.996401692816036632262361122384718799965573818714...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 3 1 3 = 1/A334478).

Cf. A002476, A175646, A334425, A334427, A334477.

A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

A334478 * A334480 = 108/(91*zeta(3)).

More digits from Vaclav Kotesovec, Jun 27 2020

A024408 Perimeters of more than one primitive Pythagorean triangle.

Original entry on oeis.org

1716, 2652, 3876, 3960, 4290, 5244, 5700, 5720, 6900, 6930, 8004, 8700, 9300, 9690, 10010, 10788, 11088, 12180, 12876, 12920, 13020, 13764, 14280, 15252, 15470, 15540, 15960, 16380, 17220, 17480, 18018, 18060, 18088, 18204, 19092, 19320, 20592, 20868

Offset: 1

David W. Wilson

nonn

a(23) = 14280 is the first perimeter of 3 primitive Pythagorean triangles: {119, 7080, 7081}, {168, 7055, 7057} and {3255, 5032, 5993}. - Jean-François Alcover, Mar 14 2012

			a(1) = 1716 with precisely two primitive Pythagorean triangles (with increasing entries): {195, 748, 773} and {364, 627, 725}. From Ron Knott's link. This is the first example of the family of perimeters 12*b(k)*(b(k) + 2) with b(k) = A007528(k), for k >= 2. See the Bernstein link, p. 234, Theorem 5. a). - _Wolfdieter Lang_, Sep 24 2019

David A. Corneth, Table of n, a(n) for n = 1..10000
Leon Bernstein, Primitive Pythagorean Triples, The Fibonacci Quarterly 20.3 (1982) 227-241.
Ron Knott, Pythagorean Triples and Online Calculators
Lindsay Witcosky, Perimeters of primitive Pythagorean triangles

Cf. A007528, A024364, A103606.

A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 18, 18, 19, 21, 22, 23, 23, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 35, 37, 38, 38, 39, 40, 40, 42, 43, 44, 45, 45, 46, 47, 47, 49, 51, 52, 52, 53, 55, 56, 58, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n-th prime > 3 into a sum of 2's or 3's (inclusive or).
From Wolfdieter Lang, Mar 13 2012: (Start)
The primes of the form 6*k+1 are given in A002476.
For n >= 1 such that prime(n+2) is from A002476, one has 8*T(prime(n+2)-1) + 1 = r(n)^2, n >= 1, with the triangular numbers T(n) = A000217(n) and r(n) = A208296(n). Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=2: prime(4)=7, a(2)=1, 8*21 + 1 = 13^2 = A208296(2)^2 = 24*7*1 + 1.
The primes of the form 6*k-1 are given in A007528.
For n >= 1 such that prime(n+2) is from A007528, one has 8*T(prime(n+2)) + 1 = r(n)^2. For T and r see the preceding comment. Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=1, prime(3)=5, a(1)=1, 8*15 + 1 = 11^2 = A208296(1)^2 = 24*5*1 + 1.
(End)

[(NthPrime(n+2)+3) div 6: n in [1..80]]; // Vincenzo Librandi, Sep 06 2016

From R. J. Mathar, May 02 2010: (Start)
A103221 := proc(n) a := 0 ; for t from 0 do if 2*t > n then return a; end if; if n-2*t mod 3 = 0 then a := a+1 ; end if; end do : end proc:
A024699 := proc(n) A103221(ithprime(n+2)) ; end proc: seq(A024699(n),n=1..120) ; (End)

pi6[n_]:=Module[{p=Prime[n+2],c},c=(p-1)/6;If[IntegerQ[c],c,(p+1)/6]]; Array[pi6,80] (* Harvey P. Dale, Aug 19 2013 *)
Table[Floor[(Prime[n + 2] + 3) / 6], {n, 100}] (* Vincenzo Librandi, Sep 06 2016 *)

a(n) = (prime(n+2)+3)\6; \\ Michel Marcus, Sep 06 2016; after Wolfdieter Lang

a(n) = A103221(prime(n+2)). - R. J. Mathar, May 02 2010

a(n) = floor((prime(n+2)+3)/6), n >= 1, prime(n)=A000040(n). Consider the two cases prime(n+2) == 1 (mod 6) and == -1 (mod 6) separately. See the formula above. - Wolfdieter Lang, Mar 15 2012

A111863 a(n) is the smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.

Original entry on oeis.org

5, 11, 17, 23, 29, 5, 41, 47, 53, 59, 5, 71, 11, 83, 89, 5, 101, 107, 113, 17, 5, 131, 137, 11, 149, 5, 23, 167, 173, 179, 5, 191, 197, 29, 11, 5, 17, 227, 233, 239, 5, 251, 257, 263, 269, 5, 281, 41, 293, 23, 5, 311, 317, 17, 47, 5, 11, 347, 353, 359, 5, 53, 29, 383, 389, 5

Offset: 1

Reinhard Zumkeller, Aug 20 2005

From Robert Israel, Jan 18 2023: (Start)
a(n) = 5 if n == 1 (mod 5).
a(n) = 6*n - 1 if n is in A024898. (End)

			For n = 13, 6*n - 1 = 77 = 7*11; 7 == 1 (mod 6), but 11 == 5 (mod 6), so a(13) = 11.

G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 2, Section 2, Problem 96.

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

Cf. A007528, A016969, A024898, A107745.

f:= n -> min(select(p -> p mod 6 = 5, numtheory:-factorset(6*n-1))):
map(f, [$1..100]); # Robert Israel, Jan 18 2023

for(k=1,60,my(f=factor(6*k-1)[,1]);for(j=1,#f,if(f[j]%6==5,print1(f[j],", ");break))) \\ Hugo Pfoertner, Dec 25 2019

A138691 Numbers of the form 68+p^2 (where p is a prime).

Original entry on oeis.org

72, 77, 93, 117, 189, 237, 357, 429, 597, 909, 1029, 1437, 1749, 1917, 2277, 2877, 3549, 3789, 4557, 5109, 5397, 6309, 6957, 7989, 9477, 10269, 10677, 11517, 11949, 12837, 16197, 17229, 18837, 19389, 22269, 22869, 24717, 26637, 27957, 29997

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

68 + p^2 is divisible by 3 for any prime p > 3. - M. F. Hasler

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

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Table[2*34 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[50]]^2+68 (* Harvey P. Dale, Oct 19 2011 *)

forprime(p=1, 1e2, print1(68+p^2, ", ")) \\ Felix Fröhlich, Jul 07 2014

A138692 Numbers of the form 86+p^2 (where p is a prime).

Original entry on oeis.org

90, 95, 111, 135, 207, 255, 375, 447, 615, 927, 1047, 1455, 1767, 1935, 2295, 2895, 3567, 3807, 4575, 5127, 5415, 6327, 6975, 8007, 9495, 10287, 10695, 11535, 11967, 12855, 16215, 17247, 18855, 19407, 22287, 22887, 24735, 26655, 27975, 30015

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

86+p^2 is divisible by 3, for any prime p>3. - M. F. Hasler

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Table[2*43 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[40]]^2+86 (* Harvey P. Dale, Jan 31 2020 *)

A138693 Numbers of the form 110 + p^2. (where p is a prime).

Original entry on oeis.org

114, 119, 135, 159, 231, 279, 399, 471, 639, 951, 1071, 1479, 1791, 1959, 2319, 2919, 3591, 3831, 4599, 5151, 5439, 6351, 6999, 8031, 9519, 10311, 10719, 11559, 11991, 12879, 16239, 17271, 18879, 19431, 22311, 22911, 24759, 26679, 27999, 30039

Offset: 1

Artur Jasinski, Mar 26 2008

110+p^2 is divisible by 3, for any prime p>3. - M. F. Hasler

Cf. A000040, A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

A138693:=n->110+ithprime(n)^2: seq(A138693(n), n=1..50); # Wesley Ivan Hurt, Sep 13 2014

Table[2*55 + Prime[p + 1]^2, {p, 0, 100}]
#^2+110&/@Prime[Range[40]] (* Harvey P. Dale, Jan 21 2012 *)

a(n) = 110 + prime(n)^2; \\ Michel Marcus, Sep 14 2014

a(n) = 110 + A000040(n)^2. - Wesley Ivan Hurt, Sep 13 2014

a(n) = 110 + A001248(n). - Michel Marcus, Sep 14 2014

A307390 Primes p such that 2*p-1 is not prime.

Original entry on oeis.org

5, 11, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 223, 227, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 347, 349, 353, 359, 373, 383, 389, 397, 401, 409, 419, 421

Offset: 1

Robert Israel, Apr 17 2019

nonn

Primes not in A005382.

			a(3) = 13 is in the sequence because 13 is prime but 2*13-1 = 25 is not.

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

Cf. A005382, A053176, A109274.

Includes A007528.

select(t -> isprime(t) and not isprime(2*t-1), [seq(i,i=3..1000,2)]);

a(n) = A109274(n) + 1. - Bhavik Mehta, Aug 14 2024

A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 33, 34, 17, 10, 5, 35, 7, 12, 36, 37, 24, 22, 7, 16, 13, 38, 5, 39, 7, 12, 40, 21, 41, 27, 7, 42, 43, 10, 5, 44, 33, 12, 13, 26, 5, 45, 46, 16, 17, 10, 24, 47, 7, 48, 28, 49, 5, 22, 7, 34

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010875(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Many of the same comments as given in A319717 apply also here, except for this filter, the "blind spot" area (where only unique values are possible for a(n)) is different, and contains at least all numbers in A070003. Because presence of 2 or 3 in the prime factorization of n do not force the value of a(n) unique, this is substantially less lax (i.e., more exact) filter than A319717. Here among the first 100000 terms, only 2393 have a unique value, compared to 74355 in A319717.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319690(i) = A319690(j).

			For n = 15 (3*5) and n = 33 (3*11), the mod 6 residue of the largest prime factor is 5, also in both cases it is unitary (A319988(n) = 1), and the quotient n/A006530(n) is equal, in this case 3. Thus a(15) and a(33) are alloted the same running count (13 in this case) by rgs-transform.
For n = 2275 (5^2 * 7 * 13), n = 3325 (5^2 * 7 * 19), 5425 (5^2 * 7 * 31) and 6475 (5^2 * 7 * 37), the largest prime factor = 1 (mod 6), and A052126(n) = 175, thus these numbers are allotted the same running count (394 in this case) by rgs-transform.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A006530, A052126, A319988, A319717.
Cf. A007528 (positions of 5's), A002476 (of 7's), A112774 (after its initial term gives the position of 10's in this sequence).
Cf. also A319994 (modulo 4 analog).

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319996aux(n) = if(1==n,0,[A006530(n)%6, A052126(n), A319988(n)]);
v319996 = rgs_transform(vector(up_to,n,A319996aux(n)));
A319996(n) = v319996[n];

cp's OEIS Frontend

A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

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A334478 Decimal expansion of Product_{k>=1} (1 - 1/A002476(k)^3).

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A024408 Perimeters of more than one primitive Pythagorean triangle.

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A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

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A111863 a(n) is the smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.

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A138691 Numbers of the form 68+p^2 (where p is a prime).

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Mathematica

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A138692 Numbers of the form 86+p^2 (where p is a prime).

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Mathematica

A138693 Numbers of the form 110 + p^2. (where p is a prime).

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