A084920 -id:A084920 - cp's OEIS frontend

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

513, 20708, 584874, 4714408, 72449100, 200562418, 1012788198, 1953009460, 6172747128, 24788658690, 37242612640, 107770200778, 198936710910, 265200653548, 449592659568, 931777815258, 1775665528380, 2155635964450, 3812897562148, 5368106367720, 6351988507678

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), A297493 (m=8), this sequence (m=10).

Cf. A000594, A259825, A295645, A297127.

Let b(n) = 42*n^6 - 90*n^4 - 75*n^3 - 35*n^2 - 9*n - 1.
a(n) = b(prime(n)) - tau(prime(n)) where tau(n)=A000594(n) is Ramanujan's tau function.
So tau(prime(n)) + 1 == -a(n) (mod prime(n)).

A056813 Largest non-unitary prime factor of LCM(1,...,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Labos Elemer, Aug 28 2000

For n>0, prime(n) appears {(prime(n+1))^2 - (prime(n))^2} times [from n=(prime(n))^2 to n=(prime(n+1))^2 - 1], that is, A000040(n) appears A069482(n) times (from n=A001248(n) to n=A084920(n+1)). - Lekraj Beedassy, Mar 31 2005
a(n) is the largest prime factor of A045948(n). [Matthew Vandermast, Oct 29 2008]
Alternative definition: a(n) = largest prime <= sqrt(n) (considering 1 as prime for this occasion, see A008578 for the 19th century definition of primes). - Jean-Christophe Hervé, Oct 29 2013

			The j-th prime appears at the position of its square, at n = prime(j)^2.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Cf. A054041, A056170.

Cf. A000040, A001248, A008578, A069482.

Table[f = Transpose[FactorInteger[LCM @@ Range[n]]]; pos = Position[f[[2]], ?(# > 1 &)]; If[pos == {}, 1, f[[1, pos[[-1]]]][[1]]], {n, 100}] (* _T. D. Noe, Oct 30 2013 *)

a(n) = prime(w) if prime(w)^2 <= n < prime(w+1)^2.

To get the sequence, repeat 1 three times, and then for any k >= 1, repeat A000040(k) A069482(k) times; or equivalently, for any k >= 1, repeat A008578(k) a number of times equal to A008578(k+1)^2 - A008578(k)^2. - Jean-Christophe Hervé, Oct 29 2013

Corrected offset by Jean-Christophe Hervé, Oct 29 2013

A087713 Greatest prime factor of the product of the neighbors of the n-th prime.

Original entry on oeis.org

3, 2, 3, 3, 5, 7, 3, 5, 11, 7, 5, 19, 7, 11, 23, 13, 29, 31, 17, 7, 37, 13, 41, 11, 7, 17, 17, 53, 11, 19, 7, 13, 23, 23, 37, 19, 79, 41, 83, 43, 89, 13, 19, 97, 11, 11, 53, 37, 113, 23, 29, 17, 11, 7, 43, 131, 67, 17, 139, 47, 71, 73, 17, 31, 157, 79, 83, 13, 173, 29, 59

Offset: 1

Reinhard Zumkeller, Sep 28 2003

Apparently a(n) = A024710(n) for n>4. - Georg Fischer, Oct 06 2018

Conjecture: The record values are A120628 \ {2}. - Jason Yuen, Jan 19 2025

			a(10) = A006530(prime(10)^2 - 1) = A006530(29*29-1) = A006530(840) = A006530(7*5*3*2^3) = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Davide Rotondo and others, A120628 and record numbers of a087713, Conversation in the SeqFan Google Group, Dec 25 2024.

Cf. A006530, A084920, A120628.

a087713 = a006530 . a084920  -- Reinhard Zumkeller, Aug 27 2013

FactorInteger[#][[-1,1]]&/@((#-1)(#+1)&/@Prime[Range[80]]) (* Harvey P. Dale, Oct 26 2019 *)

a(n) = my(p=prime(n)); vecmax(factor((p-1)*(p+1))[, 1]); \\ Michel Marcus, Jan 20 2025

from sympy import prime, primefactors
def A087713(n): p = prime(n); return max(primefactors(p*p-1))  # Ya-Ping Lu, Mar 07 2025

a(n) = A006530((A000040(n)-1)*(A000040(n)+1)) = A006530(A006093(n)*A008864(n)) = A006530(A084920(n)).

a(n) <= (prime(n)+1)/2, n > 1. - Ya-Ping Lu, Apr 10 2025

Definition clarified by Harvey P. Dale, Oct 26 2019

A275630 a(n) = product of distinct primes dividing prime(n)^2 - 1.

Original entry on oeis.org

3, 2, 6, 6, 30, 42, 6, 30, 66, 210, 30, 114, 210, 462, 138, 78, 870, 930, 1122, 210, 222, 390, 1722, 330, 42, 510, 1326, 318, 330, 798, 42, 4290, 2346, 4830, 1110, 570, 6162, 246, 3486, 7482, 2670, 2730, 570, 582, 462, 330, 11130, 1554, 12882, 13110, 2262, 3570, 330, 210, 258, 8646, 2010

Offset: 1

N. J. A. Sloane, Aug 07 2016

nonn

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

Subsequence of A007947.

Cf. A077063, A077066, A084920.

a[n_] := Times @@ FactorInteger[Prime[n]^2 - 1][[;; , 1]]; Array[a, 60] (* Amiram Eldar, Jan 30 2021 *)

a(n) = factorback(factorint(prime(n)^2-1)[, 1]); \\ Michel Marcus, Jan 30 2021

a(n) = A007947(A084920(n)). - Michel Marcus, Jan 30 2021

a(n) = A077063(n)*A077066(n)/2, for n > 1. - Amiram Eldar, Jan 30 2021

A281958 Primes p such that p^2 - 1 is not a totient number (A002202).

Original entry on oeis.org

2, 173, 317, 509, 709, 773, 787, 947, 1307, 1447, 1579, 1613, 1627, 1867, 2347, 2467, 2693, 3307, 3413, 3547, 3803, 3923, 4007, 4243, 4567, 4597, 4723, 4793, 4813, 4937, 4973, 5227, 5261, 5387, 5483, 5557, 5653, 5717, 5827, 5843, 6277, 6397, 6547, 6653, 6793, 6907

Offset: 1

Altug Alkan, Feb 03 2017

nonn

Corresponding values of p^2 - 1 are 3, 29928, 100488, 259080, 502680, 597528, 619368, 896808, 1708248, ...

			Prime number 173 is a term because 173^2 - 1 = 29928 is not a totient number.

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

Cf. A000010, A002202, A007617, A084920.

is(n) = isprime(n) && !istotient(n^2-1);

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A355433 Numbers k such that k is sqrt(k)-smooth and k+1 is sqrt(k+1)-smooth.

Original entry on oeis.org

8, 24, 48, 49, 63, 80, 120, 125, 168, 175, 195, 224, 242, 288, 324, 350, 351, 360, 363, 374, 384, 399, 440, 441, 455, 475, 494, 512, 528, 539, 560, 575, 594, 624, 675, 714, 728, 735, 759, 832, 840, 874, 896, 935, 960, 968, 1000, 1014, 1023, 1044, 1053, 1088, 1104

Offset: 1

Amiram Eldar, Jul 02 2022

nonn

Numbers k such that k and k+1 are both in A048098.

This sequence is infinite: if p is an odd prime then p^2-1 is a term.

			8 is a term since 8 is sqrt(8)-smooth (2^2 <= 8) and 9 is sqrt(9)-smooth (3^2 <= 9).

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

Cf. A048098, A355434.

Subsequences: A084920 \ {3}, A060355, A348119.

smQ[n_] := FactorInteger[n][[-1, 1]]^2 <= n; Select[Range[1000], smQ[#] && smQ[# + 1] &]

A318766 a(0) = 1; for n > 0, a(n) = (prime(n)^2 - 1) * a(n-1).

Original entry on oeis.org

1, 3, 24, 576, 27648, 3317760, 557383680, 160526499840, 57789539942400, 30512877089587200, 25630816755253248000, 24605584085043118080000, 33660439028338985533440000, 56549537567609495696179200000, 104503545424942348046539161600000

Offset: 0

Seiichi Manyama, Sep 03 2018

nonn

The limit of A061742(n)/a(n) is zeta(2) (cf. A013661).

Seiichi Manyama, Table of n, a(n) for n = 0..195

Product_{k=1..n} (prime(k)^m - 1): A005867 (m=1), this sequence (m=2).

Cf. A000040, A013661, A061742, A084920.

a[n_]:=Product[Prime[k]^2-1, {k, 1, n}]; Join[{1},Array[a, nmax]] (* Stefano Spezia, Sep 03 2018 *)

PARI
```
{a(n) = prod(k=1, n, prime(k)^2-1)}
```

a(n) = A084920(n) * a(n-1) for n > 0.

a(n) = Product_{k=1..n} (prime(k)^2 - 1).

A323278 Numbers of the form p^2-1 that have a record-breaking number of divisors, where p is prime.

Original entry on oeis.org

3, 8, 24, 48, 120, 288, 360, 840, 1680, 5040, 11880, 32760, 143640, 201600, 491400, 776160, 2042040, 3500640, 7447440, 9480240, 17297280, 34234200, 143256960, 514337040, 555120720, 569729160, 1656408600, 4283571600, 8148853440, 10951831800, 35415099720, 51437786400

Offset: 1

G. L. Honaker, Jr., Jan 11 2019

nonn

a(11)-a(26) from Chuck Gaydos.

			a(7) = 360 because 360 has a record-breaking 24 divisors and 360 = p^2-1, where p = 19 is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..63
Chris Caldwell and G. L. Honaker, Jr., Prime Curio for 23869

Cf. A000005, A084920.

Block[{s = Prime[Range[10^5]]^2 - 1, t}, t = DivisorSigma[0, s]; Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Jan 19 2019 *)

lista(nn) = {my(m = 0, p = 2, np); for (n=1, nn, np = p^2-1; if (((nd = numdiv(np)) > m), print1(np, ", "); m = nd); p = nextprime(p+1););} \\ Michel Marcus, Jan 12 2019

from sympy import divisor_count, nextprime
A323278_list, p, nmax = [], 2 , -1
while len(A323278_list) < 100:
    n = divisor_count(p**2-1)
    if n > nmax:
        nmax = n
        A323278_list.append(p**2-1)
    p = nextprime(p) # Chai Wah Wu, Feb 09 2019

a(27)-a(32) from Daniel Suteu, Jan 12 2019

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

			1.25361794500723589001447178...

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Mathematica
```
RealDigits[N[3465/2764, 105]][[1]]
```

default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.

cp's OEIS Frontend

A297494 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^10*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

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A056813 Largest non-unitary prime factor of LCM(1,...,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).

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A087713 Greatest prime factor of the product of the neighbors of the n-th prime.

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A275630 a(n) = product of distinct primes dividing prime(n)^2 - 1.

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Mathematica

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A281958 Primes p such that p^2 - 1 is not a totient number (A002202).

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PARI

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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Mathematica

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A355433 Numbers k such that k is sqrt(k)-smooth and k+1 is sqrt(k+1)-smooth.

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Mathematica

A318766 a(0) = 1; for n > 0, a(n) = (prime(n)^2 - 1) * a(n-1).

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A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.