A079079 -id:A079079 - cp's OEIS frontend

A079085 Number of distinct prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 2, 3, 4, 3, 2, 3, 4, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 3, 3, 4, 5, 4, 4, 3, 3, 4, 4, 4, 5, 4, 3, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 5, 4, 3, 4, 3, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 3, 4, 5, 5, 3, 4, 5, 5, 4, 3, 4, 5, 4, 4, 4, 4, 4, 4, 5, 4, 4, 3, 3, 4, 4, 5, 6, 4

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A079086.

a079085 = a001221 . a079079  -- Reinhard Zumkeller, Oct 08 2012

Table[PrimeNu[(Prime[n] + 1)*(Prime[n + 1] + 1)/4], {n, 1, 50}] (* G. C. Greubel, Apr 25 2017 *)

for(n=1,50, print1(omega((prime(n) + 1)*(prime(n + 1) + 1)/4), ", ")) \\ G. C. Greubel, Apr 25 2017

a(n) = A001221(A079079(n)).

A079086 Total number of prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

1, 2, 3, 4, 3, 3, 4, 5, 5, 6, 5, 3, 4, 6, 7, 6, 4, 3, 6, 5, 5, 7, 6, 5, 4, 5, 7, 6, 4, 8, 9, 5, 5, 6, 6, 4, 3, 6, 6, 6, 6, 8, 7, 4, 7, 6, 7, 8, 5, 5, 8, 7, 6, 6, 6, 8, 8, 5, 3, 4, 5, 6, 7, 5, 3, 4, 4, 5, 6, 5, 7, 9, 6, 5, 10, 10, 4, 3, 4, 6, 5, 7, 8, 6, 7, 7, 5, 4, 7, 8, 10, 9, 6, 7, 9, 8, 6, 5, 3, 3, 5, 6, 6, 6

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A079079, A079085.

a079086 = a001222 . a079079  -- Reinhard Zumkeller, Oct 08 2012

a[n_] := PrimeOmega[(Prime[n]+1) * (Prime[n+1]+1) / 4]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

a(n) = bigomega((prime(n)+1)*(prime(n+1)+1)/4); \\ Amiram Eldar, Apr 06 2025

a(n) = A001222(A079079(n)).

A079080 a(n) = gcd((prime(n)+1)(prime(n+1)+1)/4, prime(n)prime(n+1)+1).

Original entry on oeis.org

1, 2, 12, 6, 6, 3, 18, 6, 4, 60, 4, 3, 42, 6, 4, 2, 30, 2, 6, 36, 8, 6, 2, 3, 3, 204, 6, 54, 3, 48, 6, 2, 138, 6, 300, 4, 2, 6, 4, 2, 90, 12, 96, 3, 396, 10, 14, 6, 114, 3, 8, 120, 6, 2, 4, 4, 540, 4, 3, 282, 6, 6, 6, 156, 3, 6, 2, 6, 174, 3, 4, 6, 4, 2, 6, 4, 3, 3, 15, 6, 210, 12, 216, 4, 6, 2

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A023523, A079079, A079081, A079082.

a079080 n = a079079 n `gcd` a023523 (n + 1)
-- Reinhard Zumkeller, Oct 09 2012

a[n_] := Module[{p = Prime[n], q}, q = NextPrime[p]; GCD[(p+1) * (q+1) / 4, p*q + 1]]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

a(n) = my(p = prime(n), q = nextprime(p+1)); gcd((p+1)*(q+1)/4, p*q+1); \\ Amiram Eldar, Apr 06 2025

a(n) = gcd(A079079(n), A023523(n+1)).

A079081 Numerator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).

Original entry on oeis.org

3, 3, 1, 4, 7, 21, 5, 20, 45, 4, 76, 133, 11, 88, 162, 405, 31, 527, 204, 37, 185, 280, 945, 735, 833, 13, 468, 55, 1045, 76, 704, 2277, 35, 875, 19, 1501, 3239, 1148, 1827, 3915, 91, 728, 97, 3201, 25, 1060, 848, 2128, 115, 4485, 1755, 121, 2541, 8127, 4257, 4455

Offset: 1

Reinhard Zumkeller, Dec 22 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Denominator = A079082.

Cf. A023523, A079079, A079080.

a079081 n = a079081_list !! (n-1)
a079081_list = zipWith div a079079_list a079080_list
-- Reinhard Zumkeller, Oct 09 2012

a[n_] := Module[{p = Prime[n], q}, q = NextPrime[p]; Numerator[(p+1) * (q+1) / (4 * (p*q + 1))]]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

a(n) = my(p = prime(n), q = nextprime(p+1)); numerator((p+1) * (q+1) / (4 * (p*q + 1))); \\ Amiram Eldar, Apr 06 2025

a(n) = numerator(A079079(n)/A023523(n+1)).

a(n) = A079079(n)/A079080(n).

A079082 Denominator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).

Original entry on oeis.org

7, 8, 3, 13, 24, 74, 18, 73, 167, 15, 287, 506, 42, 337, 623, 1564, 120, 2044, 793, 144, 721, 1093, 3694, 2878, 3266, 51, 1837, 216, 4106, 299, 2773, 8974, 138, 3452, 75, 5927, 12796, 4537, 7223, 15484, 360, 2881, 384, 12674, 99, 4199, 3361, 8437, 456

Offset: 1

Reinhard Zumkeller, Dec 22 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Numerator = A079081.

Cf. A023523, A079079, A079080.

a079082 n = a079082_list !! (n-1)
a079082_list = zipWith div (tail a023523_list) a079080_list
-- Reinhard Zumkeller, Oct 09 2012

((#[[1]]+1)(#[[2]]+1))/(4(Times@@#+1))&/@Partition[Prime[Range[50]],2,1]//Denominator (* Harvey P. Dale, Jan 01 2018 *)

a(n) = my(p = prime(n), q = nextprime(p+1)); denominator((p+1) * (q+1) / (4 * (p*q + 1))); \\ Amiram Eldar, Apr 06 2025

a(n) = denominator(A079079(n)/A023523(n+1)).

a(n) = A023523(n+1)/A079080(n).

A079089 Sum of divisors of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

4, 12, 28, 60, 96, 104, 234, 360, 546, 744, 620, 640, 1152, 1488, 1815, 2178, 2304, 1728, 3510, 3458, 3420, 5952, 5760, 4446, 4104, 7056, 8400, 8640, 5760, 10160, 12240, 11232, 13824, 14976, 17360, 11200, 10080, 20160, 21840, 21600, 26208

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A079079, A079088, A079090, A079091.

a079089 = a000203 . a079079  -- Reinhard Zumkeller, Oct 08 2012

a[n_] := DivisorSigma[1, (Prime[n]+1) * (Prime[n+1]+1) / 4]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

a(n) = sigma((prime(n)+1)*(prime(n+1)+1)/4); \\ Amiram Eldar, Apr 06 2025

a(n) = A000203(A079079(n)).

A079090 Sum of distinct prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

3, 5, 5, 5, 12, 10, 10, 10, 10, 10, 21, 29, 23, 16, 5, 10, 41, 50, 22, 42, 44, 17, 17, 15, 27, 35, 18, 21, 38, 24, 16, 39, 40, 17, 29, 100, 122, 53, 41, 39, 30, 25, 102, 111, 21, 60, 62, 31, 52, 44, 23, 21, 23, 55, 59, 21, 27, 158, 189, 123, 83, 23, 36, 175, 213, 141, 98, 47

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008472, A079079, A079085, A079089, A079091.

a079090 = a008472 . a079079  -- Reinhard Zumkeller, Oct 09 2012

Total[Transpose[FactorInteger[((#[[1]]+1)(#[[2]]+1))/4]][[1]]]&/@ Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Apr 24 2015 *)

a(n) = vecsum(factor((prime(n)+1)*(prime(n+1)+1)/4)[, 1]); \\ Amiram Eldar, Apr 07 2025

a(n) = A008472(A079079(n)).

A079091 Sum of all prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

3, 5, 7, 9, 12, 13, 13, 14, 15, 16, 27, 29, 23, 22, 18, 19, 41, 50, 29, 47, 48, 23, 23, 25, 34, 37, 28, 27, 38, 34, 28, 42, 40, 27, 36, 102, 122, 57, 46, 45, 33, 33, 110, 114, 31, 69, 70, 39, 52, 47, 33, 36, 37, 61, 64, 32, 37, 162, 189, 123, 90, 37, 40, 177, 213, 141, 111, 60

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001414, A079079, A079086, A079089, A079090.

a079091 = a001414 . a079079  -- Reinhard Zumkeller, Oct 09 2012

Total[Times@@@FactorInteger[#]]&/@(Times@@(#+1)/4&/@Partition[ Prime[Range[100]],2,1]) (* Harvey P. Dale, May 26 2011 *)

a(n) = my(f = factor((prime(n)+1)*(prime(n+1)+1)/4)); f[, 1]~ * f[, 2]; \\ Amiram Eldar, Apr 07 2025

a(n) = A001414(A079079(n)).

A079093 Squarefree kernel of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

3, 6, 6, 6, 42, 21, 30, 30, 30, 30, 38, 399, 462, 66, 6, 30, 930, 1054, 102, 222, 370, 210, 210, 105, 357, 1326, 78, 330, 3135, 114, 66, 1518, 4830, 210, 570, 3002, 6478, 1722, 1218, 870, 2730, 546, 582, 3201, 330, 530, 742, 798, 13110, 4485, 390, 330, 462, 1806

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007947, A079079, A079095.

a079093 = a007947 . a079079  -- Reinhard Zumkeller, Oct 09 2012

Times@@Transpose[FactorInteger[#]][[1]]&/@(Times@@@Partition[ Prime[ Range[ 60]]+1,2,1]/4) (* Harvey P. Dale, Jan 06 2013 *)

a(n) = factorback(factor((prime(n)+1)*(prime(n+1)+1)/4)[,1]); \\ Michel Marcus, Feb 14 2018

a(n) = A007947(A079079(n)).

A079084 Greatest prime factor of (prime(n)+1)*(prime(n+1)+1)/4.

Original entry on oeis.org

3, 3, 3, 3, 7, 7, 5, 5, 5, 5, 19, 19, 11, 11, 3, 5, 31, 31, 17, 37, 37, 7, 7, 7, 17, 17, 13, 11, 19, 19, 11, 23, 23, 7, 19, 79, 79, 41, 29, 29, 13, 13, 97, 97, 11, 53, 53, 19, 23, 23, 13, 11, 11, 43, 43, 11, 17, 139, 139, 71, 71, 11, 13, 157, 157, 83, 83, 29, 29, 59, 59, 23, 23, 19

Offset: 1

Reinhard Zumkeller, Dec 22 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A006530, A079079.

a079084 = a006530 . a079079  -- Reinhard Zumkeller, Oct 08 2012

a[n_]:=FactorInteger[(Prime[n]+1)(Prime[n+1]+1)/4][[-1,1]]; Table[a[n],{n,1,74}] (* Jean-François Alcover, Mar 18 2011 *)
FactorInteger[(First[#]+1) (Last[#]+1)/4][[-1,1]]&/@Partition[Prime[ Range[100]],2,1] (* Harvey P. Dale, Nov 14 2011 *)

a(n) = my(f=factor((prime(n)+1)*(prime(n+1)+1)/4)); f[#f~,1]; \\ Michel Marcus, Nov 12 2023

a(n) = A006530(A079079(n)).

cp's OEIS Frontend

A079085 Number of distinct prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

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A079086 Total number of prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

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A079080 a(n) = gcd((prime(n)+1)*(prime(n+1)+1)/4, prime(n)*prime(n+1)+1).

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A079081 Numerator of (prime(n)+1)*(prime(n+1)+1)/(4*(prime(n)*prime(n+1)+1)).

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A079082 Denominator of (prime(n)+1)*(prime(n+1)+1)/(4*(prime(n)*prime(n+1)+1)).

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A079089 Sum of divisors of (prime(n)+1)*(prime(n+1)+1)/4.

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Haskell

Mathematica

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A079090 Sum of distinct prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

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Haskell

Mathematica

PARI

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A079091 Sum of all prime factors of (prime(n)+1)*(prime(n+1)+1)/4.

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A079080 a(n) = gcd((prime(n)+1)(prime(n+1)+1)/4, prime(n)prime(n+1)+1).

A079081 Numerator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).

A079082 Denominator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).