A077063 -id:A077063 - cp's OEIS frontend

A077064 Squarefree numbers of form prime - 1.

1, 2, 6, 10, 22, 30, 42, 46, 58, 66, 70, 78, 82, 102, 106, 130, 138, 166, 178, 190, 210, 222, 226, 238, 262, 282, 310, 330, 346, 358, 366, 382, 418, 430, 438, 442, 462, 466, 478, 498, 502, 546, 562, 570, 586, 598, 606, 618, 642, 646, 658, 682, 690, 718, 742

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

This sequence is infinite and its relative density in the sequence of primes is equal to Artin's constant (A005596): Product_{p prime} (1-1/(p*(p-1))) = 0.373955... (Victorovich, 2013). - Amiram Eldar, Dec 29 2020

			A005117(44) = 70 = 2*5*7 is a term as 70 = A000040(20)-1 = 71-1.

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222. See section 3.4, p. 210.
G. D. Victorovich, On additive property of arithmetic functions (in Russian), Thesis, Moscow State University, 2013.

Cf. A005596, A006093, A005117, A000040, A077063, A077067.

Select[Prime[Range[200]]-1,SquareFreeQ] (* Harvey P. Dale, Feb 09 2015 *)

isok(n) = issquarefree(n) && isprime(n+1); \\ Michel Marcus, Mar 22 2016

lista(nn) = forprime(p=2, nn, if (issquarefree(p-1), print1(p-1, ", "))); \\ Michel Marcus, Mar 22 2016

Wrong formula removed by Amiram Eldar, Dec 29 2020

A077066 Squarefree kernel of prime(n) + 1.

Original entry on oeis.org

3, 2, 6, 2, 6, 14, 6, 10, 6, 30, 2, 38, 42, 22, 6, 6, 30, 62, 34, 6, 74, 10, 42, 30, 14, 102, 26, 6, 110, 114, 2, 66, 138, 70, 30, 38, 158, 82, 42, 174, 30, 182, 6, 194, 66, 10, 106, 14, 114, 230, 78, 30, 22, 42, 258, 66, 30, 34, 278, 282, 142, 42, 154, 78, 314, 318, 166

Offset: 1

Reinhard Zumkeller, Oct 23 2002

			a(25) = rad(prime(25)+1) = rad(97+1) = rad(2*7^2) = 14.

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

Cf. A007947, A008864, A077063, A077067.

a077066 = a007947 . a008864  -- Reinhard Zumkeller, Sep 04 2013

a[n_] := Times @@ FactorInteger[Prime[n] + 1][[;;, 1]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

a(n)=my(f=factor(prime(n)+1)[,1]); prod(i=1,#f,f[i]) \\ Charles R Greathouse IV, Aug 21 2013

a(n) = A007947(A008864(n)).

a(A077067(n)) = A077067(n).

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

A356474 a(n) = phi(rad(prime(n)-1)), where phi = A000010 and rad = A007947.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 1, 2, 10, 6, 8, 2, 4, 12, 22, 12, 28, 8, 20, 24, 2, 24, 40, 10, 2, 4, 32, 52, 2, 6, 12, 48, 16, 44, 36, 8, 24, 2, 82, 42, 88, 8, 72, 2, 6, 20, 48, 72, 112, 36, 28, 96, 8, 4, 1, 130, 66, 8, 44, 24, 92, 72, 32, 120, 24, 78, 80, 12, 172, 56, 10, 178, 120, 60

Offset: 1

Jianing Song, Aug 09 2022

			prime(8) = 19, so a(8) = phi(rad(18)) = phi(6) = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A007947, A077063, A173557.

Array[EulerPhi[Times @@ FactorInteger[Prime[#] - 1][[All, 1]]] &, 74] (* Michael De Vlieger, Aug 09 2022 *)

a(n) = my(f=factor(prime(n)-1)[, 1]); prod(k=1, #f, f[k]-1) \\ following Michel Marcus's program for A173557

a(n) = Product_{primes p dividing prime(n)-1} (p-1).

a(n) = phi(A077063(n)) = A173557(prime(n)-1).

A356489 a(n) = A000265(rad(prime(n)-1)), rad = A007947.

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 1, 3, 11, 7, 15, 3, 5, 21, 23, 13, 29, 15, 33, 35, 3, 39, 41, 11, 3, 5, 51, 53, 3, 7, 21, 65, 17, 69, 37, 15, 39, 3, 83, 43, 89, 15, 95, 3, 7, 33, 105, 111, 113, 57, 29, 119, 15, 5, 1, 131, 67, 15, 69, 35, 141, 73, 51, 155, 39, 79, 165, 21, 173, 87, 11, 179

Offset: 1

Jianing Song, Aug 09 2022

			prime(8) = 19, so a(8) = A000265(rad(18)) = A000265(6) = 3.
prime(11) = 31, so a(11) = A000265(rad(30)) = A000265(30) = 15.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000265, A007947, A077063, A057023, A204455.

Array[#/2^IntegerExponent[#, 2] &[Times @@ FactorInteger[Prime[#] - 1][[All, 1]]] &, 72] (* Michael De Vlieger, Aug 09 2022 *)

a(n) = factorback(setminus(factorint(prime(n)-1)[, 1]~, [2]))

a(n) = A000265(rad(prime(n)-1)) = rad(A000265(prime(n)-1)).
a(n) = Product_{odd primes p dividing prime(n)-1} p.
a(n) = A000265(A077063(n)) = rad(A057023(n)) = A204455(prime(n)-1).

cp's OEIS Frontend

A077064 Squarefree numbers of form prime - 1.

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A077066 Squarefree kernel of prime(n) + 1.

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

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A356474 a(n) = phi(rad(prime(n)-1)), where phi = A000010 and rad = A007947.

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A356489 a(n) = A000265(rad(prime(n)-1)), rad = A007947.

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