Thomas Kellar - cp's OEIS frontend

A175907 Known friendly squarefree numbers.

6, 30, 42, 66, 78, 102, 114, 138, 174, 186, 210, 222, 246, 258, 273, 282, 318, 330, 354, 366, 390, 402, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 642, 654, 678, 690, 714, 762, 786, 798, 806, 822, 834, 858, 870, 894, 906, 930, 942, 966, 978, 1002, 1038

Offset: 1

Thomas Kellar, Oct 14 2010, Oct 15 2010

nonn

From Walter Nissen, May 28 2011: (Start)
As with most aspects of friendly and solitary numbers, this sequence is not known to be complete. A friend could possibly be found for 10, for example; same doubtful status as an odd perfect number.
Note that not all friendly numbers will be found among the primitive friendly numbers listed in link "Primitive Friendly Pairs", and this would be true even if those were not limited to small examples.
Other terms are 1330, 1995, and 49166.
(End)

			6, being 2 * 3, is squarefree. Having abundancy = 2, 6 is friendly with all the other perfect numbers. Ergo, it is in the sequence. ( 1 ), 2, 3, and 5, being prime powers, are solitary. 4 is a square. Ergo, a(1) is 6.

Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp. 96-100.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Primitive Friendly Pairs
Walter Nissen, Abundancy: some resources
Eric Weisstein's World of Mathematics, Solitary Number

Cf. A005117, A014567, A074902 (known friendly numbers), A095751, A096366, A140688.

{ for (j=1,2000, if (issquarefree(j), t=sigma(j)/j; for (i=1,1000000, p=sigma(i)/i; if(p == t && j != i, print(j," ",i); ); ); ); ); quit; } \\ provides useful suggestions, but not definitive, Walter Nissen, May 28 2011

Added 273 as it is friendly with 2876211; 273 is a counterexample to the conjecture that 6 divides a(n). - Walter Nissen, May 28 2011

Added 806 as it is friendly with 2449562488893. - Suyash Pandit, Jan 24 2024

A180617 Sum of divisors of the product of two consecutive primes.

Original entry on oeis.org

12, 24, 48, 96, 168, 252, 360, 480, 720, 960, 1216, 1596, 1848, 2112, 2592, 3240, 3720, 4216, 4896, 5328, 5920, 6720, 7560, 8820, 9996, 10608, 11232, 11880, 12540, 14592, 16896, 18216, 19320, 21000, 22800, 24016, 25912, 27552, 29232, 31320, 32760, 34944, 37248, 38412

Offset: 1

Thomas Kellar, Sep 12 2010

nonn

			a(1) = sigma(2*3) = 12, a(2) = sigma(3*5) = 24.

A distant relative of A054640.

Cf. A000203, A001043, A006094, A126199.

[(1+NthPrime(n))*(1+NthPrime(n+1)): n in [1..50]]; // Vincenzo Librandi, Feb 16 2015

DivisorSigma[1,#]&/@(Times@@@Partition[Prime[Range[50]],2,1]) (* Harvey P. Dale, Apr 04 2015 *)
Table[Prime[n]*Prime[n+1]+Prime[n]+Prime[n+1]+1,{n,1,30}] (* Metin Sariyar, Dec 08 2019 *)

for (n=1,10, i=prod(x=n,n+1,prime(x)); p=sigma(i); print1(p, ", "); )

a(n)=my(p=prime(n)); (p+1)*(nextprime(p+1)+1) \\ Charles R Greathouse IV, Feb 16 2015

a(n) = A000203(A006094(n)). - Omar E. Pol, Dec 08 2019
a(n) = A006094(n) + A001043(n) + 1. - Metin Sariyar, Dec 08 2019
a(n) = A126199(n) + 1 (after above formula). - Omar E. Pol, Dec 08 2019

More terms from Vincenzo Librandi, Feb 16 2015

Name simplified by Omar E. Pol, Dec 08 2019

A038802 Factor 2n+1 = (2^m1)(3^m2)(5^m3)*...; a(n) = number of initial zero exponents.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 1, 6, 7, 1, 8, 2, 1, 9, 10, 1, 2, 11, 1, 12, 13, 1, 14, 3, 1, 15, 2, 1, 16, 17, 1, 2, 18, 1, 19, 20, 1, 3, 21, 1, 22, 2, 1, 23, 3, 1, 2, 24, 1, 25, 26, 1, 27, 28, 1, 29, 2, 1, 3, 4, 1, 2, 30, 1, 31, 3, 1, 32, 33, 1, 4, 2, 1, 34, 35, 1, 2, 36

Offset: 1

Thomas Kellar

nonn

			9 = (2^0)*(3^2), thus a(4) = 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

A038802 := proc(n) numtheory[factorset](2*n+1) ; min(%); numtheory[pi](%)-1 ; end proc: # R. J. Mathar, Mar 01 2011

Table[f = FactorInteger[2 n + 1]; PrimePi[f[[1, 1]]] - 1, {n, 100}] (* T. D. Noe, Apr 23 2013 *)

lpf(n)=factor(n)[1,1]
a(n)=primepi(lpf(2*n+1))-1 \\ Charles R Greathouse IV, Jul 29 2016

a(n) = A049084(A020639(2n+1))-1. - R. J. Mathar, Mar 01 2011

a(69) corrected by Rick G. Rosner, Apr 22 2013

A045536 Primes p such that p+2 and 2p+1 are also prime.

Original entry on oeis.org

3, 5, 11, 29, 41, 179, 191, 239, 281, 419, 431, 641, 659, 809, 1019, 1031, 1049, 1229, 1289, 1451, 1481, 1931, 2129, 2141, 2339, 2549, 2969, 3299, 3329, 3359, 3389, 3539, 3821, 3851, 4019, 4271, 4481, 5231, 5279, 5441, 5501, 5639, 5741, 5849, 6131

Offset: 1

Thomas Kellar

Intersection of A001359 and A005384. - Zak Seidov, Feb 28 2017

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001359, A005384.

[p: p in PrimesUpTo(6200) | IsPrime(p+2) and IsPrime(2*p+1)]; // Vincenzo Librandi, Apr 08 2013

select(t -> isprime(t) and isprime(t+2) and isprime(2*t+1), [3, seq(t,t=5..10000,6)]); # Robert Israel, Feb 28 2017

Select[Prime[Range[1000]], PrimeQ[#+2] && PrimeQ[2#+1]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011*)

is(n)=isprime(n) && isprime(n+2) && isprime(2*n+1) \\ Charles R Greathouse IV, Feb 25 2014

Corrected by Jud McCranie, Dec 30 2000

Name changed and incorrect comment and program removed by T. D. Noe, Aug 05 2010

A049472 a(n) = floor(n/sqrt(2)).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47

Offset: 0

Thomas Kellar

For n > 0: A006337(n) = number of repeating n's. - Reinhard Zumkeller, Jul 04 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Index entries for sequences related to Beatty sequences

First differences give A080764.

Cf. A006337, A049474.

a049472 = floor . (/ sqrt 2) . fromIntegral
-- Reinhard Zumkeller, Jul 04 2015

[Floor(n/Sqrt(2)): n in [0..70] ]; // Vincenzo Librandi, Aug 23 2011

A049472:=n->floor(n/sqrt(2)): seq(A049472(n), n=0..100); # Wesley Ivan Hurt, Jun 25 2016

Floor[Range[0,70]/Sqrt[2]] (* Harvey P. Dale, Aug 22 2011 *)

a(n)=sqrtint(n^2\2) \\ Charles R Greathouse IV, Sep 02 2015

A038869 Primes p such that both p-2 and 2p-1 are prime.

Original entry on oeis.org

7, 19, 31, 139, 199, 229, 271, 601, 619, 661, 811, 829, 1279, 1429, 1609, 2029, 2089, 2131, 2311, 2551, 2791, 3169, 3331, 3391, 3529, 3769, 4051, 4159, 4231, 4261, 4339, 4639, 4801, 5419, 5479, 5659, 5851, 6271, 6301, 6361, 6691, 6961, 7561, 7951, 8539

Offset: 1

Thomas Kellar

nonn

Primes p such that A(2*p) - 3*A(p) = 3 (7, 31, 661, 811, 2551, ...) and primes p such that 7*A(p) - A(2*p) = 21 (19, 139, 619, 1429, ...), where A=A288814, are both subsequences of A038869. - David James Sycamore, Aug 07 2017

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A005382, A118071.

[n: n in [0..10000]|IsPrime(n) and IsPrime(n-2) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 18 2010

Transpose[Select[Partition[Prime[Range[1200]],2,1],#[[2]]-#[[1]]==2 && PrimeQ[2#[[2]]-1]&]][[2]] (* Harvey P. Dale, Jun 19 2014 *)

is(n)=n%6==1 && isprime(n-2) && isprime(n) && isprime(2*n-1) \\ Charles R Greathouse IV, Aug 09 2017

cp's OEIS Frontend

User: Thomas Kellar

A175907 Known friendly squarefree numbers.

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A180617 Sum of divisors of the product of two consecutive primes.

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Magma

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A038802 Factor 2n+1 = (2^m1)*(3^m2)*(5^m3)*...; a(n) = number of initial zero exponents.

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A045536 Primes p such that p+2 and 2p+1 are also prime.

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Magma

Maple

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A049472 a(n) = floor(n/sqrt(2)).

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Haskell

Magma

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A038869 Primes p such that both p-2 and 2p-1 are prime.

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Magma

Mathematica

PARI

A038802 Factor 2n+1 = (2^m1)(3^m2)(5^m3)*...; a(n) = number of initial zero exponents.