A053193 -id:A053193 - cp's OEIS frontend

A036689 Product of a prime and the previous number.

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332, 1640, 1806, 2162, 2756, 3422, 3660, 4422, 4970, 5256, 6162, 6806, 7832, 9312, 10100, 10506, 11342, 11772, 12656, 16002, 17030, 18632, 19182, 22052, 22650, 24492, 26406, 27722, 29756, 31862, 32580, 36290, 37056, 38612, 39402, 44310

Offset: 1

Felice Russo

Records in A002618. - Artur Jasinski, Jan 23 2008

Also records in A174857. - Vladimir Shevelev, Mar 31 2010

			2*1, 3*2, 5*4, 7*6, 11*10, 13*12, 17*16, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to prime powers

Cf. A000040, A001248, A002618, A005596, A053650, A053192, A053193, A053650, A082695, A117495, A136141.
Twice the terms of A008837.
Subsequence of A002378 (oblong numbers).
Column 1 of A257251. (Row 1 of A257252.)
Column 2 of A379010.

a036689 n = a036689_list !! (n-1)
a036689_list = zipWith (*) a000040_list $ map pred a000040_list
-- Reinhard Zumkeller, Sep 17 2011

[ n*(n-1): n in PrimesUpTo(220) ]; // Bruno Berselli, Apr 11 2011

A036689 := proc(n) local p ; p := ithprime(n) ; p*(p-1) ; end proc: # R. J. Mathar, Apr 11 2011

Table[Prime[n] EulerPhi[Prime[n]], {n, 100}] (* Artur Jasinski, Jan 23 2008 *)
Table[Prime[n] (Prime[n] - 1), {n, 1, 50}] (* Bruno Berselli, Apr 22 2014 *)
#(#-1)&/@Prime[Range[50]] (* Harvey P. Dale, Sep 08 2019 *)

forprime(p=2,1e3,print1(p^2-p", ")) \\ Charles R Greathouse IV, Jun 10 2011

A036689(n) = ((p->(p-1)*p)(prime(n))); \\ Antti Karttunen, Dec 14 2024

(define (A036689 n) (* (A000040 n) (- (A000040 n) 1))) ;; Antti Karttunen, May 01 2015

a(n) = prime(n) * (prime(n) - 1).
a(n) = phi(prime(n)^2) = A000010(A001248(n)).
a(n) = prime(n) * phi(prime(n)). - Artur Jasinski, Jan 23 2008
From Reinhard Zumkeller, Sep 17 2011: (Start)
a(n) = A000040(n) * A006093(n) = A001248(n) - A000040(n).
A006530(a(n)) = A000040(n). (End)
a(n) = A009262(prime(n)). - Enrique Pérez Herrero, May 12 2012
a(n) = prime(n)! mod (prime(n)^2). - J. M. Bergot, Apr 10 2014
a(n) = 2*A008837(n). - Antti Karttunen, May 01 2015
Sum_{n>=1} 1/a(n) = A136141. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)*zeta(3)/zeta(6) (A082695).
Product_{n>=1} (1 - 1/a(n)) = A005596. (End)

Deleted two incorrect comments. - N. J. A. Sloane, May 07 2020

A053650 Cototient function of n^2.

Original entry on oeis.org

0, 2, 3, 8, 5, 24, 7, 32, 27, 60, 11, 96, 13, 112, 105, 128, 17, 216, 19, 240, 189, 264, 23, 384, 125, 364, 243, 448, 29, 660, 31, 512, 429, 612, 385, 864, 37, 760, 585, 960, 41, 1260, 43, 1056, 945, 1104, 47, 1536, 343, 1500, 969, 1456, 53, 1944, 825, 1792, 1197

Offset: 1

Labos Elemer, Feb 18 2000

Seems to be invertible like n*Phi(n). Compare with A002618, A038040.

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

Cf. A000005, A038040.

Cf. A001248, A002618, A053650, A053192, A053193, A036689, A229099.

List([1..60], n-> n*(n- Phi(n)) ); # G. C. Greubel, May 18 2019

a053650 = a051953 . a000290  -- Reinhard Zumkeller, Jan 21 2014

[n*(n-EulerPhi(n)): n in [1..60]]; // Vincenzo Librandi, Jul 27 2016

Table[n(n-EulerPhi[n]), {n, 60}] (* Michael De Vlieger, Jul 26 2016 *)

a(n) = n^2 - eulerphi(n^2) \\ Michel Marcus, Jul 27 2013

[n*(n - euler_phi(n)) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = n*(n - phi(n)) = n^2 - n*phi(n) = Cototient(n^2) = A051953(A000290(n)).
a(n) = n^2 - A002618(n).
For p prime, Cototient(p)=1 and a(p)=p.
a(n) = n*cototient(n) = n*A051953(n). - Omar E. Pol, Nov 22 2012
Dirichlet g.f.: zeta(s-2)*(1 - 1/zeta(s-1)). - Ilya Gutkovskiy, Jul 26 2016
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023

A299714 Irregular triangle read by rows: row n contains numbers k such that 1<=k<=2n+1 and gcd(n-k, 2n+1) != 1.

Original entry on oeis.org

1, 2, 3, 1, 4, 7, 5, 6, 1, 2, 4, 7, 10, 12, 13, 8, 9, 1, 3, 4, 7, 10, 13, 16, 17, 19, 11, 2, 7, 12, 17, 22, 1, 4, 7, 10, 13, 16, 19, 22, 25, 14, 15, 1, 4, 5, 7, 10, 13, 16, 19, 22, 25, 27, 28, 31, 2, 3, 7, 10, 12, 17, 22, 24, 27, 31, 32, 18, 1, 4, 6, 7, 10, 13, 16, 19, 22, 25, 28, 31, 32, 34, 37, 20, 21, 1, 2, 4, 7, 10, 12, 13

Offset: 1

Joerg Arndt, Feb 17 2018

Row n contains 2*n+1 - phi(2*n+1) = A053193(n) terms. Row n has just one term (namely n) if 2*n+1 is prime.

If we extend each row [r,s, ..., z] to infinite length by appending [j*N*r,j*N*s, ..., j*N*z] for j>=1 and N=2*n+1 we obtain a relation on pairs of positive numbers (u,v). Write R(u,v) if v appears in row u (the relation is 1 on the pair (u,v)). Then R(u,u), R(u,v) ==> R(v, u), R(u,v) ==> R(u, v mod (2*u+1) ), and R(u,v) and v > u ==> R(u-(v-u), v-(v-u)) = R(2*u - v, u).

			Triangle starts:
[01]: [1]
[02]: [2]
[03]: [3]
[04]: [1, 4, 7]
[05]: [5]
[06]: [6]
[07]: [1, 2, 4, 7, 10, 12, 13]
[08]: [8]
[09]: [9]
[10]: [1, 3, 4, 7, 10, 13, 16, 17, 19]
[11]: [11]
[12]: [2, 7, 12, 17, 22]
[13]: [1, 4, 7, 10, 13, 16, 19, 22, 25]
[14]: [14]
[15]: [15]
[16]: [1, 4, 5, 7, 10, 13, 16, 19, 22, 25, 27, 28, 31]
[17]: [2, 3, 7, 10, 12, 17, 22, 24, 27, 31, 32]
[18]: [18]
...

Paolo Xausa, Table of n, a(n) for n = 1..11968 (rows 1..250 of the triangle, flattened)

Cf. A300288.

T:= n-> select(k-> igcd(n-k, 2*n+1)<>1, [$1..2*n+1])[]:
seq(T(n), n=1..25);  # Alois P. Heinz, Mar 09 2018

A299714row[n_]:=With[{q=2n+1},If[PrimeQ[q],{n},Select[Range[q],GCD[n-#,q]!=1&]]];Array[A299714row,20] (* Paolo Xausa, Nov 10 2023 *)

is(n,k)= ( gcd(n-k, 2*n+1)!=1 );
for (n=1, 33, for (k=1, 2*n+1, if ( is(n,k), print1(k,", ") ); ); );

T(n,k) = A300288(n,k) - n.

A300288 Irregular triangle read by rows: row n lists the numbers k such that -n <= k <= n and gcd(k, 2*n+1) != 1.

Original entry on oeis.org

0, 0, 0, -3, 0, 3, 0, 0, -6, -5, -3, 0, 3, 5, 6, 0, 0, -9, -7, -6, -3, 0, 3, 6, 7, 9, 0, -10, -5, 0, 5, 10, -12, -9, -6, -3, 0, 3, 6, 9, 12, 0, 0, -15, -12, -11, -9, -6, -3, 0, 3, 6, 9, 11, 12, 15, -15, -14, -10, -7, -5, 0, 5, 7, 10, 14, 15, 0, -18, -15, -13, -12, -9, -6, -3, 0, 3, 6, 9, 12, 13, 15, 18, 0

Offset: 1

Joerg Arndt, Mar 02 2018

Row n contains 2*n+1 - phi(2*n+1) = A053193(n) terms. Row n has just one term (namely 0) if 2*n+1 is prime.

			Triangle starts:
[01]:  0,
[02]:  0,
[03]:  0,
[04]: -3, 0, 3,
[05]:  0,
[06]:  0,
[07]: -6, -5, -3, 0, 3, 5, 6,
[08]:  0,
[09]:  0,
[10]: -9, -7, -6, -3, 0, 3, 6, 7, 9,
[11]:  0,
[12]: -10, -5, 0, 5, 10,
[13]: -12, -9, -6, -3, 0, 3, 6, 9, 12,
[14]:  0,
[15]:  0,
[16]: -15, -12, -11, -9, -6, -3, 0, 3, 6, 9, 11, 12, 15,
[17]: -15, -14, -10, -7, -5, 0, 5, 7, 10, 14, 15,
[18]:  0,
[19]: -18, -15, -13, -12, -9, -6, -3, 0, 3, 6, 9, 12, 13, 15, 18,
[20]:  0,
...

Paolo Xausa, Table of n, a(n) for n = 1..11968 (rows 1..250 of the triangle, flattened)

Cf. A053193, A299714.

A300288row[n_]:=With[{q=2n+1},If[PrimeQ[q],{0},Select[Range[-n, n],GCD[#, q]!=1&]]];Array[A300288row,20] (* Paolo Xausa, Nov 10 2023 *)

is(n, k)= ( gcd(k, 2*n+1)!=1 );
for (n=1, 33, for (k=-n, +n, if (is(n, k), print1(k, ", ") ); ); );

T(n,k) = A299714(n,k) - n.

cp's OEIS Frontend

A036689 Product of a prime and the previous number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Scheme

Formula

Extensions

A053650 Cototient function of n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Magma

Mathematica

PARI

Sage

Formula

A299714 Irregular triangle read by rows: row n contains numbers k such that 1<=k<=2*n+1 and gcd(n-k, 2*n+1) != 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A300288 Irregular triangle read by rows: row n lists the numbers k such that -n <= k <= n and gcd(k, 2*n+1) != 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A299714 Irregular triangle read by rows: row n contains numbers k such that 1<=k<=2n+1 and gcd(n-k, 2n+1) != 1.