A008837 -id:A008837 - 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

A371201 a(n) = Sum_{k=prime(n)..prime(n+1)-1} k, with a(0) = 1.

Original entry on oeis.org

1, 2, 7, 11, 34, 23, 58, 35, 82, 153, 59, 201, 154, 83, 178, 297, 333, 119, 381, 274, 143, 453, 322, 513, 740, 394, 203, 418, 215, 442, 1673, 514, 801, 275, 1435, 299, 921, 957, 658, 1017, 1053, 359, 1855, 383, 778, 395, 2454, 2598, 898, 455, 922, 1413, 479, 2455, 1521, 1557, 1593

Offset: 0

Raul Prisacariu, Mar 15 2024

The sequence can be obtained graphically using the following grid walk rules. From an origin the first movement iteration consists of moving 1 unit in any direction. The n-th movement iteration consists of moving in the same direction n units. If n is a prime number, the movement iteration consists of first changing the movement direction by 90 degrees and then moving n units in the new direction. If n is a nonprime number, the movement iteration consists of moving n units in the same direction as the previous movement iteration. The sequence is obtained by measuring the length of each 90-degree turn.

a(0) is the length of the grid segment before doing any 90-degree turns and a(1) is the length of the first 90-degree turn.

			a(0) = 1.
a(1) = 2.
a(2) = 3 + 4 = 7.
a(3) = 5 + 6 = 11.
a(4) = 7 + 8 + 9 + 10 = 34.
a(5) = 11 + 12 = 23.
a(6) = 13 + 14 + 15 + 16 = 58.
a(7) = 17 + 18 = 35.
The natural numbers are summed in groups where each prime begins a new group,
  primes     v   v       v       v
         1   2   3   4   5   6   7   8   9  10  ...
        \-/ \-/ \-----/ \-----/ \-------------/
  a(n) = 1   2     7       11          34
    n  = 0   1     2       3           4

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000040, A054265, A138383.

Cf. A008837 (partial sums).

ithprime(0):=1:
a:= n-> ((j, k)-> (k-1+j)*(k-j)/2)(map(ithprime, [n, n+1])[]):
seq(a(n), n=0..56);  # Alois P. Heinz, Mar 16 2024

Join[{1},Table[Prime[n]+(Prime[n+1]+Prime[n])*(Prime[n+1]-Prime[n]-1)/2,{n,56}]] (* James C. McMahon, Apr 20 2024 *)

first(n) = {
	my(res = primes(n), t = 0);
	for(i = 1, n,
		res[i] = binomial(res[i],2) - t;
		t+=res[i];
	);
	res	
} \\ David A. Corneth, Mar 16 2024

from sympy import nextprime, prime
def A371201(n):
    if n == 0: return 1
    q = nextprime(p:=prime(n))
    return (q-p)*(p+q-1)>>1 # Chai Wah Wu, Jun 01 2024

For n > 0, a(n) = A138383(n) - (prime(n+1) - prime(n)).
a(n) = binomial(prime(n+1), 2) - Sum_{k=0..n-1} a(k). - David A. Corneth, Mar 15 2024
a(n) = prime(n) + A054265(n), for n >= 1. - Michel Marcus, Mar 15 2024
a(n) = (prime(n+1)-prime(n))*(prime(n+1)+prime(n)-1)/2 for n>=1. - Chai Wah Wu, Jun 01 2024

More terms from Michel Marcus, Mar 15 2024

A126995 a(n) = binomial(prime(n+2), 3).

Original entry on oeis.org

1, 10, 35, 165, 286, 680, 969, 1771, 3654, 4495, 7770, 10660, 12341, 16215, 23426, 32509, 35990, 47905, 57155, 62196, 79079, 91881, 113564, 147440, 166650, 176851, 198485, 209934, 234136, 333375, 366145, 419220, 437989, 540274, 562475, 632710, 708561, 762355

Offset: 0

Artur Jasinski, Jan 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A008837, A058077, A125550, A126993, A126994, A126996, A126997, A126998.

[Binomial(NthPrime(n), 3): n in [3..40]]; // Vincenzo Librandi, May 10 2017

Table[Binomial[Prime[n + 2], Prime[2]], {n, 1, 40}]
Table[Binomial[Prime[n], 3], {n, 3, 40}] (* Vincenzo Librandi, May 10 2017 *)

a(n)=binomial(prime(n+2),3) \\ Charles R Greathouse IV, May 10 2017

[binomial(nth_prime(n+2), 3) for n in (1..40)] # G. C. Greubel, May 29 2019

a(n) ~ (n log n)^3 / 6. - Charles R Greathouse IV, May 10 2017

Missing n=0 term added by N. J. A. Sloane, May 17 2020

A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

Original entry on oeis.org

5, 39, 68, 203, 333, 410, 689, 915, 1314, 1958, 2328, 2525, 2943, 3164, 4658, 5513, 6123, 7439, 8145, 9264, 9653, 13053, 13514, 14460, 16448, 18023, 19113, 19670, 21389, 24414, 25043, 28308, 30363, 31064, 34689, 37733, 39303, 40100, 41718, 44205, 46764, 50288

Offset: 1

Juri-Stepan Gerasimov, Dec 11 2009

nonn

The halves of even numbers of the form p(p-1)/2 for p prime.

Sum of the quadratic residues of primes of the form 4k + 1. For example, a(3)=68 because 17 is the 3rd prime of the form 4k + 1 and the quadratic residues of 17 are 1, 4, 9, 16, 8, 2, 15, 13 which sum to 68. This sum is also the sum of the quadratic nonresidues. Cf. A230077. - Geoffrey Critzer, May 07 2015

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.21 p. 110.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A005098, A007742, A008837.

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Table[Table[Mod[a^2, p], {a, 1, (p - 1)/2}] // Total, {p,
Select[Prime[Range[100]], Mod[#, 4] == 1 &]}] (* Geoffrey Critzer, May 07 2015 *)
Select[(# (#-1))/4&/@Prime[Range[100]],IntegerQ] (* Harvey P. Dale, Dec 24 2022 *)

lista(nn) = forprime(p=2, nn, if ((p % 4)==1, print1(p*(p-1)/4, ", "))); \\ Michel Marcus, Mar 23 2016

Corrected (16448 inserted, 25043 inserted) by R. J. Mathar, May 22 2010

A126996 a(n) = binomial(prime(3+n), prime(3)).

Original entry on oeis.org

1, 21, 462, 1287, 6188, 11628, 33649, 118755, 169911, 435897, 749398, 962598, 1533939, 2869685, 5006386, 5949147, 9657648, 13019909, 15020334, 22537515, 29034396, 41507642, 64446024, 79208745, 87541245, 106308566, 116828271, 140364532, 254231775

Offset: 0

Artur Jasinski, Jan 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A008837, A058077, A125550, A126993, A126995, A126997, A126998.

[Binomial(NthPrime(n+3), 5): n in [0..30]]; // Vincenzo Librandi, May 21 2019

Table[Binomial[Prime[n + 3], Prime[3]], {n, 0, 30}]
Binomial[Prime[Range[3,40]],5] (* Harvey P. Dale, Mar 20 2021 *)

vector(30, n, binomial(prime(n+3), 5)) \\ G. C. Greubel, May 29 2019

[binomial(nth_prime(n+3), 5) for n in (1..30)] # G. C. Greubel, May 29 2019

Missing n=0 term added by N. J. A. Sloane, May 17 2020

A126997 a(n) = binomial(prime(4+n), prime(4)).

Original entry on oeis.org

1, 330, 1716, 19448, 50388, 245157, 1560780, 2629575, 10295472, 22481940, 32224114, 62891499, 154143080, 341149446, 436270780, 869648208, 1329890705, 1629348612, 2898753715, 4151918628, 6890268572, 12846240784, 17199613200, 19813501785, 26075972546, 29796772356, 38620298376

Offset: 0

Artur Jasinski, Jan 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A058077, A125550, A126993, A126993, A008837, A126995, A126996, A126998.

[Binomial(NthPrime(n+4), 7): n in [1..30]]; // Vincenzo Librandi, May 21 2019

Table[Binomial[Prime[n + 4], Prime[4]], {n, 1, 30}]

vector(30, n, binomial(prime(n+4), prime(4)) ) \\ G. C. Greubel, May 29 2019

[binomial(nth_prime(n+4), 7) for n in (1..30)] # G. C. Greubel, May 29 2019

Terms a(24) onward added by G. C. Greubel, May 30 2019

Missing n=0 term added by N. J. A. Sloane, May 17 2020

A126998 a(n) = binomial(prime(n+5), prime(5)).

Original entry on oeis.org

1, 78, 12376, 75582, 1352078, 34597290, 84672315, 854992152, 3159461968, 5752004349, 17417133617, 76223753060, 279871768995, 418094152866, 1285063345176, 2560547383576, 3558497368608, 9036996468045, 16141841823510, 36519676207704, 99468442390512, 158940114100040

Offset: 0

Artur Jasinski, Jan 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A058077, A125550, A126993, A126993, A008837, A126995, A126996, A126997.

[Binomial(NthPrime(n+5), NthPrime(5)): n in [1..30]]; // G. C. Greubel, May 29 2019

Table[Binomial[Prime[n+5], Prime[5]], {n, 1, 30}]

vector(30, n, binomial(prime(n+5), prime(5)) ) \\ G. C. Greubel, May 29 2019

[binomial(nth_prime(n+5), nth_prime(5)) for n in (1..30)] # G. C. Greubel, May 29 2019

Terms a(19) onward added by G. C. Greubel, May 30 2019

Missing n=0 term added by N. J. A. Sloane, May 17 2020

A138426 a(n) = ((prime(n))^5-prime(n))/5.

Original entry on oeis.org

6, 48, 624, 3360, 32208, 74256, 283968, 495216, 1287264, 4102224, 5725824, 13868784, 23171232, 29401680, 45868992, 83639088, 142984848, 168919248, 270025008, 360845856, 414614304, 615411264, 787808112, 1116811872

Offset: 1

Artur Jasinski, Mar 19 2008

Number of monic irreducible polynomials of degree 5 over GF(prime(n)). - Robert Israel, Jan 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wikipedia, Necklace polynomial

Cf. A008837, A127919, A138420, A208536.

[(NthPrime((n))^5 - NthPrime((n)))/5: n in [1..30] ]; // Vincenzo Librandi, Jun 18 2011

seq((ithprime(i)^5-ithprime(i))/5, i = 1 .. 50); # Robert Israel, Jan 07 2015

a = {}; Do[p = Prime[n]; AppendTo[a, (p^5 - p)/5], {n, 1, 50}]; a
(#^5-#)/5&/@Prime[Range[30]] (* Harvey P. Dale, Mar 12 2018 *)

forprime(p=2,1e3,print1((p^5-p)/5", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A138404(n)/5. - R. J. Mathar, Oct 15 2017

A217983 If n = floor(p/2) * p^e, for some (by necessity unique) prime p and exponent e > 0, then a(n) = p, otherwise a(n) = 1.

Original entry on oeis.org

1, 2, 3, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Johannes W. Meijer, Oct 25 2012

nonn

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere. - The original name of the sequence.
The a(n) are related to the prime numbers A000040 and the number of nonzero quadratic residues modulo the n-th prime A130290, see the first formula and the Maple program.
This sequence resembles the exponential of the von Mangoldt function A014963; for the latter sequence a(A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere.
Positions of the first occurrence of each successive noncomposite number (and also the records) is given by the union of {2} and A008837. - Antti Karttunen, Jan 17 2025

Antti Karttunen, Table of n, a(n) for n = 1..100949

Cf. A000040, A014963, A128060, A130290, A160479.
Cf. A000079, A000244 (after their initial 1's, the positions of 2's and 3's respectively), A020699 (positions of 5's from its third term 10 onward), A169634 (positions of 7's from the second term onward), A379956 (positions of terms > 1).
Cf. A008837, A121057.

nmax := 78: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: for n from 1 to nmax do A217983(n) := 1 od: for n from 1 to nmax do for n1 from 1 to floor(log[A000040(n)](nmax)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: seq(A217983(n), n=1..nmax);

A217983(n) = { my(f=factor(n)); for(i=1,#f~,if((n/(f[i,1]^f[i,2])) == (f[i,1]\2), return(f[i,1]))); (1); }; \\ Antti Karttunen, Jan 16 2025

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n)= 1 elsewhere.

a(n) = (A160479(n+1) * A128060(n+1))/(2*n+1) for n >= 2.

Definition simplified, original definition moved to comments; more terms added by Antti Karttunen, Jan 16 2025

A006308 Coefficients of period polynomials.

Original entry on oeis.org

3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050

Offset: 2

N. J. A. Sloane

nonn

Conjecture: a(n) = A008837(n) = p*(p-1)/2 = Sum_{k=0..p-1} mod(k^3, p) where p = prime(n). - Michael Somos, Feb 17 2020

D. H. and Emma Lehmer, Cyclotomy for nonsquarefree moduli, pp. 276-300 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 243.

Sean A. Irvine, The polynomials f(p,x) for primes p=3..101

Cf. A008837. [From R. J. Mathar, Oct 28 2008]

For an odd prime p, let g be a primitive root of p^2, q=g^p, and zeta=exp(2*pi*i/p^2). Define h(p,k) = Sum_{j=0..p-2} zeta^((q+k*p)*q^j) and a polynomial f(p,x) = Product_{k=0..p-1} (x-h(p,k)). Finally, a(n) = -[x^(p-2)] f(p,x) where p = A000040(n) is the n-th prime. - Sean A. Irvine, Mar 07 2017

More terms and offset corrected by Sean A. Irvine, Mar 07 2017

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

A371201 a(n) = Sum_{k=prime(n)..prime(n+1)-1} k, with a(0) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A126995 a(n) = binomial(prime(n+2), 3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A126996 a(n) = binomial(prime(3+n), prime(3)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Extensions

A126997 a(n) = binomial(prime(4+n), prime(4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica