A008864 -id:A008864 - cp's OEIS frontend

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.

Cf. A065463.

a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]

a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...
and gamma is the Euler-Mascheroni constant A001620. (End)

A008335 Number of distinct primes dividing p+1 as p runs through the primes.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 1, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 4, 2, 3, 3, 4, 2, 2, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 2

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001221, A008864, A023514.

for i from 1 to 500 do if isprime(i) then print(nops(factorset(i+1))); fi; od;

a[n_] := PrimeNu[Prime[n]+1]; Array[a, 100] (* Amiram Eldar, Sep 10 2024 *)

a(n) = omega(prime(n)+1); \\ Michel Marcus, Mar 29 2016

a(n) = A001221(A008864(n)). - Michel Marcus, Mar 29 2016

A123134 a(n) = prime(n)*(prime(n+1) + 1).

Original entry on oeis.org

8, 18, 40, 84, 154, 234, 340, 456, 690, 928, 1178, 1554, 1804, 2064, 2538, 3180, 3658, 4148, 4824, 5254, 5840, 6636, 7470, 8722, 9894, 10504, 11124, 11770, 12426, 14464, 16764, 18078, 19180, 20850, 22648, 23858, 25748, 27384, 29058, 31140, 32578

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 30 2006

nonn

All terms are even. - Michel Marcus, Apr 02 2017

			a(1) = 2*(3+1) = 8, a(2) = 3*(5+1) = 18, a(3) = 5*(7+1) = 40, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Frank & P. Jacqueroux, International Contest, 2001. Item 21

Cf. A000040, A008864, A286624.

[NthPrime(n)*(NthPrime(n+1) +1): n in [1..40]]; // G. C. Greubel, Aug 04 2021

a[n_]:=Prime[n](Prime[n+1]+1); Array[a, 80] (* Giovanni Resta, Apr 02 2017 *)
#[[1]](#[[2]]+1)&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Jan 06 2019 *)

for(n=1,100,print1(prime(n)*(prime(n+1)+1),","))

from sympy import prime
def a(n): return prime(n) * (prime(n + 1) + 1) # Indranil Ghosh, Apr 02 2017

a(n) = A000040(n)*A008864(n+1). - Zak Seidov, Apr 02 2017

a(n) = A286624(n) - 1. - Antti Karttunen, Jul 06 2017

A135731 a(1) = 3; thereafter a(n+1) = a(n) + nextprime(a(n)) - prevprime(a(n)).

Original entry on oeis.org

3, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278

Offset: 1

Enoch Haga, Nov 26 2007

Essentially the same as A008864. [From R. J. Mathar, Oct 28 2008]

Only the first term is prime, the rest are even, and between any pair of adjacent terms a(n) and a(n+1), there is just one prime, namely prime(n+2). - David James Sycamore, Dec 07 2018

			a(1) = 3, so a(2) = 3 + (5-2)  = 6,
a(3) = 6 + (7-5) = 8,
a(4) = 8 + (11-7) = 12; etc.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001223, A008864, A135732.

NestList[#+NextPrime[#]-NextPrime[#,-1]&,3,60] (* Harvey P. Dale, Oct 13 2024 *)

a(n+1) = a(n) + A001223(n+1) for n>1. - David James Sycamore, Dec 07 2018

Definition corrected and entry revised by David James Sycamore, Dec 07 2018

A136151 Composites n with exactly two distinct prime divisors and of the form n=1+(any prime).

Original entry on oeis.org

6, 12, 14, 18, 20, 24, 38, 44, 48, 54, 62, 68, 72, 74, 80, 98, 104, 108, 152, 158, 164, 192, 194, 200, 212, 224, 242, 272, 278, 284, 314, 332, 338, 368, 384, 398, 422, 432, 458, 464, 488, 500, 524, 542, 548, 578, 608, 614, 632, 648, 662, 674, 692, 734, 752, 758

Offset: 1

Enoch Haga, Dec 16 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A136152, A136153, A136154, A136155.

isA136151 := proc(n) if isprime(n-1) then if nops(numtheory[factorset](n)) =2 then true; else false ; fi ; else false ; fi ; end: for i from 1 to 200 do n := ithprime(i)+1 ; if isA136151( n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Feb 01 2008

Select[Range[800],PrimeNu[#]==2&&PrimeQ[#-1]&] (* Harvey P. Dale, Jun 22 2018 *)

A008864 INTERSECT A007774. - R. J. Mathar, Feb 01 2008

Edited by R. J. Mathar, Feb 01 2008

A136152 Composites one larger than a prime and with exactly three distinct prime factors.

Original entry on oeis.org

30, 42, 60, 84, 90, 102, 110, 114, 132, 138, 140, 150, 168, 174, 180, 182, 198, 228, 230, 234, 240, 252, 258, 264, 270, 282, 294, 308, 312, 318, 348, 350, 354, 360, 374, 380, 402, 410, 434, 440, 444, 450, 468, 480, 492, 504, 522, 558, 564, 572, 588, 594, 600

Offset: 1

Enoch Haga, Dec 16 2007

			a(0)=30 because 30 follows the prime 29 and has three factors 2, 3 and 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A136151, A136153, A136154, A136155.

isA008864 := proc(n) if n -prevprime(n) = 1 then true ; else false ; fi ; end: isA033992 := proc(n) if nops(numtheory[factorset](n)) = 3 then true ; else false ; fi ; end: isA136152 := proc(n) isA008864(n) and isA033992(n) ; end: for n from 1 do p := ithprime(n) ; if isA136152(p+1) then print(p+1) ; fi ; od: # R. J. Mathar, Feb 20 2008

Select[Prime[Range[110]]+1,PrimeNu[#]==3&] (* Harvey P. Dale, Apr 08 2012 *)

Find primes followed by N with exactly three prime factors, without repetition.

Equals A008864 INTERSECT A033992. - R. J. Mathar, Feb 20 2008

Edited by R. J. Mathar, Feb 20 2008

A136153 Composites one larger than a prime, with exactly four distinct prime factors.

Original entry on oeis.org

390, 420, 462, 510, 570, 660, 770, 798, 840, 858, 930, 1020, 1050, 1092, 1110, 1218, 1230, 1260, 1290, 1302, 1320, 1410, 1428, 1430, 1482, 1554, 1560, 1610, 1638, 1710, 1722, 1848, 1890, 1914, 1932, 1950, 1974, 1980, 2030, 2040, 2070, 2090, 2100, 2130

Offset: 1

Enoch Haga, Dec 16 2007

			a(0)=390 because 30 follows the prime 29 and has four prime factors 2, 3, 5 and 13.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A136151, A136152, A136154, A136155.

Select[Prime[Range[400]]+1,PrimeNu[#]==4&] (* Harvey P. Dale, Aug 15 2021 *)

isok(n) = (omega(n)==4) && isprime(n-1); \\ Michel Marcus, Jun 08 2014

Equals A008864 INTERSECT A033993. - R. J. Mathar, Feb 20 2008

Edited by R. J. Mathar, Feb 20 2008

A136155 Composites one larger than a prime and with exactly two or three distinct prime factors.

Original entry on oeis.org

6, 12, 14, 18, 20, 24, 30, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294

Offset: 1

Enoch Haga, Dec 16 2007

			a(1)=6, which is one larger than the prime 5 and has 2 distinct prime factors (namely 2 and 3).
60 is in the sequence because 59 is prime and 60 = 2^2*3*5 has three distinct prime factors.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A136151, A136152, A136153, A136154.

A001221 := proc(n) nops(numtheory[factorset](n)) ; end: isA136155 := proc(n) if isprime(n-1) then RETURN( A001221(n)=2 or A001221(n)= 3) ; else RETURN(false) ; fi ; end: for n from 1 to 300 do if isA136155(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 03 2008

okQ[n_] := PrimeQ[n-1] && (PrimeNu[n]==2 || PrimeNu[n]==3);
Select[Range[6, 300, 2], okQ] (* Jean-François Alcover, Feb 04 2023 *)

Union of A136151 and A136152. Subset of A008864. - R. J. Mathar, Apr 24 2008

A136151 UNION A136152. - R. J. Mathar, May 03 2008

Edited by R. J. Mathar and Jens Kruse Andersen, Apr 24 2008

A160696 Largest k such that k^2 divides prime(n)+1.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 2, 2, 1, 4, 1, 1, 2, 4, 3, 2, 1, 2, 6, 1, 4, 2, 3, 7, 1, 2, 6, 1, 1, 8, 2, 1, 2, 5, 2, 1, 2, 2, 1, 6, 1, 8, 1, 3, 10, 2, 4, 2, 1, 3, 4, 11, 6, 1, 2, 3, 4, 1, 1, 2, 7, 2, 2, 1, 1, 2, 13, 2, 5, 1, 6, 4, 1, 2, 8, 1, 1, 1, 1, 2, 1, 12, 1, 2, 2, 15, 1, 1, 4, 6, 4, 2, 2, 10, 6, 1, 3, 2, 1, 2, 3, 2

Offset: 1

Reinhard Zumkeller, May 24 2009

nonn

A160697 and A160698 give record values and where they occur.

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

Cf. A000188, A008864, A049097, A049098.

a(n) = core(prime(n)+1, 1)[2]; \\ Michel Marcus, Nov 06 2022

a(A049097(n)) = 1; a(A049098(n)) > 1;

a(n) = A000188(A008864(n)).

A175216 The first nonprimes after the primes.

Original entry on oeis.org

4, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272

Offset: 1

Jaroslav Krizek, Mar 06 2010

Essentially the same as A135731, A055670, A028815 and A008864. [R. J. Mathar, Mar 13 2010]

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A008864, A028815, A055670, A135731.

[n eq 1 select 4 else NthPrime(n) +1: n in [1..100]]; // G. C. Greubel, Aug 06 2024

Table[Prime[n] +1 +Boole[n==1], {n,100}] (* G. C. Greubel, Aug 06 2024 *)

def A175216(n): return nth_prime(n) +1 +int(n==1)
[A175216(n) for n in range(1,101)] # G. C. Greubel, Aug 06 2024

a(1) = 4, for n >= 2, a(n) = A008864(n) = A000040(n) + 1.

cp's OEIS Frontend

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

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A008335 Number of distinct primes dividing p+1 as p runs through the primes.

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A123134 a(n) = prime(n)*(prime(n+1) + 1).

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A135731 a(1) = 3; thereafter a(n+1) = a(n) + nextprime(a(n)) - prevprime(a(n)).

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A136151 Composites n with exactly two distinct prime divisors and of the form n=1+(any prime).

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Maple

Mathematica

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Extensions

A136152 Composites one larger than a prime and with exactly three distinct prime factors.

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Maple

Mathematica

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Extensions

A136153 Composites one larger than a prime, with exactly four distinct prime factors.

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