A322976 -id:A322976 - cp's OEIS frontend

A067513 Number of divisors d of n such that d+1 is prime.

1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 2, 1, 4, 1, 4, 1, 4, 1, 3, 1, 5, 1, 2, 1, 4, 1, 5, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 4, 1, 4, 1, 3, 1, 8, 1, 2, 1, 4, 1, 5, 1, 3, 1, 4, 1, 8, 1, 2, 1, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 2, 1, 5, 1, 6, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1

Offset: 1

Amarnath Murthy, Feb 12 2002

1, 2 and 4 are the only numbers such that for every divisor d, d+1 is a prime.
These and only these primes appear as prime divisors of any term of InvPhi(n) set if n is not empty, i.e., if n is from A002202. - Labos Elemer, Jun 24 2002
a(n) is the number of integers k such that n = k - k/p where p is one of the prime divisors of k. (See, e.g., A064097 and A333123, which are related to the mapping k -> k - k/p.) - Robert G. Wilson v, Jun 12 2022

			a(12) = 5 as the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the corresponding primes are 2,3,5,7 and 13. Only 3+1 = 4 is not a prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty, arXiv:2208.06704 [math.NT], 2022.
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty II, arXiv:2209.01087 [math.NT], 2022-2023.
Steve Fan and Carl Pomerance, Shifted-prime divisors, arXiv:2401.10427 [math.NT], 2024.
Mikhail R. Gabdullin, Moments of the shifted prime divisor function, arXiv preprint (2025). arXiv:2505.24050 [math.NT]
M. Ram Murty and V. Kumar Murty, A variant of the Hardy-Ramanujan theorem, Hardy-Ramanujan Journal, Vol. 44 (2021), pp. 32-40.
Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.

Even-indexed terms give A046886.
Cf. A000005, A001221, A001222, A002202, A027750, A064097, A141197, A185633, A202727, A202728, A322312, A322976, A333123, A346467, A355452.
Cf. A005408 (positions of 1's), A051222 (of 2's).

a067513 = sum . map (a010051 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Jul 31 2012

A067513 := proc(n)
    local a,d;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isprime(d+1) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A067513(n),n=1..100) ; # R. J. Mathar, Aug 07 2022

a[n_] := Length[Select[Divisors[n]+1, PrimeQ]]
Table[Count[Divisors[n],?(PrimeQ[#+1]&)],{n,110}] (* _Harvey P. Dale, Feb 29 2012 *)
a[n_] := DivisorSum[n, 1 &, PrimeQ[# + 1] &]; Array[a, 100] (* Amiram Eldar, Jan 11 2025 *)

a(n)=sumdiv(n,d,isprime(d+1)) \\ Charles R Greathouse IV, Dec 23 2011

from sympy import divisors, isprime
def a(n): return sum(1 for d in divisors(n, generator=True) if isprime(d+1))
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Jul 12 2022

a(n) = 2 iff Bernoulli number B_{n} has denominator 6 (cf. A051222). - Vladeta Jovovic, Feb 13 2002
a(n) <= A141197(n). - Reinhard Zumkeller, Oct 06 2008
a(n) = A001221(A027760(n)). - Enrique Pérez Herrero, Dec 23 2011
a(n) = Sum_{k = 1..A000005(n)} A010051(A027750(n,k)+1). - Reinhard Zumkeller, Jul 31 2012
a(n) = A001221(A185633(n)) = A001222(A322312(n)). - Antti Karttunen, Jul 12 2022
Sum_{k=1..n} a(k) = n * (log(log(n)) + B) + O(n/log(n)), where B is a constant (Prachar, 1955). - Amiram Eldar, Jan 11 2025

Edited by Dean Hickerson, Feb 12 2002

A322977 Number of even divisors d of n such that d-1 is prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 3, 0, 0, 0, 4, 0, 1, 0, 3, 0, 3, 0, 2, 0, 0, 0, 6, 0, 0, 0, 1, 0, 3, 0, 3, 0, 0, 0, 6, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 4, 0, 0, 0, 6, 0, 0, 0, 3, 0, 4, 0, 1, 0, 0, 0, 7, 0, 2, 0, 2, 0, 2, 0, 3, 0

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Equally: Number of divisors d of n such that d-1 is an odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..15120
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A059841, A072627, A183063, A322975, A322976, A322978 (bisection).

Array[DivisorSum[#, 1 &, And[EvenQ@ #, PrimeQ[# - 1]] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322977(n) = sumdiv(n, d, (!(d%2))*isprime(d-1));

a(n) = Sum_{d|n, d>1} A059841(d)*A010051(d-1).

a(n) <= A183063(n).

A322975 Number of divisors d of n such that d-2 is prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 0, 2, 1, 1, 4, 0, 0, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 0, 0, 3, 1, 1, 4, 1, 2, 1, 0, 1, 1, 2, 0, 2, 1, 0, 4, 2, 1, 2, 0, 2, 2, 0, 0, 3, 2, 1, 0, 1, 0, 4, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 0, 0, 1, 2, 5

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is subtracted from each, as 1-2 = -1, 3-2 = 1, 5-2 = 3, etc, the only differences that are primes are: [3, 5, 7, 13, 19, 31, 43, 53, 61, 97, 103, 163, 229, 313, 383, 691, 1153, 1483, 3463], thus (a10395) = 19.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A062301, A067513, A072627, A322358, A322976, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# - 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322975(n) = sumdiv(n, d, isprime(d-2));

a(n) = Sum_{d|n, d>2} A010051(d-2).

a(A000040(n)) = A062301(n).

A360326 a(n) is the number of divisors of n that have only prime-indexed prime factors.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 2, 2, 1, 2, 3, 1, 4, 1, 1, 4, 2, 1, 4, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 2, 6, 1, 1, 2, 1, 3, 4, 1, 1, 4, 4, 1, 2, 1, 2, 4, 1, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 3, 1, 1, 6, 1, 2, 2, 1, 2, 5, 2, 2, 2, 4, 1, 2

Offset: 1

Amiram Eldar, Feb 03 2023

First differs from A322976 at n = 21.
Equivalently, a(n) is the number of divisors of the largest divisor of n that has only prime-indexed prime factors.
The asymptotic mean of this sequence is Product_{p in A006450} p/(p-1) > 3. See A076610 for a numerical estimate of the value of this product.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A006450, A076610, A320628, A322976, A360325, A360327.

f[p_, e_] := If[PrimeQ[PrimePi[p]], e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), e[i]+1, 1));}

a(n) = 1 if and only if n is in A320628.
a(n) = A000005(n) if and only if n is in A076610.
a(n) = A000005(A360325(n)).
Multiplicative with a(p^e) = e+1 if p is a prime-indexed prime (A006450), and 1 otherwise.

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A067513 Number of divisors d of n such that d+1 is prime.

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A322977 Number of even divisors d of n such that d-1 is prime.

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A322975 Number of divisors d of n such that d-2 is prime.

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A360326 a(n) is the number of divisors of n that have only prime-indexed prime factors.

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