A141197 -id:A141197 - 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

A185208 Numbers having no divisors d > 1 such that d + 1 are prime powers.

Original entry on oeis.org

1, 5, 11, 13, 17, 19, 23, 25, 29, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 209

Offset: 1

Reinhard Zumkeller, Nov 01 2012

nonn

A141197(a(n)) = A049073(a(n)) = 1.
Contains all primes except for 2 and Mersenne primes A000668. - Jon Perry, Nov 11 2012
A composite number is in the sequence iff all its factors are. - Jon Perry, Nov 11 2012

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

Cf. A000961, A010055, A027750.

Cf. A000668.

a185208 n = a185208_list !! (n-1)
a185208_list =  filter ((== 1) . a141197) [1..]

Select[Range[210], Select[Divisors[#] // Rest, PrimeNu[# + 1] == 1 &] == {} &] (* Jean-François Alcover, Aug 17 2013 *)

A049073 LCM of all divisors of d of n such that d+1 is a prime power.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 3, 28, 1, 30, 31, 16, 3, 2, 7, 36, 1, 2, 3, 40, 1, 42, 1, 44, 15, 46, 1, 48, 7, 10, 3, 52, 1, 18, 1, 56, 3, 58, 1, 60, 1, 62, 63, 16, 1, 66, 1, 4, 3, 70, 1, 72, 1, 2, 15, 4, 7, 78, 1, 80, 3, 82, 1, 84, 1

Offset: 1

David E. Daykin

a(A185208(n)) = 1. - Reinhard Zumkeller, Nov 01 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A141197, A027750, A010055.

a049073 = foldl lcm 1 . filter ((== 1) . a010055 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Nov 01 2012

a[n_] := LCM @@ Select[Divisors[n], PrimeNu[# + 1] == 1 &]; Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Aug 17 2013 *)

More terms from Erich Friedman, Jun 03 2001

A141198 a(n) is the number of divisors of n that are each one more than a power of a prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 0, 3, 2, 3, 0, 5, 0, 2, 2, 3, 1, 5, 0, 5, 1, 1, 0, 7, 1, 2, 2, 4, 0, 6, 0, 4, 2, 2, 1, 7, 0, 2, 1, 6, 0, 5, 0, 3, 3, 1, 0, 8, 0, 4, 2, 3, 0, 6, 1, 5, 1, 1, 0, 10, 0, 2, 2, 4, 2, 4, 0, 4, 1, 4, 0, 10, 0, 2, 2, 3, 0, 4, 0, 7, 2, 2, 0, 9, 2, 1, 1, 4, 0, 9, 0, 2, 1, 1, 1, 9, 0, 3, 3, 6, 0, 5, 0

Offset: 1

Leroy Quet, Jun 12 2008

nonn

1 is considered here to be a power of a prime. 0 is not considered here to be a power of a prime.

			The divisors of 9 are 1, 3 and 9. 1 is one more than 0, not a power of a prime. 3 is one more than 2, a power of a prime. And 9 is one more than 8, a power of a prime. There are therefore 2 such divisors that are each one more than a power of a prime. So a(9) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1049 from Diana Mecum)

Cf. A000961, A141197.

a[n_] := DivisorSum[n, 1 &, # == 2 || PrimePowerQ[#-1] &]; Array[a, 100] (* Amiram Eldar, Jun 22 2025 *)

a(n) = sumdiv(n, d, d == 2 || isprimepower(d - 1)); \\ Amiram Eldar, Jun 22 2025

Corrected and extended by Diana L. Mecum, Jul 05 2007

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

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A185208 Numbers having no divisors d > 1 such that d + 1 are prime powers.

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A049073 LCM of all divisors of d of n such that d+1 is a prime power.

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A141198 a(n) is the number of divisors of n that are each one more than a power of a prime.

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Examples

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Mathematica

PARI

Extensions