A023508 -id:A023508 - cp's OEIS frontend

A008334 Number of distinct primes dividing p-1, where p = n-th prime.

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

Offset: 1

N. J. A. Sloane

nonn

This is omega(p-1), i.e. A001221(A006093(n)). For Omega(p-1) = A001222(A006093(n)), see A023508. - Lekraj Beedassy, Oct 08 2004

Primes counted without multiplicity. - Harvey P. Dale, May 05 2018

N. P. Ryzhova, Asymptotic formulae in a binary problem of shifted prime numbers (in Russian), Additive problems of number theory, Interuniv. Collect. Sci. Works, Kujbyshev 1985 (1985), pp. 25-31.

T. D. Noe, Table of n, a(n) for n=1..10000
H. Halberstam, On the Distribution of Additive Number-Theoretic Functions (III), Journal of the London Mathematical Society, Vol. s1-31, No. 1 (1956), pp. 14-27.
C. B. Haselgrove, Some Theorems in the Analytic Theory of Numbers, Journal of the London Mathematical Society, Vol. s1-26, No. 4 (1951), pp. 273-277.

Cf. A001221, A001222, A006093, A023508.

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

PrimeNu[#]&/@(Prime[Range[100]]-1) (* Harvey P. Dale, May 05 2018 *)

a(n,p=prime(n))=omega(p-1) \\ Charles R Greathouse IV, Nov 29 2024

Sum_{k; prime(k)<=n} a(k) = n*log(log(n))/log(n) + O(n/log(n)) (Haselgrove, 1951; Halberstam, 1956; Ryzhova, 1985). - Amiram Eldar, Mar 05 2021 [corrected by Charles R Greathouse IV, Nov 29 2024]

Definition clarified by Harvey P. Dale, May 05 2018

A210934 Sum of prime factors of prime(n)+1 (counted with multiplicity).

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 8, 9, 9, 10, 10, 21, 12, 15, 11, 11, 12, 33, 21, 12, 39, 13, 14, 13, 16, 22, 19, 13, 18, 24, 14, 18, 28, 16, 15, 25, 81, 45, 16, 34, 15, 22, 15, 99, 19, 16, 57, 17, 26, 30, 21, 16, 24, 17, 48, 20, 16, 25, 141, 52, 75, 19, 22, 22, 159, 58, 87

Offset: 1

Paolo P. Lava, Mar 30 2012

nonn

From an idea of Michael B. Porter.

			prime(8) = 19, and 19+1 = 20 = 2*2*5, so a(8) = 2+2+5 = 9.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A023508, A023514, A210936

with(numtheory);
P:=proc(i)
local a,k,n;
for n from 1 to i do
  a:=ifactors(ithprime(n)+1)[2]; print(add(a[k][1]*a[k][2],k=1..nops(a)));
od; end:
# alternative
A210934 := proc(n)
    local p,pplus,f ;
    p := ithprime(n) ;
    pplus := ifactors(p+1)[2] ;
    add(op(1,f)*op(2,f),f=pplus) ;
end proc:
seq(A210934(n),n=1..300) ; # R. J. Mathar, May 25 2022

a(n) = A001414(A008864(n)). - Michel Marcus, Oct 05 2013

A210936 Sum of prime factors of prime(n)-1 (counted with multiplicity).

Original entry on oeis.org

0, 2, 4, 5, 7, 7, 8, 8, 13, 11, 10, 10, 11, 12, 25, 17, 31, 12, 16, 14, 12, 18, 43, 17, 13, 14, 22, 55, 13, 15, 15, 20, 23, 28, 41, 15, 20, 14, 85, 47, 91, 15, 26, 15, 18, 19, 17, 42, 115, 26, 35, 26, 16, 17, 16, 133, 71, 16, 30, 18, 52, 77, 25, 38, 22, 83

Offset: 1

Paolo P. Lava, Mar 30 2012

nonn

From an idea of Michael B. Porter.

			prime(10) = 29, and 29-1 = 28 = 2*2*7, so a(10) = 2+2+7 = 11.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A023508, A023514, A210934

with(numtheory);
P:=proc(i)
local a,k,n;
for n from 1 to i do
  a:=ifactors(ithprime(n)-1)[2]; print(add(a[k][1]*a[k][2],k=1..nops(a)));
od; end:
# alternative
A210936 := proc(n)
        local p,pplus,f ;
        p := ithprime(n) ;
        pplus := ifactors(p-1)[2] ;
        add(op(1,f)*op(2,f),f=pplus) ;
end proc:
seq(A210936(n),n=1..300) ; # R. J. Mathar, May 25 2022

a(n) = A001414(A006093(n)). - Michel Marcus, Oct 05 2013

A339902 Number of prime divisors of A339821(n), counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, 4, 5, 6, 7, 3, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 4, 5, 6, 7, 6, 7, 8, 9, 6, 7, 8, 9, 8, 9, 10, 11, 7, 8, 9, 10, 9, 10, 11, 12, 9, 10, 11, 12, 11, 12, 13, 14, 3, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 6, 7, 8, 9, 8, 9, 10, 11, 8, 9, 10, 11, 10, 11, 12, 13, 7

Offset: 0

Antti Karttunen, Dec 21 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000010, A001222, A003972, A019565, A023508, A339821, A339822, A339906.

A339902(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += bigomega(p-1)); n >>= 1); (s); };

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A023508(e1) + A023508(e2) + ... + A023508(ek).
a(n) = A001222(A339821(n)).
a(n) = A001222(A003972(A019565(n))) = A001222(A000010(A019565(2n))).
a(n) >= A339822(n).

A382789 The number of prime factors of Euler phi of the n-th primorial number, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 3, 5, 7, 10, 14, 17, 19, 22, 25, 29, 33, 36, 38, 41, 43, 47, 50, 53, 58, 61, 63, 67, 73, 77, 80, 82, 87, 92, 96, 99, 103, 106, 109, 113, 117, 122, 124, 127, 129, 134, 137, 144, 148, 152, 156, 159, 161, 165, 169, 172, 178, 182, 190, 192, 195, 200, 204

Offset: 0

Amiram Eldar, Apr 05 2025

Amiram Eldar, Table of n, a(n) for n = 0..10000

Partial sums of A023508.

Cf. A000010, A002110, A001222, A055768, A055769.

Join[{0}, Accumulate[PrimeOmega[Prime[Range[100]] - 1]]]

list(nmax) = {my(s = 0, c = 0); print1(s, ", "); forprime(p = 1, , c++; s += bigomega(p-1); print1(s, ", "); if(c == nmax, break));}

a(n) = A001222(A000010(A002110(n))).
a(n) = A001222(A005867(n)).
a(n) = Sum_{k=1..n} A023508(k).

cp's OEIS Frontend

A008334 Number of distinct primes dividing p-1, where p = n-th prime.

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A210934 Sum of prime factors of prime(n)+1 (counted with multiplicity).

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A210936 Sum of prime factors of prime(n)-1 (counted with multiplicity).

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Maple

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A339902 Number of prime divisors of A339821(n), counted with multiplicity.

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A382789 The number of prime factors of Euler phi of the n-th primorial number, counted with multiplicity.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula