A210934 -id:A210934 - cp's OEIS frontend

A023514 a(n) = sum of exponents in prime-power factorization of prime(n) + 1.

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

Offset: 1

Clark Kimberling

nonn

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

Cf. A210934.

Cf. A000040, A001222.

with(numtheory):
a:= n-> add(i[2], i=ifactors(ithprime(n)+1)[2]):
seq(a(n), n=1..100);

Array[Plus@@Last/@FactorInteger[Prime[ # ]+1]&,6! ] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)

a(n) = bigomega(prime(n)+1); \\ Michel Marcus, Jan 04 2016

a(n) = A001222(A000040(n)+1). - Zak Seidov, Jan 04 2016

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

A127305 Primes p such that p + (sum of the prime factors of p-1) + (sum of prime factors of p+1) is prime.

Original entry on oeis.org

2, 13, 47, 71, 101, 151, 193, 239, 241, 293, 331, 337, 359, 383, 397, 401, 421, 463, 487, 557, 577, 709, 773, 797, 821, 929, 1019, 1031, 1069, 1093, 1103, 1181, 1217, 1249, 1327, 1367, 1423, 1499, 1571, 1759, 1787, 1789, 1831, 1871, 1877, 1913, 1933, 2053

Offset: 1

J. M. Bergot, Mar 28 2007

nonn

The primes are taken "with multiplicity".

			2 is a term, since 2 + 0 + 3 = 5 is a prime.
13 is a term since 13 + (2+2+3) + (2+7) = 29 is prime, i.e. the prime factors are added with multiplicity.
151 is prime, 150 = 2*3*5*5, 152 = 2*2*2*19. 151 + 2+3+5+5 + 2+2+2+19 = 191 is prime, hence 151 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000 (updated by _Robert Israel_, May 25 2022)

Cf. A210934, A210936.

[ p: p in PrimesInInterval(3, 2100) | IsPrime(&+[ &+[ k[1]*k[2]: k in Factorization(n)]: n in [p-1..p+1] ] ) ]; /* Klaus Brockhaus, Apr 06 2007 */

spf:= proc(n) local t; add(t[1]*t[2], t = ifactors(n)[2]) end proc:
filter:= proc(p) isprime(p) and isprime(p+spf(p-1)+spf(p+1)) end proc:
select(filter, [$2..10000]); # Robert Israel, May 25 2022

pspfQ[n_] := PrimeQ[n + Total[Flatten[Table[#[[1]], {#[[2]]}] & /@ Flatten[FactorInteger[n + {1,-1}], 1] ] ] ]; {2}~Join~Select[Prime[Range[400]], pspfQ] (* Harvey P. Dale, Jan 08 2015, corrected by Michael De Vlieger, May 25 2022 *)

Edited and extended by Klaus Brockhaus, Apr 06 2007

Edited by N. J. A. Sloane, May 25 2022

A343416 a(n) = A001414(n) + A000203(n) + A001414(A000203(n)) + A000203(A001414(n)).

Original entry on oeis.org

1, 11, 15, 25, 22, 30, 29, 41, 44, 41, 42, 54, 50, 55, 56, 85, 61, 78, 68, 76, 70, 73, 80, 94, 90, 93, 73, 92, 99, 112, 105, 104, 97, 104, 99, 139, 134, 125, 116, 126, 137, 149, 146, 137, 119, 140, 154, 182, 117, 167, 146, 149, 172, 157, 131, 161, 151, 166, 191, 224, 218, 190, 150, 294, 155, 205

Offset: 1

J. M. Bergot and Robert Israel, Apr 14 2021

nonn

If n = prime(k) is prime, a(n) = 3*prime(k)+1+A210934(k).

			For n = 3, A001414(3) = 3, A000203(3) = A000203(A001414(3)) = 4 and A001414(A000203(3)) = 4, so a(3) = 3+4+4+4 = 15.

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

Cf. A000203, A001414, A210934.

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
f:= proc(n) local a,b;
  a:= spf(n);
  b:= numtheory:-sigma(n);
  a+b+spf(b)+numtheory:-sigma(a)
end proc:
map(f, [$1..100]);

cp's OEIS Frontend

A023514 a(n) = sum of exponents in prime-power factorization of prime(n) + 1.

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

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A127305 Primes p such that p + (sum of the prime factors of p-1) + (sum of prime factors of p+1) is prime.

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A343416 a(n) = A001414(n) + A000203(n) + A001414(A000203(n)) + A000203(A001414(n)).

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