A066911 -id:A066911 - cp's OEIS frontend

A181835 The sum of the primes <= n that are strongly prime to n.

0, 0, 0, 0, 0, 3, 0, 5, 8, 12, 7, 10, 12, 23, 19, 24, 31, 39, 36, 53, 51, 60, 54, 64, 72, 90, 80, 82, 88, 91, 90, 119, 127, 144, 127, 129, 143, 155, 139, 160, 174, 190, 185, 226, 225, 260, 248, 256

Offset: 0

Peter Luschny, Nov 17 2010

nonn

k is strongly prime to n iff k is relatively prime to n and k does not divide n-1.

			a(11) = 3 + 7 = 10.

Peter Luschny, Strong coprimality.

Cf. A181830, A181831, A181832, A181834, A181836, A066911.

with(numtheory):
Primes := n -> select(k->isprime(k),{$1..n}):
StrongCoprimes := n -> select(k->igcd(k,n)=1,{$1..n}) minus divisors(n-1):
StrongCoprimePrimes := n -> Primes(n) intersect StrongCoprimes(n):
A181835 := proc(n) local i; add(i,i=StrongCoprimePrimes(n)) end:

a[n_] := Select[Range[2, n], PrimeQ[#] && CoprimeQ[#, n] && !Divisible[n-1, #] &] // Total; Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Jun 28 2013 *)

A066913 (sum of primes < n that do not divide n) (mod n).

Original entry on oeis.org

0, 0, 2, 3, 0, 5, 3, 7, 5, 0, 6, 11, 2, 4, 3, 7, 7, 17, 1, 10, 4, 20, 8, 23, 20, 7, 16, 7, 13, 29, 5, 30, 14, 5, 8, 11, 12, 24, 25, 30, 33, 16, 23, 4, 3, 26, 46, 35, 27, 21, 2, 1, 10, 52, 35, 36, 17, 2, 27, 10, 13, 34, 50, 51, 28, 23, 32, 5, 59, 64, 0, 58, 55, 7, 29, 7, 1, 70, 1

Offset: 1

Leroy Quet, Jan 22 2002

nonn

			a(8) = (3 + 5 + 7) (mod 8) = 7 because 3, 5 and 7 are the primes < 8 that do not divide 8.

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

Cf. A066911, A071089.

Table[Mod[Total[Select[Prime[Range[PrimePi[n]]],Mod[n,#]!=0&]],n],{n,80}] (* Harvey P. Dale, Aug 06 2019 *)

a(n) = sum(i=1, n-1, if (isprime(i) && (n%i), i)) % n; \\ Michel Marcus, May 20 2014

a(n) = A066911(n) modulo n. - Michel Marcus, May 20 2014

a(prime(n)) = A071089(n). - Michel Marcus, May 20 2014

A143655 Triangle read by rows, primes not dividing n; A054521 * (A061397 * 0^(n-k)), 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 3, 0, 5, 0, 0, 0, 0, 3, 0, 5, 0, 7, 0, 0, 2, 0, 0, 5, 0, 7, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 7, 0, 0, 0, 11, 0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 0, 0, 0, 3, 0, 5, 0, 0, 0, 0, 0, 11, 0

Offset: 1

Gary W. Adamson, Aug 28 2008

Row sums = A066911: (0, 0, 2, 3, 5, 5, 10, 15, 14,....)

			First few rows of the triangle =
0;
0, 0;
0, 2, 0;
0, 0, 3, 0;
0, 2, 3, 0, 0;
0, 0, 0, 0, 5, 0;
0, 2, 3, 0, 5, 0, 0;
0, 0, 3, 0, 5, 0, 7, 0;
...
Row 8 has 3 primes < 8 not dividing 8: (3, 5, 7); where (3 + 5 + 7) = A066911(8).

Cf. A061397, A066911, A054521.

Triangle read by rows, A054521 * (A061397 * 0^(n-k)), 1<=k<=n. T(n,k) = prime if k is prime but does not divide n. A054521 = a triangle with row sums phi(n). A061397 = (0, 2, 3, 0, 5, 0, 7,...)

A115333 Sum of primes that do not divide n and are less than the largest prime dividing n.

Original entry on oeis.org

0, 0, 2, 0, 5, 0, 10, 0, 2, 3, 17, 0, 28, 8, 2, 0, 41, 0, 58, 3, 7, 15, 77, 0, 5, 26, 2, 8, 100, 0, 129, 0, 14, 39, 5, 0, 160, 56, 25, 3, 197, 5, 238, 15, 2, 75, 281, 0, 10, 3, 38, 26, 328, 0, 12, 8, 55, 98, 381, 0, 440, 127, 7, 0, 23, 12, 501, 39, 74, 3, 568, 0, 639, 158, 2, 56, 10

Offset: 1

Leroy Quet, Mar 05 2006

nonn

When n is prime, n = largest prime dividing n; hence a(n) is the sum of all primes less than n = A034387(n)-n. a(n) = SUM{p such that p is in A000040 AND NOT(p|n) AND p < A006530(n)}. - Jonathan Vos Post, Mar 08 2006

The zeros give A055932: All prime divisors are consecutive primes starting at 2. - Robert G. Wilson v, May 01 2006

			The primes < 7 and coprime to 7 are 2, 3 and 5. So a(7) = 2+3+5 = 10.

Cf. A000040, A006530, A034387, A066911, A083720.

f[n_] := Plus @@ Complement[Prime@ Range@ PrimePi[ Max[First /@ FactorInteger@n] - 1], First /@ FactorInteger@n]; Array[f, 77] (* Hans Havermann, Mar 06 2006 *)

More terms from Hans Havermann, Mar 06 2006

A349555 a(n) = Sum_{p<=n, p prime} p^floor(1/gcd(n/p)).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 11, 16, 15, 12, 18, 25, 29, 34, 35, 40, 42, 55, 59, 72, 69, 66, 78, 97, 96, 87, 98, 93, 101, 122, 130, 159, 148, 143, 150, 157, 161, 178, 183, 192, 198, 229, 239, 270, 275, 258, 282, 325, 322, 323, 310, 315, 329, 378, 367, 374, 361, 352, 382, 433, 441, 470

Offset: 1

Wesley Ivan Hurt, Nov 21 2021

nonn

For each prime number p less than or equal to n, add 1 if p|n, otherwise add p (see example).

			a(9) = 15; The primes less than or equal to 9 are 2, 3, 5, 7 and only 3|9. We then have, respectively, a(9) = 2 + 1 + 5 + 7 = 15.

Cf. A001221, A066911.

nterms=100;Table[Total[Map[If[Mod[n,#]==0,1,#]&,Prime[Range[PrimePi[n]]]]],{n,nterms}] (* Paolo Xausa, Nov 22 2021 *)

a(n) = A001221(n) + A066911(n).

cp's OEIS Frontend

A181835 The sum of the primes <= n that are strongly prime to n.

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A066913 (sum of primes < n that do not divide n) (mod n).

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A143655 Triangle read by rows, primes not dividing n; A054521 * (A061397 * 0^(n-k)), 1<=k<=n.

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A115333 Sum of primes that do not divide n and are less than the largest prime dividing n.

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A349555 a(n) = Sum_{p<=n, p prime} p^floor(1/gcd(n/p)).

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