A278373 -id:A278373 - cp's OEIS frontend

A278374 a(n) = smallest k with sigma(k) + phi(k) - 2k = A278373(n).

1, 4, 6, 8, 27, 125, 16, 12, 18, 20, 28, 50, 63, 32, 44, 98, 52, 24, 68, 42, 242, 40, 70, 338, 54, 36, 56, 116, 78, 578, 165, 130, 722, 135, 48, 164, 114, 1058, 104, 188, 285, 138, 100, 80, 230, 1682, 136, 128, 60, 1922, 244, 152, 741, 112, 225, 351, 72, 284, 2738, 90, 84, 651, 222, 3362, 711, 332, 3698, 459, 196, 96, 162, 356

Offset: 1

David W. Wilson, Nov 19 2016

nonn

David W. Wilson, Table of n, a(n) for n = 1..10000
n = 24 is the smallest number with sigma(n) + phi(n) - 2n = 20 = A278383(18), so a(18) = 24.

Cf. A000010 (phi), A000203 (sigma).

Function[w, Map[Position[w, #][[1, 1]] &, Take[#, 72]] &@ Union@ w]@ Table[DivisorSigma[1, n] + EulerPhi@ n - 2 n, {n, 10^4}] (* Michael De Vlieger, Nov 30 2016 *)

A051709 a(n) = sigma(n) + phi(n) - 2n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 7, 0, 9, 0, 10, 2, 2, 0, 20, 1, 2, 4, 12, 0, 20, 0, 15, 2, 2, 2, 31, 0, 2, 2, 26, 0, 24, 0, 16, 12, 2, 0, 44, 1, 13, 2, 18, 0, 30, 2, 32, 2, 2, 0, 64, 0, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0

Offset: 1

Jud McCranie and Carlos Rivera

nonn

Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013
Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - T. D. Noe, Aug 01 2002
It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - Eric Desbiaux, Jul 04 2013
a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - Jianing Song, Jun 27 2021

			a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.)
Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).

Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=sigma(n)+eulerphi(n)-2*n \\ Charles R Greathouse IV, Jul 05 2013

A051709(n) = -sumdiv(n,d,(dAntti Karttunen, Mar 02 2018

Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002
From Antti Karttunen, Mar 02 2018: (Start)
a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]
a(n) = -Sum_{d|n, dA008683(n/d)*A001065(d). (End)
Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 03 2023

A056996 Positive integers not of the form sigma(k) + phi(k) - 2k for some k.

Original entry on oeis.org

5, 11, 19, 21, 23, 27, 33, 35, 39, 43, 45, 47, 51, 53, 55, 59, 67, 69, 71, 79, 81, 83, 87, 95, 99, 101, 103, 105, 107, 113, 115, 117, 119, 123, 129, 131, 135, 139, 141, 143, 147, 151, 153, 155, 159, 165, 167, 171, 173, 175, 177, 179, 185, 187, 189, 191, 193, 195

Offset: 1

David W. Wilson, Sep 06 2000

nonn

Under the assumption that every even integer >= 18 is the sum of three distinct primes, all elements of this sequence are odd. See A278373 for more details.

David W. Wilson, Table of n, a(n) for n = 1..10000

Complement is A278373.

Cf. A000010 (phi), A000203 (sigma).

Take[#, 59] &@ Complement[Range@ #, Union@ Table[DivisorSigma[1, n] + EulerPhi@ n - 2 n, {n, #}]] &[10^5] (* Michael De Vlieger, Nov 30 2016 *)

A278922 Largest p such that n = p + q + r where p < q < r are all prime, or 0 if no such primes p, q, r exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 3, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 7, 2, 5, 2, 7, 2, 7, 2, 7, 2, 11, 2, 11, 2, 5, 2, 11, 2, 13, 2, 11, 2, 13, 2, 13, 2, 11, 2, 17, 2, 13, 2, 13, 2, 17, 2, 17, 2, 17, 2, 19, 2, 19, 2, 13, 2, 17, 2, 19, 2, 17, 2, 23, 2, 19, 2, 19, 2, 23, 2, 23, 2, 23, 2, 23, 2, 29, 2, 23, 2, 29, 2, 29, 2, 23, 2, 29, 2, 31, 2, 31, 2, 29, 2, 31, 2, 29, 2, 31, 2, 37

Offset: 1

M. F. Hasler, Dec 01 2016

nonn

Empirically, a(n) >= 2 for all n >= 18. Since a(2n) = 2 unless it is zero, the terms with even indices are less interesting, and the terms with odd indices are listed in A278923.
For even n, the existence of the three primes reduces to a slightly strengthened* variant of Goldbach's conjecture. For odd n, is a slightly strengthened* variant of the weak (a.k.a. odd, or ternary) Goldbach conjecture, considered to be proved since 2013. (*) In both cases, the strengthening consists of requiring that the three primes must be distinct.
From Robert G. Wilson v, Dec 02 2016: (Start)
The first occurrence of the n-th prime: 10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, 471, 487, 503, ..., .
Conjecture: primes appear in their natural order. (End)

David W. Wilson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Cf. A278923.

Cf. A278373, complement of A056996.

f[n_] := If[OddQ@n || n < 18, Block[{p = 0, q = 3, r = 5}, While[q < r, r = NextPrime@ q; While[r < n - q - 1, If[n < 2q + r && PrimeQ[n - r - q], p = Max[p, n - r - q]; Break[]]; r = NextPrime@ r]; q = NextPrime@ q]; p], 2]; Array[f, 121] (* Robert G. Wilson v, Dec 02 2016 *)

a(n,p=if(bittest(n,0),n\3-1,3))=while(p=precprime(p-1),forprime(q=p+1,(n-p-1)\2,isprime(n-p-q)&&return(p)))

a(2n) = 2 (for n > 4), since one of the three primes must necessarily be even, and that can only be p = 2.

a(n) = 0 for n < 2 + 3 + 5 = 10, and for odd n < 3 + 5 + 7 = 15.

A278923 Largest p such that 2n-1 = p + q + r with three primes p < q < r, or 0 if no such primes exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 3, 5, 5, 3, 5, 7, 5, 7, 7, 7, 11, 11, 5, 11, 13, 11, 13, 13, 11, 17, 13, 13, 17, 17, 17, 19, 19, 13, 17, 19, 17, 23, 19, 19, 23, 23, 23, 23, 29, 23, 29, 29, 23, 29, 31, 31, 29, 31, 29, 31, 37, 29, 37, 37, 31, 41, 37, 31, 41, 37, 41, 43, 41, 41, 43, 43, 41, 43, 43, 47, 47, 43, 41, 47, 43, 47, 53, 47, 47, 53, 53, 53, 53, 59, 47

Offset: 1

M. F. Hasler, Dec 01 2016

nonn

Empirically, such primes always exist for n >= 10, i.e., 2n-1 >= 19, whence a(n) >= 3 for all n >= 10. Again empirically, a(n) ~ 2n/3 as n -> oo and a(n) is always close to 2n/3.

This is the bisection (every other term) of A278922, whose terms with even indices are all equal to 2 (or 0).

David W. Wilson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Cf. A278922.

Cf. A278373, complement of A056996.

a(n,p=(n=2*n-1)\3-1)=while(p=precprime(p-1),forprime(q=p+1,(n-p-1)\2,isprime(n-p-q)&&return(p)))

cp's OEIS Frontend

A278374 a(n) = smallest k with sigma(k) + phi(k) - 2k = A278373(n).

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A051709 a(n) = sigma(n) + phi(n) - 2n.

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A056996 Positive integers not of the form sigma(k) + phi(k) - 2k for some k.

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A278922 Largest p such that n = p + q + r where p < q < r are all prime, or 0 if no such primes p, q, r exist.

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A278923 Largest p such that 2n-1 = p + q + r with three primes p < q < r, or 0 if no such primes exist.

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