The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615. 19
 1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702. A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative. If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703. Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17. LINKS Antti Karttunen, Table of n, a(n) for n = 1..20000 Index entries for sequences related to sigma(n) FORMULA a(n) = gcd(A000203(n), A001615(n)). For prime p, a(p^e) = (p+1)^(e mod 2). For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n). a(A007913(n)) | a(n). a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)). a(n) = A000203(n) / A344696(n). a(n) = A001615(n) / A344697(n). MATHEMATICA Table[GCD[DivisorSigma[1, n], DivisorSum[n, MoebiusMu[n/#]^2*#&]], {n, 100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *) PROG (PARI) A001615(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615 A344695(n) = gcd(sigma(n), A001615(n)); (Python 3.8+) from math import prod, gcd from sympy import primefactors, divisor_sigma def A001615(n): plist = primefactors(n) return n*prod(p+1 for p in plist)//prod(plist) def A344695(n): return gcd(A001615(n), divisor_sigma(n)) # Chai Wah Wu, Jun 03 2021 CROSSREFS Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1). Cf. also A048250, A291750, A291751, A340070, A323363. Subsets of range: A008864, A065091 (conjectured). Sequence in context: A366795 A092261 A367991 * A348503 A348047 A348984 Adjacent sequences: A344692 A344693 A344694 * A344696 A344697 A344698 KEYWORD nonn AUTHOR Antti Karttunen and Peter Munn, May 26 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 12 20:44 EDT 2024. Contains 373360 sequences. (Running on oeis4.)