%I #32 Jul 09 2022 18:31:43

%S 1,2,10,34,106,120,216,260,340,408,440,580,672,696,820,1060,1272,1666,

%T 1780,1940,2136,2340,2464,3320,3576,3960,4280,4536,5280,5380,5860,

%U 6456,6960,7520,8746,8840,9120,9632,10040,10776,12528,12640,13464,14560,16180,16660,17400,17620,19040,19416,19992,21320,22176,22968

%N Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

%C Numbers k for which A351546(k) is a unitary divisor of k.

%C The condition guarantees that A351555(k) = 0, therefore this is a subsequence of A351554.

%C The condition is also a necessary condition for A349745, therefore it is a subsequence of this sequence.

%C All six known 3-perfect numbers (A005820) are included in this sequence.

%C All 65 known 5-multiperfects (A046060) are included in this sequence.

%C Not all multiperfects (A007691) are present (only 587 of the first 1600 are), but all 23 known terms of A323653 are terms, while none of the (even) terms of A046061 or A336702 are.

%H Antti Karttunen, <a href="/A351551/b351551.txt">Table of n, a(n) for n = 1..20223</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) [= 13365 = 3^5 * 5^1 * 11^1] is 2^5 * 7^1 = 224, therefore A351546(672) is a unitary divisor of 672, and 672 is included in this sequence.

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };

%o isA351551(n) = { my(u=A351546(n)); (!(n%u) && 1==gcd(u,n/u)); };

%Y Cf. A000203, A000396, A003961, A007691, A046061, A065997, A336702, A351546, A351555, A353633 (characteristic function).

%Y Subsequence of A351552 and of A351554.

%Y Cf. A349745, A351550 (subsequences), A005820, A046060, A323653 (very likely subsequences).

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 16 2022