A351542 - OEIS

%I #16 Feb 16 2022 17:05:45

%S 196,200,364,588,760,950,1000,1092,1148,1160,1274,1358,1372,1400,1450,

%T 1490,1568,1764,1782,1900,1990,2156,2200,2324,2360,2600,2716,2900,

%U 2912,2950,2980,3042,3160,3200,3276,3332,3388,3400,3430,3444,3490,3560,3564,3724,3822,3892,3950,3980,4004,4018,4074,4102,4116,4360

%N Even numbers k that have an odd prime factor p such that p also divides sigma(k), but valuation(k,p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

%C Even numbers k that have an odd prime factor prime(i) such that it is also a prime factor of sigma(k), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351543 in that A286561(k, prime(i)) must be > 0.

%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 196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.

%e 364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

%t Select[Range[2, 4400, 2], Function[{k, s, facs, t}, AnyTrue[DeleteCases[facs[[All, 1]], 2], And[Mod[s, #] == 0, IntegerExponent[s, #] != IntegerExponent[k, #], Mod[t, #] != 0] &]] @@ {#1, #2, #3, Apply[Times, (NextPrime[#1])^#2 & @@@ #3]} & @@ {#, DivisorSigma[1, #], FactorInteger[#]} &] (* _Michael De Vlieger_, Feb 16 2022 *)

%o (PARI)

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

%o Aux351542(n) = { my(f=factor(n),s=sigma(n),u=A003961(n),v); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), v=valuation(s,f[k,1]); (v>0) && (v!=f[k,2]), 0)); };

%o isA351542(n) = (!(n%2) && Aux351542(n)>0);

%Y Cf. A000203, A003961, A286561.

%Y Cf. A351541 (subsequence).

%Y Subsequence of A351543.

%K nonn

%O 1,1

%A _Antti Karttunen_, Feb 16 2022