A351542 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.

196, 200, 364, 588, 760, 950, 1000, 1092, 1148, 1160, 1274, 1358, 1372, 1400, 1450, 1490, 1568, 1764, 1782, 1900, 1990, 2156, 2200, 2324, 2360, 2600, 2716, 2900, 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

Offset: 1

Author

Antti Karttunen, Feb 16 2022

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Links

• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A286561.

Cf. A351541 (subsequence).

Subsequence of A351543.

Programs

Mathematica

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 *)

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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)); };
isA351542(n) = (!(n%2) && Aux351542(n)>0);