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A351542
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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