OFFSET
1,1
COMMENTS
LINKS
EXAMPLE
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.
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 *)
PROG
(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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 16 2022
STATUS
approved