A348938 - OEIS

%I #11 Nov 04 2021 20:47:39

%S 153,245,261,369,425,477,637,725,801,833,845,873,909,981,1017,1025,

%T 1233,1325,1341,1377,1421,1557,1573,1629,1773,1805,1813,2009,2057,

%U 2061,2097,2169,2225,2313,2349,2421,2425,2525,2529,2597,2637,2645,2725,2853,2873,2989,3141,3177,3321,3357,3425,3501,3509,3577,3609,3681

%N Odd numbers k for which A064989(sigma(k)) < A064989(k), and which are of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

%C Obviously, any hypothetical odd perfect number would be neither in this sequence nor in A348939.

%C Of the numbers in range 1..2^20, 9644 reside in this sequence and 3865 in A348939. Of the numbers <= 2^25, 229480 are in this sequence, and 88270 in A348939.

%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>

%t q[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; odde = Select[e, OddQ]; Length[e] > 1 && Length[odde] == 1 && Divisible[odde[[1]] - 1, 4] && Divisible[p[[Position[e, odde[[1]]][[1, 1]]]] - 1, 4]]; f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 4000, 2], q[#] && s[DivisorSigma[1, #]] < s[#] &] (* _Amiram Eldar_, Nov 04 2021 *)

%o (PARI)

%o A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };

%o isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));

%o isA348748(n) = ((n%2)&&(A064989(sigma(n)) < A064989(n)));

%o isA348938(n) = (isA228058(n)&&isA348748(n));

%Y Intersection of A228058 and A348748.

%Y Cf. A064989, A326042, A348939.

%K nonn

%O 1,1

%A _Antti Karttunen_, Nov 04 2021