cp's OEIS Frontend

A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

%I A386430 #28 Aug 26 2025 15:13:03
%S A386430 1,3,7,15,21,27,31,33,57,69,87,91,93,105,127,141,177,189,195,217,231,
%T A386430 237,273,285,301,381,399,447,465,483,495,513,567,573,597,609,627,651,
%U A386430 717,775,819,837,861,889,903,987,1023,1029,1149,1185,1239,1311,1365,1419,1431,1437,1455,1497,1561,1653,1659,1687,1743
%N A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.
%C A386430 Conjecture: After the initial 1, and apart from any hypothetical odd perfect numbers, all other terms are in A248150, i.e., sigma(k) == 0 (mod 4). This would imply (with the same caveat), that this sequence has no common terms with A228058 and no squares larger than one. This is true at least for the first 709203 terms (terms in range [1..2^34]).
%C A386430 Terms k such that A162642(k) = 1 are rare: 3, 7, 27, 31, 127, 567, 775, 8191, 27783, 131071, 524287, 2147483647, ... (odd terms of A387160).
%H A386430 Antti Karttunen, <a href="/A386430/b386430.txt">Table of n, a(n) for n = 1..10000</a>
%H A386430 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A386430 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H A386430 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%F A386430 {k | k odd, A351555(k) = 0}.
%e A386430 a(386548) = 5919068925 = 3^4 * 5^2 * 7^2 * 11^2 * 17 * 29. sigma(5919068925) = 15355618740 = 2^2 * 3^4 * 5 * 7 * 11^2 * 19^2 * 31. The "don't care primes" is given by A003961(A007947(5919068925))) = 2947945 = 5*7*11*13*19*31, thus only odd prime factor that matters here is 3, which in case has the same exponent (4) in both n = 5919068925 and sigma(n). In a way, this number is very close to satisfying Euler's criterion for odd perfect numbers (A228058), except that it has two unitary prime factors of the form 4k+1, instead of just one, apart from the square factor. Both n/17 and n/29 are in A228058.
%o A386430 (PARI)
%o A386430 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A386430 A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
%o A386430 isA386430(n) = ((n%2) && (0==A351555(n)));
%Y A386430 Odd terms of A351554.
%Y A386430 Cf. A000203, A003961, A056169, A162642, A228058, A248150, A351555, A386424, A387160.
%Y A386430 Cf. A349169 (subsequence).
%K A386430 nonn,new
%O A386430 1,2
%A A386430 _Antti Karttunen_, Aug 22 2025