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A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).

%I A387411 #20 Sep 01 2025 23:43:24
%S A387411 1,3,4,7,10,15,18,23,27,31,47,57,63,95,119,127,255,348,383,415,447,
%T A387411 511,575,695,767,959,1023,1054,1071,1535,1919,2047,2626,3471,3839,
%U A387411 4095,4415,6815,8191,8703,13823,16383,31743,32767,39895,42367,48127,64607,65535,68727,74495,81919,92159,98303,113535,124671,131071
%N A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).
%C A387411 Like in many sequences of this type, the criterion seems to select for numbers with a long tail of trailing 1-bits. Terms that are not in A004767 are: 1, 4, 10, 18, 57, 348, 1054, 2626, 675348, 1869741, 12371554, 14070141, 1158654378, 1673018314, etc.
%H A387411 Antti Karttunen, <a href="/A387411/b387411.txt">Table of n, a(n) for n = 1..159</a>
%H A387411 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%t A387411 a3961[x_] := Apply[Times, Prime[PrimePi[#1] + 1]^#2 & @@@ FactorInteger[x]] - Boole[x == 1];
%t A387411 a265[x_] := x/2^IntegerExponent[x, 2];
%t A387411 Select[Range[2^17], Divisible[1 + a3961[#], a265[# + 1] ] &] (* _Michael De Vlieger_, Sep 01 2025 *)
%o A387411 (PARI)
%o A387411 A000265(n) = (n>>valuation(n,2));
%o A387411 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A387411 isA387411(n) = !((1+A003961(n))%A000265(1+n));
%Y A387411 Cf. A000265, A003961, A004767.
%Y A387411 Subsequences: A000225, A348514 (which is also a subsequence of A387414).
%Y A387411 For similar sequences, see A336700, A387410, A387415, A387410, A387418, A387419.
%K A387411 nonn,new
%O A387411 1,2
%A A387411 _Antti Karttunen_, Sep 01 2025