A356455 Numbers m = A005940(n) such that A005940(n) < (n-1), listed in order of appearance in A005940.

7, 11, 14, 13, 22, 33, 28, 17, 26, 39, 44, 65, 66, 56, 19, 34, 51, 52, 85, 78, 117, 88, 119, 130, 132, 112, 23, 38, 57, 68, 95, 102, 153, 104, 133, 170, 255, 156, 234, 176, 209, 238, 260, 264, 224, 247, 361, 29, 46, 69, 76, 115, 114, 171, 136, 161, 190, 285, 204

Offset: 1

Author

Michael De Vlieger, Aug 07 2022

Keywords

• nonn

Comments

Primes p > 5 are in the sequence. We may rewrite the k-th prime p instead as prime(k)^1, and thus convert it to a single run of 1 ones followed by (k-1) zeros, which becomes a binary number consisting of 1 followed by k-1 zeros. It is clear that for p > 5, prime(k) < 2^(k-1).

Examples

			7 is in the sequence since 7 = prime(3+1)^1, which we write as 1 following 3 zeros when approached from the least significant digit, i.e., "1000"_2 = 8, thus A005940(8) = 7; and 7 < 8.
5 is not in the sequence since 5 = prime(2+1)^1 -> "100"_2 = 4, and 5 > 4.
14 is in the sequence since 14 = prime(0+1)^1 * prime(3+1)^1, which we can express as a binary number with singleton 1s following 0 and 3 zeros, i.e., "10001"_2 = 17, hence A005940(17) = 14 and we see 14 < 17.
33 is in the sequence since 33 = prime(1+1)^1 * prime(4+1)^1 = A005940("100010"_2) = A005940(34) = 33, and we see 33 < 34.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..13175 (terms in A005940 for n < 2^21)
• Michael De Vlieger, Fan style binary tree diagram of b(n) for n = 1..2^14-1, where b(n) = A005940(n), highlighting terms such that b(n+1) < n in red, b(n+1) = n in ochre, and b(n+1) > n in blue. Terms shown in red appear in this sequence, while b(A029747(n)+1) = A029747(n) appears in ochre.

Crossrefs

Cf. A005940, A029747, A356450.

Programs

Mathematica

nn = 2^10; a[0] = 1; Reap[Do[k = Prime[1 + DigitCount[n, 2, 0]]*a[n - 2^Floor@ Log2@ n]; Set[a[n], k]; If[k < n, Sow[k]], {n, nn}]][[-1, -1]]