A293715 Numbers k such that A007755(k) is prime.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 18, 19, 21, 23, 24, 27, 28, 31, 33, 43, 51, 53, 54, 57, 60, 61, 62, 65, 67, 68, 69, 71, 73, 76, 79, 81, 83, 84, 89, 91, 110, 111, 115, 116, 118, 121, 124, 126, 129, 131, 132, 138, 139, 144, 145, 147, 149, 150, 153, 156

Offset: 1

Author

Amiram Eldar, Oct 15 2017

Keywords

• nonn

Comments

Shapiro conjectured that A007755(n) is prime for all n > 1, and verified it up to n = 10. Mills showed that A007755(34)=(2^16+1)^2 is composite.

The least number n such that Omega(A007755(n)) = 1, 2, 3, ... is 2, 13, 30, 58, 74, 90, 106, 122, 146, 162, 178, 194, 210, 226, ... (Omega is the number of prime factors with multiplicity, A001222).

Examples

			The first 11 values of A007755(n) after n=1 are the primes: 2, 3, 5, 11, 17, 41, 83, 137, 257, 641, 1097, 2329, therefore 2-12 are in the sequence.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, B41, p. 148.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..289
• W. H. Mills, Iteration of the phi function, The American Mathematical Monthly, Vol. 50.9 (1943), pp. 547-549.
• Harold Shapiro, An arithmetic function arising from the phi function, The American Mathematical Monthly, Vol. 50, No. 1 (1943), pp. 18-30.

Crossrefs

Cf. A007755, A092873.

Programs

Mathematica

s = Import[b007755.txt", "Data"][[All, 2]]; a = Flatten[Position[s, _?(PrimeQ[#] &)]] (* using the b-File from A007755 *)