A343293 a(n+1) is the smallest preimage k such that A008477(k) = a(n) with a(1) = 36.

36, 64, 81, 512, 196, 16384, 1089, 8589934592, 3844, 4611686018427387904, 31329, 191561942608236107294793378393788647952342390272950272, 478864

Offset: 1

Author

Bernard Schott, Apr 11 2021

Keywords

• nonn
• more

Comments

Equivalently, when g is the reciprocal map of f = A008477 as defined in the Name, the terms of this sequence are the successive terms of the infinite iterated sequence {m, g(m), g(g(m)), g(g(g(m))), ...} that begins with m = a(1) = 36, hence f(a(n)) = a(n-1).

Why choose 36? Because it is the smallest integer for which there exists such an infinite iterated sequence, with g(36) = 64; then f(36) = 32 with the periodic sequence (32, 25, 32, 25, ...) (see A062307). Explanation: 36 is the first nonsquarefree number in A342973 that is also squareful. The nonsquarefree terms < 36: 12, 18, 20, 24, 28 in A342973 are not squareful (A332785), so they have no preimage by f.

When a(n-1) has several preimages by f, as a(n) is the smallest preimage, this sequence is well defined (see examples).

All the terms are nonsquarefree but also powerful, hence they are in A001694.

a(n) < a(n+2) (last comment in A008477) but a(n) < a(n+1) or a(n) > a(n+1).

Prime factorizations from a(1) to a(13): 2^2*3^2, 2^6, 3^4, 2^9, 2^2*7^2, 2^14, 3^2*11^2, 2^33, 2^2*31^2, 2^62, 3^2*59^2, 2^177, 2^4*173^2.

It appears that a(2m) = 2^q for some q>1 and a(2m+1) = r^2 for some r>1.

a(14) <= 2^692.

Examples

			a(1) = 36; 64 = 2^6 so f(64) = 6^2 = 36, also 192 = 2^6*3^1 and f(192) = 6^2*1^3 = 36 we have f(64) = f(192) = 36; but as 64 < 192, hence g(36) = 64 and a(2) = 64.
a(2) = 64 = f(81) = f(256), but as 81 < 256, g(64) = 81 and a(3) = 81.
a(4) = 512 = f(196) = f(400), but as 196 < 400, g(512) = 196 and a(5) = 196.
		

Links

• Annales Concours Général, Sujet Concours Général 2012 (in French, problems).
• Annales Concours Général, Corrigé Concours Général 2012 (in French, solutions).

Crossrefs

Cf. A001694, A008477, A062307, A332785, A342973.

Extensions

a(10)-a(13) from Bert Dobbelaere, Apr 13 2021