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Solutions to sigma(k) + 7 = sigma(k+7).
2

%I #34 Nov 27 2022 11:16:16

%S 74,531434,387420482,2541865828322

%N Solutions to sigma(k) + 7 = sigma(k+7).

%C No other solutions < 4290000000. Sequence A063679 shows how to generate more solutions, but there may be solutions other than those produced by A063679.

%C No others < 10^17. - _Seth A. Troisi_, Oct 25 2022

%C k or k+7 must be a square or twice a square (A028982). See comment in A015886. - _Seth A. Troisi_, Oct 26 2022

%C From _Jon E. Schoenfield_, Oct 26 2022: (Start)

%C Each of the first 4 terms of the sequence is of the form k = 9^j - 7:

%C 74 = 9^2 - 7,

%C 531434 = 9^6 - 7,

%C 387420482 = 9^9 - 7,

%C 2541865828322 = 9^13 - 7.

%C The next terms of this form are 9^53 - 7 and 9^82 - 7.

%C Does the sequence contain any terms that are not of this form?

%C (End)

%e sigma(74) + 7 = 121 = sigma(74+7), so 74 is in the sequence.

%o (PARI) isok(k) = sigma(k) + 7 == sigma(k+7); \\ _Michel Marcus_, Oct 25 2022

%Y Cf. A000203, A063679.

%Y Cf. A054799, A015913, A015914, A015915, A015916, A015917, A054905.

%K nonn,bref,hard,more

%O 1,1

%A _Jud McCranie_, Jul 28 2001

%E a(4) from _Seth A. Troisi_, Oct 24 2022