A333038 Numbers m such that sigma(m) <= sigma(m-1).

5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 106, 107, 109, 111, 113, 115, 117, 118, 119, 121, 123

Offset: 1

Author

Bernard Schott, Mar 06 2020

Keywords

• nonn

Comments

This sequence is infinite because all primes p >= 5 are terms with sigma(p) < sigma(p-1).

The integer m is a term iff A053222(m-1) <= 0.

The numbers m such that sigma(m) = sigma(m-1) are in A231546.

Examples

			Sigma(9) = 1+3+9 = 13 < sigma(8) = 1+2+4+8 = 15 so 9 is a term.
Sigma(15) = 1+3+5+15 = 24 = sigma(14) = 1+2+7+14 = 24 so 15 is a term.
Sigma(63) = 1+3+7+9+21+63 = 104 > sigma(62) = 1+2+31+62 = 96 and 63 is not a term.
		

References

• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004

Crossrefs

Cf. A000203, A053222, A231546 (subsequence: sigma(m) = sigma(m-1)).

Cf. A053224 (sigma(m) < sigma(m+1)), A053226 (sigma(m) > sigma(m+1)).

Programs

Maple

filter:= m -> sigma(m) <= sigma(m-1): select(filter, [$1..500]);

Mathematica

Select[Range[2, 123], DivisorSigma[1, #] <= DivisorSigma[1, # - 1] &] (* Amiram Eldar, Mar 06 2020 *)
Flatten[Position[Partition[DivisorSigma[1,Range[200]],2,1],?(#[[2]]<= #[[1]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Mar 28 2020 *)

PARI

isok(m) = (m>1) && (sigma(m) <= sigma(m-1)); \\ Michel Marcus, Mar 09 2020