A343302 Numbers k such that k through k+4 are all deficient (in A005100).

1, 7, 13, 31, 43, 49, 61, 73, 91, 115, 121, 127, 133, 145, 151, 163, 169, 181, 187, 211, 229, 235, 241, 247, 253, 265, 283, 289, 295, 313, 325, 331, 343, 355, 373, 385, 403, 409, 421, 427, 433, 451, 469, 481, 505, 511, 523, 535, 553, 565, 583, 589, 595

Offset: 1

Author

Jianing Song, Apr 11 2021

Keywords

• nonn

Comments

Since every multiple of 6 (other than 6 itself) is an abundant number, the maximum length of consecutive runs of deficient numbers is 5.

All terms are congruent to 1 modulo 6.

This is a proper subset of A231626, with the smallest missing term being 347: here only the first members of 5 consecutive deficient numbers in arithmetic progression with common difference 1 are allowed. Terms of A231626 that are not here are listed in A343303.

Examples

			115 is a term since all of 115, 116, 117, 118 and 119 are deficient.
2989 is not a term since 2989 + 3 = 2992 is an abundant number.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005100 (deficient numbers), A316099, A343301.

Set difference of A231626 by A343303.

Programs

Mathematica

Position[Partition[DivisorSigma[-1, Range[600]], 5, 1], ?(Max[#] < 2 &), 1] // Flatten (* _Amiram Eldar, Mar 21 2024 *)
SequencePosition[Table[If[DivisorSigma[1,n]<2n,1,0],{n,600}],{1,1,1,1,1}][[;;,1]] (* Harvey P. Dale, Jul 16 2025 *)

PARI

isA343302(k) = if(k%6!=1, 0, for(i=0, 4, if( sigma(k+i) >= 2*(k+i), return(0) )); 1)