A188597 Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

1, 3, 9, 15, 45, 105, 315, 1155, 26325, 33705, 449295, 1805475, 10240425, 13800465, 16029405, 16286445, 21003885, 32062485, 132701205, 594397485, 815634435, 29169504045, 40833636525, 295612416135, 636988686495, 660733931655, 724387847085, 740099543085, 1707894294975, 4439852974095, 7454198513685

Offset: 1

Author

T. D. Noe, Apr 05 2011

Keywords

• nonn

Comments

The abundancy of a number k is defined as A(k) = sigma(k)/k. Deficient numbers have an abundancy less than 2. This sequence has terms in common with A171929. Sequence A188263, which deals with abundant numbers, approaches 2 from above. The similar sequence for even numbers consists of the powers of 2.

a(29) > 10^12. - Donovan Johnson, Apr 08 2011

This sequence is finite iff there is an odd perfect number (which would have abundancy 2). Otherwise, one always has a subsequent term a(n+1) <= a(n)*p where p is the smallest prime not dividing a(n) and larger than 1/(2/A(a(n))-1). Indeed, such an a(n)*p is still deficient but has abundancy larger than a(n), thus closer to 2. - M. F. Hasler, Feb 22 2017

From M. F. Hasler, Jan 25 2020: (Start)

The upper bounds a(n)*p mentioned above are often terms of the sequence, but not the subsequent but a later one: e.g., 9*5 = 45, 15*7 = 105, 45*7 = 315, 105*11 = 1155, 315*107 = 33705, 1155*389 = 449295, 26325*389 = 10240425, ...

Is 9 the largest term not divisible by 15? Only 7 of the 26 terms listed after 45 are not multiples of 7: is this subsequence finite? (End)

Links

• Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)

Crossrefs

Cf. A171929 (odd numbers whose abundancy is closer to 2 than any smaller odd number).

Programs

Mathematica

k = 1; minDiff = 1; Join[{k}, Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; 2 - abun > minDiff || abun => 2, k = k + 2]; minDiff = 2 - abun; k, {10}]]

Extensions

a(22)-a(28) from Donovan Johnson, Apr 08 2011