A064001 Odd abundant numbers not divisible by 5.

81081, 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837837, 891891, 908523, 960687, 999999, 1024947, 1054053, 1072071

Offset: 1

Author

Harvey P. Dale, Sep 17 2001

Keywords

• nonn

Comments

Or, odd abundant numbers that do not end in 5.

All terms below 2000000 are divisible by 21 (so by 3). Moreover, except for a few, most are divisible by 231. - Labos Elemer, Sep 15 2005 [The least term that is not divisible by 21 is a(908) = 28683369. - Amiram Eldar, Jan 27 2025]

An odd abundant number (see A005231) not divisible by 3 nor 5 must have at least 15 distinct prime factors (e.g., 61#/5#*7^2*11*13*17, where # is primorial) and be >= 67#/5#*77 = A047802(3) ~ 2.0*10^25. -- The smallest non-primitive abundant number (cf. A006038) in this sequence is 7*a(1) = 567567 = a(14). - M. F. Hasler, Jul 27 2016

There are 26 terms less than 10^6 and a surprising fact is that 18 of them are doublets (cf. A020338). - Omar E. Pol, Jan 17 2025

The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 26, 290, 3071, 31600, 320948, 3174762, 31693948, ... . Apparently, the asymptotic density of this sequence equals 0.000031... . Therefore, the least term not divisible by 3 that was mentioned above is a(~6*10^20) = 20169691981106018776756331. - Amiram Eldar, Jan 27 2025

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. ed. 1997, p. 169.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
• Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.
• Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.

Crossrefs

Intersection of A005231 and A047201.

Cf. A006038, A047802, A110585.

Cf. A020338.

Programs

Mathematica

Select[ Range[ 1, 10^6, 2 ], DivisorSigma[ 1, # ] - 2# > 0 && Mod[ #, 5 ] != 0 & ]
ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&!Equal[Mod[n, 2], 0]&& !Equal[Mod[n, 5], 0], Print[n];ta=Append[ta, n]], {n, 1, 2000000}] ta=Delete[ta, 1] (* Labos Elemer, Sep 15 2005 *)

PARI

{ n=0; forstep (m=1, 10^9, 2, if (m%5 && sigma(m) > 2*m, write("b064001.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 05 2009

Extensions

More terms from Robert G. Wilson v, Sep 28 2001

Further terms from Labos Elemer, Sep 15 2005

Entry revised by N. J. A. Sloane, Mar 28 2006