A110585 -id:A110585 - cp's OEIS frontend

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

Harvey P. Dale, Sep 17 2001

nonn

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

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

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.

Intersection of A005231 and A047201.
Cf. A006038, A047802, A110585.
Cf. A020338.

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 *)

{ 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

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

A112640 Odd abundant numbers divisible neither by 5 nor 7.

Original entry on oeis.org

28683369, 36165987, 38660193, 86050107, 108497961, 115980579, 131339637, 138428433, 140397543, 153393669, 160876287, 258150321, 277272567, 296394813, 315517059, 325493883, 347941737, 353761551, 372883797, 373715199, 392006043

Offset: 1

Labos Elemer, Sep 15 2005

nonn

Subsequence of A064001 which itself is a subsequence of A005231. All 500 terms in b-file are divisible by 99. Cf. also A047802. - Zak Seidov, Mar 30 2011
From Amiram Eldar, Aug 15 2024: (Start)
The least term that is not divisible by 99 is a(1718) = 21097921689.
The least term that is not divisible by 3 is 149#/7# = Product_{k=5..35} prime(k) = 7105630242567996762185122555313528897845637444413640621. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Zak Seidov)
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.

Cf. A005231, A047802, A064001, A110585.

Do[If[Mod[n,5]>0&&Mod[n,7]>0&&DivisorSigma[1, n]>2n, Print[n]], {n, 28683369, 6*10^9,2}] (* Zak Seidov, Mar 30 2011 *)

is(n)=gcd(n,70)==1 && sigma(n,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

Extended by Ray Chandler, Sep 19 2005

A112642 Primorial number quotients arising in A007684: a(n) = A002110(A007684(n))/A002110(n-1).

Original entry on oeis.org

6, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621, 1924344668948998025181489521338230544342953524990122861050411878226909135705454891961917517

Offset: 1

Labos Elemer, Sep 19 2005

nonn

These numbers are (perhaps the smallest) squarefree solutions to Puzzle 329 of Rivera; a(n) is abundant, not divisible by the first n-1 prime numbers, i.e., the least prime divisor of a(n) is the n-th prime number.

Duplicate of A007702.

			The corresponding sigma(a(n))/a(n) abundance ratios are as follows: 2, 2.14825, 2.00097, 2.01433, 2.00587, 2.00101, ...;
the terms have 2,3,5,7,11,... as least prime divisors.

Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.

Cf. A007702, A002110, A007684, A064001, A110585, A112640.

a(n) = A002110(A007684(n))/A002110(n-1).

Term a(2) and name corrected by Andrey Zabolotskiy, Jul 16 2022

cp's OEIS Frontend

A064001 Odd abundant numbers not divisible by 5.

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A112640 Odd abundant numbers divisible neither by 5 nor 7.

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A112642 Primorial number quotients arising in A007684: a(n) = A002110(A007684(n))/A002110(n-1).

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