A110464 -id:A110464 - cp's OEIS frontend

A110465 Prime numbers that when multiplied in order yield the sequence of oddly colossally abundant numbers A110464.

3, 5, 3, 7, 11, 13, 3, 5, 17, 19, 23, 29, 31, 7, 3, 37, 41, 43, 47, 53, 59, 5, 61, 67, 71, 73, 11, 79, 83, 3, 89, 97, 13, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 7, 157, 163, 167, 17, 173, 179, 181, 191, 193, 197, 199, 19, 211, 3, 223, 227, 229, 5, 233, 239

Offset: 1

T. D. Noe, Jul 21 2005

nonn

This is sequence A073751 without the "2" terms.

Cf. A004490 (colossally abundant numbers).

maxN=100; f={{3, 1}, {5, 0}}; primes=1; lst={3}; x=Table[pFactor[f[[i]]], {i, primes+1}]; For[n=2, n<=maxN, n++, i=Position[x, Max[x]][[1, 1]]; AppendTo[lst, f[[i, 1]]]; f[[i, 2]]++; If[i>primes, primes++; AppendTo[f, {Prime[i+2], 0}]; AppendTo[x, pFactor[f[[ -1]]]]]; x[[i]]=pFactor[f[[i]]]]; lst

A119239 Oddly superabundant numbers: odd n with sigma(n)/n > sigma(k)/k for all odd k < n.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 135135, 225225, 405405, 675675, 2027025, 2297295, 3828825, 6891885, 11486475, 34459425, 43648605, 72747675, 130945815, 218243025, 654729075, 1003917915, 1527701175

Offset: 1

T. D. Noe, May 09 2006

nonn

Every oddly colossally abundant number (A110464) is in this sequence.

a(8) = 945 is the first term with abundancy > 2, a(41) = 1018976683725 is the first term with abundancy > 3, and a(141) = 1853070540093840001956842537745897243375 is the first term with abundancy > 4. See A119240. - Antti Karttunen, Jul 21 2025

T. D. Noe, Table of n, a(n) for n = 1..1000
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Abundancy : Some Resources

Cf. A004394 (superabundant numbers), A005231 (odd abundant numbers), A053624 (highly composite odd numbers), A119240.

Cf. also A171929, A228059, A386423.

rec=0; lst={}; Do[abun=DivisorSigma[1,n]/n; If[abun>rec, rec=abun; AppendTo[lst,n]], {n,1,10^6,2}]; lst

r=0;forstep(n=1,1e6,2,t=sigma(n)/n;if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Nov 27 2013

Definition clarified by Jonathan Sondow, Dec 08 2011

A353076 Odd positive integers k such that sigma(k) > exp(gamma) * k * log(log(k))/2.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 81, 87, 93, 99, 105, 117, 135, 147, 153, 165, 171, 189, 195, 207, 225, 231, 255, 273, 285, 297, 315, 345, 351, 357, 375, 399, 405, 435, 441, 465, 495, 525, 555, 567, 585

Offset: 1

Amiram Eldar, Apr 22 2022

nonn

The first 23 oddly colossally abundant numbers (A110464) are in this sequence.

According to a proof by Washington and Yang (2021), the Riemann hypothesis is equivalent to the statement that all the terms of this sequence are smaller than A110464(24) = 18565284664427130919514350125.

			3 is in the sequence since 3 is odd and sigma(3) = 4 > exp(gamma) * 3 * log(log(3))/2 = 0.251... .

Amiram Eldar, Table of n, a(n) for n = 1..470
Lawrence C. Washington and Ambrose Yang, Analogues of the Robin-Lagarias criteria for the Riemann hypothesis, International Journal of Number Theory, Vol. 17, No. 4 (2021), pp. 843-870; arXiv preprint, arXiv:2008.04787 [math.NT], 2020.

Cf. A000203 (sigma), A067698, A073004 (exp(gamma)), A110464.

Select[Range[3, 600, 2], DivisorSigma[1, #] > Exp[EulerGamma] * # * Log[Log[#]]/2 &]

from sympy import divisor_sigma, EulerGamma, E, log
print([k for k in range(3, 600, 2) if divisor_sigma(k) > (E**EulerGamma * k * log(log(k)) / 2)]) # Karl-Heinz Hofmann, Apr 22 2022

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A110465 Prime numbers that when multiplied in order yield the sequence of oddly colossally abundant numbers A110464.

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A119239 Oddly superabundant numbers: odd n with sigma(n)/n > sigma(k)/k for all odd k < n.

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A353076 Odd positive integers k such that sigma(k) > exp(gamma) * k * log(log(k))/2.

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