A333039 -id:A333039 - cp's OEIS frontend

A333040 Even numbers m such that sigma(m) < sigma(m-1).

46, 106, 118, 166, 226, 274, 298, 316, 346, 358, 406, 466, 514, 526, 562, 586, 622, 694, 706, 766, 778, 826, 838, 862, 886, 946, 1006, 1114, 1126, 1156, 1186, 1198, 1282, 1306, 1366, 1396, 1426, 1486, 1522, 1546, 1576, 1594, 1618, 1702, 1726, 1756

Offset: 1

Bernard Schott, Mar 22 2020

nonn

The even terms of A333039 represent about only 7% of the data, so they are proposed in this sequence. Some of these integers are semiprimes with for example these two families:
1) m = 2*p with p prime of the form k^2+k+3 is in A027753. The first few terms are: 46, 118, 226, 766, ... but not all the integers of this form are terms; the first 3 counterexamples are 6, 10, 1018 (see examples).
2) m = 2*p with p prime = (r*s*t+1)/2 and 2A234103. The first few terms are: 106, 166, 274, 346, 358, ... but not all the integers of this form are terms; the first 3 counterexamples are 386, 898 and 958 (see examples).
There is also this subsequence of even m = 2^2*p where p prime, congruent to 34 mod 45, is in A142330. The first few terms are: 316, 1396, 1756, 2416, ... but not all the integers of this form are terms; the first counterexample that comes from the 8th term of A142330 is 5716.
Even (and odd) numbers such that sigma(m) = sigma(m-1) are in A231546.

			166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.
386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.
766 = 2*383 where 383 = 19^2+19+3 and 765 = 3^2*5*13, as 1152 = sigma(766) < sigma(765) = 1404, hence 766 is a term.
1018 = 2*509 where 509 = 22^2+22+3, and 1017 = 3^2*113, as 1530 = sigma(1018) > sigma(1017) = 1482, hence 1018 is not a term.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A333039.
Subsequence of A333038.
Cf. A231546.

filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):
select(filter, [seq(i,i=2..2000,2)]); # Robert Israel, Mar 29 2020

Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* Amiram Eldar, Mar 24 2020 *)

isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 22 2020

A333041 Odd numbers m such that sigma(m) > sigma(m-1).

Original entry on oeis.org

3, 63, 75, 135, 147, 195, 255, 315, 399, 405, 459, 483, 495, 525, 555, 567, 615, 627, 663, 675, 693, 735, 759, 765, 795, 819, 855, 915, 945, 975, 999, 1035, 1095, 1125, 1155, 1215, 1239, 1287, 1323, 1395, 1455, 1515, 1539, 1575, 1647, 1659, 1683, 1755, 1785, 1815, 1827, 1845, 1875

Offset: 1

Bernard Schott, Apr 14 2020

nonn

The odd terms of A333038 [sigma(m) <= sigma(m-1)] represent about 95% of the data, so the odd integers that do not satisfy this relation are proposed here.
Except for 3, there are no prime powers in this sequence.
It appears that most of the terms are divisible by 3; the two smallest exceptions are 13475 and 17255 (see A323726).
Odd (and even) numbers such that sigma(m) = sigma(m-1) are in A231546.

			sigma(63) = 1+3+7+9+21+63 = 104 > sigma(62) = 1+2+31+62=96 and 63 is in the sequence.
sigma(77) = 1+7+11+77 = 96 < sigma(76) = 1+2+4+19+38+76 = 140 and 77 is not a term.

A323726 is a subsequence.
Apart from the first term, a subsequence of A334117.
Cf. A000203, A053222, A231546, A333038, A333039, A333040.

Select[2 * Range[1000] + 1, DivisorSigma[1, #] > DivisorSigma[1, # - 1] &] (* Amiram Eldar, Apr 14 2020 *)

is(n)=n%2 && sigma(n)>sigma(n-1) \\ Charles R Greathouse IV, Apr 14 2020

cp's OEIS Frontend

A333040 Even numbers m such that sigma(m) < sigma(m-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI