A323726 -id:A323726

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A359666

Integers k such that sigma(k) <= sigma(k+1) <= sigma(k+2) <= sigma(k+3), where sigma is the sum of divisors.

+10
1

1, 13, 61, 73, 133, 145, 193, 205, 253, 397, 457, 481, 493, 553, 565, 613, 625, 661, 673, 733, 757, 793, 817, 853, 913, 973, 997, 1033, 1093, 1213, 1237, 1285, 1321, 1333, 1453, 1513, 1537, 1633, 1645, 1657, 1681, 1813, 1825, 1873, 1933, 2077, 2113, 2173, 2233, 2245, 2293, 2413, 2497

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

EXAMPLE

73 is a term because sigma(73)=74 <= sigma(74)=114 <= sigma(75)=124 <= sigma(76)=140.

MATHEMATICA

Position[OrderedQ /@ Partition[DivisorSigma[1, Range[2500]], 4, 1], True] // Flatten (* Amiram Eldar, Feb 28 2023 *)

PROG

(PARI) isok(n)=sigma(n)<=sigma(n+1) && sigma(n+1)<=sigma(n+2) && sigma(n+2)<=sigma(n+3)

CROSSREFS

Cf. A000203, A053224, A323726, A364662.

KEYWORD

nonn,easy

AUTHOR

Alexandru Petrescu, Feb 28 2023

STATUS

approved

A333041

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..53.

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

A323726 is a subsequence.

Apart from the first term, a subsequence of A334117.

Cf. A000203, A053222, A231546, A333038, A333039, A333040.

KEYWORD

nonn

AUTHOR

Bernard Schott, Apr 14 2020

STATUS

approved

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