A066110 -id:A066110 - cp's OEIS frontend

A054546 First differences of nonprimes (including 0 and 1, A002808).

1, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

G. L. Honaker, Jr., Apr 09 2000

Sum of first n terms equals n-th nonprime number.

First differences of A141468. - Omar E. Pol, Oct 21 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A141468, A066110, A001223

t=Flatten[Position[Table[PrimeQ[w], {w, 2, 256}], False]]+1 Delete[t-RotateRight[t], 1]
Differences[Select[Range[0,200],!PrimeQ[#]&]] (* Harvey P. Dale, May 27 2018 *)

a(n) = A018252(n) - A141468(n). - Omar E. Pol, Oct 21 2011

More terms from James Sellers, Apr 11 2000

A066111 Prime powers m such that sigma_4(m^2)/sigma_2(m^2) is prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 31, 43, 61, 83, 109, 121, 125, 131, 229, 239, 257, 263, 269, 311, 313, 343, 361, 443, 463, 503, 571, 593, 599, 619, 641, 647, 653, 659, 701, 797, 811, 853, 953, 967, 1009, 1031, 1039, 1063, 1123, 1373, 1459, 1483, 1499, 1663, 1669, 1693

Offset: 1

Labos Elemer, Dec 06 2001

nonn

Numbers m = p^w such that A001159(m^2)/A001157(m^2) is prime, i.e., m^2 is in A066109.

Also m is the square root of a term from A066109 (omitting the term 20). Apart from 20, up to 10000000 A066109 consists of squares of prime powers.

			m=125: m^2 = 15625 = A066109(13), sigma_4(15625) = 59700165039453751, sigma_2(15625) = 254313151, sigma_4/sigma_2 = 234750601 = A066110(13) is prime. Observe also that sigma_2 is close to sigma_4/sigma_2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157, A001159, A066109, A066110.

isok(m) = isprimepower(m) && isprime(sigma(m^2, 4)/sigma(m^2, 2)); \\ Michel Marcus, Apr 06 2020

A066112 Numbers k such that sigma_4(k)/sigma_2(k) is an integer but not a prime.

Original entry on oeis.org

1, 16, 36, 48, 49, 64, 81, 100, 121, 144, 162, 180, 196, 225, 245, 256, 324, 361, 400, 432, 441, 484, 500, 529, 576, 605, 625, 648, 676, 729, 784, 841, 900, 931, 980, 1024, 1089, 1156, 1200, 1225, 1280, 1296, 1369, 1444, 1521, 1600, 1620, 1681, 1764, 1805

Offset: 1

Labos Elemer, Dec 06 2001

nonn

			The sequence includes squares, twice squares (such as 162 and 648), and other numbers (such as 48 and 180). The sigma_4/sigma_2 quotients usually have more than one distinct prime factor. Exception: sigma_4(48)/sigma_2(48) = 5732210/3410 = 1681 = 41^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001157, A001159, A046871, A066109, A066110, A066111.

Do[s=DivisorSigma[4, n]; z=DivisorSigma[2, n]; If[IntegerQ[s/z]&&!PrimeQ[s/z], Print[n]], {n, 1, 10000}]

isok(k) = { my(f=sigma(k, 4)/sigma(k, 2)); !frac(f) && !isprime(f) } \\ Harry J. Smith, Feb 01 2010

Edited by Jon E. Schoenfield, Dec 24 2016

A066134 Numbers from A066112 that are neither square nor twice a square, i.e., are not in A028982 but are in A028983.

Original entry on oeis.org

48, 180, 245, 432, 500, 605, 931, 980, 1200, 1280, 1620, 1805, 2205, 2352, 2420, 3380, 3724, 3888, 3920, 4500, 4655, 5445, 5780, 5808, 6125, 6845, 7203, 7220, 7936, 8112, 8379, 8405, 8820, 9072, 9251, 9680, 10580, 10800, 11520, 11760, 12500

Offset: 1

Labos Elemer, Dec 06 2001

nonn

			180 is neither square nor twice a square, but sigma_4(180)/sigma_2(180) = 1135275414/49686 = 22849 = 73*313.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001157, A001159, A028982, A028983, A046871, A066109, A066110, A066111, A066112.

q[k_] := !IntegerQ[Sqrt[k]] && !IntegerQ[Sqrt[k/2]] && IntegerQ[r = DivisorSigma[4, k]/DivisorSigma[2, k]] && !PrimeQ[r]; Select[Range[12500], q] (* Amiram Eldar, Feb 23 2025 *)

{ n=0; for (m=1, 10^9, if (issquare(m) || issquare(m/2), next); if (frac(f=sigma(m, 4)/sigma(m, 2)), next); if (!isprime(f), write("b066134.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 02 2010

cp's OEIS Frontend

A054546 First differences of nonprimes (including 0 and 1, A002808).

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A066111 Prime powers m such that sigma_4(m^2)/sigma_2(m^2) is prime.

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A066112 Numbers k such that sigma_4(k)/sigma_2(k) is an integer but not a prime.

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A066134 Numbers from A066112 that are neither square nor twice a square, i.e., are not in A028982 but are in A028983.

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