A129246 -id:A129246 - cp's OEIS frontend

A056637 a(n) is the least prime of class n-, according to the Erdős-Selfridge classification of primes.

2, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107, 1432349099, 22111003847, 110874748763

Offset: 1

Robert G. Wilson v, Jan 31 2001

A prime p is in class 1- if p-1 has no prime factor larger than 3. If p-1 has other prime factors, p is in class (c+1)-, where c- is the largest class of its prime factors. See also A005109.
1432349099 < a(16) <= 25782283783.
a(18) <= 619108107719, a(19) <= 19811459447009, a(20) <= 152772264735359. These upper limits can be found by generating class (n+1)- primes from a list of n- class primes; if the latter is sufficiently complete, one can deduce that there is no smaller (n+1)- prime. - M. F. Hasler, Apr 05 2007

Cf. A005113, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.

Cf. A082449, A129246, A081640, A129248.

PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; a = Table[0, {15}]; a[[1]] = 2; k = 5; Do[c = ClassMinusNbr[ k]; If[ a[[c]] == 0, a[[c]] = k]; k = NextPrime[k], {n, 3, 7223000}]; a

a(n+1) >= 2*a(n)+1, since a(n+1)-1 is even and must have a factor of class n- which is odd (n>1) and >= a(n). a(n+1) <= min { p = 2*k*a(n)+1 | k=1,2,3... such that p is prime }, since a(n) is a prime of class n-. - M. F. Hasler, Apr 05 2007

Extended by Robert G. Wilson v, Mar 20 2003
More terms from Don Reble, Apr 11 2003
a(16) and a(17) from M. F. Hasler, Apr 21 2007

A090896 a(n) = sigma^n(n), where sigma^n denotes functional iteration.

Original entry on oeis.org

1, 4, 8, 24, 120, 1170, 1512, 15748, 15748, 61440, 196584, 1572840, 1098240, 15605760, 50328576, 7074432, 355532800, 2143289344, 13815410400, 224319234048, 33541980160, 109760857440, 1468475386560, 33151875434496, 1415960985600, 1510712608204800

Offset: 1

Gabriel Cunningham (gcasey(AT)mit.edu), Feb 25 2004

nonn

a(8) = a(9). Are there any other numbers m, n such that a(m) = a(n)? a(16) < a(15); a(25) < a(24). How often is a(n+1) < a(n)?

a(88) = a(89), a(108) = a(109) and a(124) = a(125). - Seiichi Manyama, May 08 2021

			a(3) = sigma^3(3) = sigma^2(4) = sigma(7) = 8

Seiichi Manyama, Table of n, a(n) for n = 1..200

Main diagonal of A129246.

Cf. A000203, A101303.

a:= n-> (numtheory[sigma]@@n)(n):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 03 2018

a[n_] := Nest[DivisorSigma[1, #]&, n, n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 30 2025 *)

a(n) = my(k=sigma(n)); for(n=2, n, k=sigma(k)); k; \\ Seiichi Manyama, May 08 2021

A175837 (2n-1)-abundant numbers.

Original entry on oeis.org

12, 18, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252

Offset: 1

Vladimir Shevelev, Dec 05 2010

nonn

A number k is (2n-1)-abundant if sum_{d|k, d 2*k-1, a specialization of the definition in A175522.
Adding 2k-1 on both sides of the condition yields the equivalent condition A129246(k) > 2*(2k-1).
Adding 2k-1 on both sides also yields sum_{d|k} (2*d-1) > 2*(2k-1), equivalent to 2*sum_{d|k}d - tau(k) > 2*(2k-1) or sigma(k) > 2k-1+tau(k)/2, equivalent to A033880(k) > tau(k)/2-1.

Cf. A175522, A033880, A000203, A005100, A005101.

aQ[n_] := DivisorSum[n, 2#-1&, # 2n-1; Select[Range[252], aQ] (* Amiram Eldar, Feb 18 2019 *)

A175837 = { n | A033880(n) > A000005(n)/2-1 }.

More terms from Amiram Eldar, Feb 18 2019

A173430 Last of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 15, 15, 15, 6, 6, 15, 15, 14, 10, 12, 12, 14, 14, 15, 104, 18, 18, 20, 20, 104, 22, 24, 24, 104, 26, 40, 28, 30, 30, 104, 104, 33, 34, 48, 91, 38, 38, 56, 40, 42, 42, 44, 44, 45, 46, 48, 48, 80, 255, 51, 52, 54, 54, 72, 56, 80, 58, 60, 60, 62, 62, 104, 255, 84, 66, 68, 68

Offset: 1

Walter Nissen, Feb 18 2010

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 4 ) ) ... ) the iterates are 4, 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 4, 7, 8, 15, so a(4) = 15 .
Here sigma ( 4 ) = 7, sigma ( sigma ( 4 ) ) = sigma ( 7 ) = 8, etc.

Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, Vol. 5, No. 2 (1996), pp. 91-100.
Leonard Eugene Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality, Carnegie Institution of Washington, 1919, Chapters II and X

Cf. A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

a[1] = 1; a[n_] := Module[{k = n}, While[CoprimeQ[k, (s = DivisorSigma[1, k])], k = s]; k]; Array[a, 68] (* Amiram Eldar, Sep 02 2019 *)

A173431 Count of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 6, 5, 4, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 2, 1, 4, 3, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 5, 2, 1, 2, 1, 1

Offset: 1

Walter Nissen, Feb 18 2010

The last of these iterates is the value in A173430.

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 7 ) ) ... ) the iterates are 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 7, 8, 15, and there are 3 of these, so a(7) = 3.
Here sigma ( 7 ) = 8, sigma ( sigma ( 7 ) ) = sigma ( 8 ) = 15, etc.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Leonard Eugene Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality, Carnegie Institution of Washington, 1919, Chapters II and X

Cf. A173430, A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

a(n)=my(t,s);if(n==1,1,while(1,s++;t=sigma(n);if(gcd(t,n)==1,n=t,return(s)))) \\ Charles R Greathouse IV, Feb 06 2012

cp's OEIS Frontend

A056637 a(n) is the least prime of class n-, according to the Erdős-Selfridge classification of primes.

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A090896 a(n) = sigma^n(n), where sigma^n denotes functional iteration.

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A175837 (2n-1)-abundant numbers.

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A173430 Last of consecutive coprime iterations of sum-of-divisors function.

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Crossrefs

Programs

Mathematica

A173431 Count of consecutive coprime iterations of sum-of-divisors function.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI