A007497 -id:A007497 - cp's OEIS frontend

A120611 Sum of previous term and preceding divisors.

1, 2, 3, 4, 7, 8, 15, 19, 20, 27, 31, 32, 47, 48, 66, 72, 90, 111, 115, 116, 123, 127, 128, 175, 183, 187, 188, 242, 245, 253, 254, 384, 610, 613, 614, 617, 618, 624, 690, 826, 836, 862, 865, 866, 869, 870, 891, 922, 925, 926, 929, 930, 982, 985, 986, 989, 990

Offset: 1

Franklin T. Adams-Watters, Jun 16 2006

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A120610, A007497.

a = {1, 2}; Do[AppendTo[a, Plus @@ Intersection[a, Divisors[a[[-1]]]]], {n, 3, 57}]; a (* Ivan Neretin, May 13 2015 *)

a(1) = 1, a(2) = 2, for n>=2, a(n+1) = a(n) + sum_{1<=k

A283166 a(0) = 0; a(1) = 1; a(2n) = sigma(a(n)), a(2n+1) = sigma(a(n)) + sigma(a(n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 4, 1, 8, 7, 11, 4, 11, 7, 8, 1, 16, 15, 23, 8, 20, 12, 19, 7, 19, 12, 20, 8, 23, 15, 16, 1, 32, 31, 55, 24, 48, 24, 39, 15, 57, 42, 70, 28, 48, 20, 28, 8, 28, 20, 48, 28, 70, 42, 57, 15, 39, 24, 48, 24, 55, 31, 32, 1, 64, 63, 95, 32, 104, 72, 132, 60, 184, 124, 184, 60, 116, 56, 80, 24, 104, 80, 176, 96, 240, 144, 200, 56, 180, 124

Offset: 0

Ilya Gutkovskiy, Mar 02 2017

nonn

A variation on Stern's diatomic sequence (A002487) and iterating the sum of the divisors function (A007497).

			a(0) = 0;
a(1) = 1;
a(2) = a(2*1) = sigma(a(1)) = sigma(1) = 1;
a(3) = a(2*1+1) = sigma(a(1)) + sigma(a(2)) = sigma(1) + sigma(1) = 1 + 1  = 2;
a(4) = a(2*2) = sigma(a(2)) =  sigma(1) = 1;
a(5) = a(2*2+1) = sigma(a(2)) + sigma(a(3)) =  sigma(1) + sigma(2) = 1 + 3 = 4, etc.

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences related to sigma(n)

Cf. A000203, A002487, A007497.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], DivisorSigma[1, a[n/2]], DivisorSigma[1, a[(n - 1)/2]] + DivisorSigma[1, a[(n + 1)/2]]]; Table[a[n], {n, 0, 90}]

a(n) = if (n<2, n, if (n%2==0, sigma(a(n/2)), sigma(a((n-1)/2))+sigma(a((n+1)/2))));
tabl(nn)={for (n=0, nn, print1(a(n), ", "); ); };
tabl(90); \\ Indranil Ghosh, Mar 03 2017

A059460 Iteration of unitary-sigma function: a(1) = 2, a(n) = usigma(a(n-1)).

Original entry on oeis.org

2, 3, 4, 5, 6, 12, 20, 30, 72, 90, 180, 300, 520, 756, 1120, 1584, 2040, 3888, 4148, 5580, 9600, 13416, 22176, 31680, 46800, 61880, 108864, 126880, 171864, 276480, 344232, 492480, 639600, 1039584, 1663200, 2306304, 3454080, 6390144

Offset: 1

Yasutoshi Kohmoto

nonn

Amiram Eldar, Table of n, a(n) for n = 1..6000

Cf. A007497, A034448.

f[n_] := Times @@ (1 + Power@@@ FactorInteger[n]); NestList[f, 2, 30] (* Amiram Eldar, Aug 11 2019 *)

Corrected by Jud McCranie, Oct 28 2001

A126850 a(n) = OrdinaryUnitarySigma(a(n-1)).

Original entry on oeis.org

2, 3, 4, 7, 8, 15, 24, 60, 168, 480, 1512, 3360, 12096, 28448, 64512, 163760, 401760, 991872, 2399040, 6858000, 13999104, 32752000, 69400800, 172186560, 517867392, 1666990080, 5662137600, 14475575296, 33946612000, 73359820800, 158022774000

Offset: 2

Yasutoshi Kohmoto, Feb 24 2007

nonn

Cf. A107749, A126849.

Cf. A107749, A126849, A007497.

A034448 := proc(n) local ifs,d ; if n = 1 then 1; else ifs := ifactors(n)[2] ; mul(1+ op(1,op(d,ifs))^op(2,op(d,ifs)),d=1..nops(ifs)) ; fi ; end: A006519 := proc(n) local i ; for i in ifactors(n)[2] do if op(1,i) = 2 then RETURN( op(1,i)^op(2,i) ) ; fi ; od: RETURN(1) ; end: A107749 := proc(n) local p2 ; p2 := A006519(n) ; numtheory[sigma](p2)*A034448(n/p2) ; end: A126850 := proc(n) option remember ; if n = 1 then 2; else A107749(A126850(n-1)) ; fi ; end: seq(A126850(n),n=1..40) ; # R. J. Mathar, Jun 15 2008

f[2, e_] := 2^(e + 1) - 1;
f[p_, e_] := p^e + 1;
A107749[n_] := If[n == 1, 1, Times @@ f @@@ FactorInteger[n]];
a[n_] := a[n] = If[n == 2, 2, A107749[a[n - 1]]];
Table[a[n], {n, 2, 32}] (* Jean-François Alcover, Jul 22 2024, after Amiram Eldar in A107749 *)

a(n)= A107749(a(n-1)). - R. J. Mathar, Jun 15 2008

Edited and extended by R. J. Mathar, Jun 15 2008

A283167 a(0) = 1; a(2n) = 2a(n), a(2*n+1) = sigma(a(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 2, 1, 8, 7, 6, 4, 4, 3, 2, 1, 16, 15, 14, 8, 12, 12, 8, 7, 8, 7, 6, 4, 4, 3, 2, 1, 32, 31, 30, 24, 28, 24, 16, 15, 24, 28, 24, 28, 16, 15, 14, 8, 16, 15, 14, 8, 12, 12, 8, 7, 8, 7, 6, 4, 4, 3, 2, 1, 64, 63, 62, 32, 60, 72, 48, 60, 56, 56, 48, 60, 32, 31, 30, 24, 48, 60, 56, 56, 48, 60, 56, 56, 32, 31, 30

Offset: 0

Ilya Gutkovskiy, Mar 02 2017

nonn

			a(0) = 1;
a(1) = a(2*0+1) = sigma(a(0)) = sigma(1) = 1;
a(2) = a(2*1) = 2*a(1) = 2*1 = 2;
a(3) = a(2*1+1) = sigma(a(1)) = sigma(1) = 1;
a(4) = a(2*2) = 2*a(2) = 2*2 = 4;
a(5) = a(2*2+1) = sigma(a(2)) = sigma(2) = 1 + 2 = 3, etc.

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences related to sigma(n)

Cf. A000203, A007497, A033879, A033880.

a[0] = 1; a[n_] := If[EvenQ[n], 2 a[n/2], DivisorSigma[1, a[(n - 1)/2]]]; Table[a[n], {n, 0, 90}]

a(n) = if (n==0, 1, if (n%2==0, 2*a(n/2),sigma(a((n-1)/2))));tabl(nn)={for (n=0,nn,print1(a(n),", "););};tabl(90) \\ Indranil Ghosh, Mar 03 2017

a(2*n) - a(2*n+1) = A033879(a(n)).

a(2*n+1) - a(2*n) = A033880(a(n)).

A283764 a(0) = 0; a(1) = 1; a(2n) = sigma(a(n)), a(2n+1) = sigma(a(n)+a(n+1)).

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 4, 7, 1, 15, 8, 12, 7, 12, 8, 15, 1, 31, 24, 24, 15, 42, 28, 20, 8, 20, 28, 42, 15, 24, 24, 31, 1, 63, 32, 72, 60, 124, 60, 56, 24, 80, 96, 144, 56, 124, 42, 56, 15, 56, 42, 124, 56, 144, 96, 80, 24, 56, 60, 124, 60, 72, 32, 63, 1, 127, 104, 120, 63, 210, 195, 336, 168, 360, 224, 360, 168, 210, 120, 186, 60, 210, 186, 372, 252, 744, 403, 465, 120, 546

Offset: 0

Ilya Gutkovskiy, Mar 16 2017

nonn

A variation on Stern's diatomic sequence (A002487) and iterating the sum of the divisors function (A007497).

Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences related to sigma(n)

Cf. A000203, A002487, A007497, A283166.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], DivisorSigma[1, a[n/2]], DivisorSigma[1, a[(n - 1)/2] + a[(n + 1)/2]]]; Table[a[n], {n, 0, 100}]

a(n) = if(n<2, n, if(Mod(n,2), sigma(a((n - 1)/2) + a((n + 1)/2)), sigma(a(n/2))));
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Mar 16 2017

A286011 a(1)=1, and for n>1, a(n) is the maximum number of iterations of sigma resulting in n, starting at some integer k; or 0 if n cannot be reached from any k.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 3, 4, 0, 0, 0, 2, 1, 2, 5, 0, 0, 1, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 1, 1, 2, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 1, 0, 0, 7, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2

Offset: 1

Michel Marcus, Apr 30 2017

nonn

a(n)=0 for n in A007369 and a(n)>0 for n in A002191.
Records are found at indices given by A007497.
The above would be correct for a(1) = 0 (in a weak sense) or rather a(1) = -1 (for infinity), but as the sequence is defined, 2 & 3 do not produce a record, so the indices of records are 1, (3), 4, 7, ... = {1} U A007497 \ {2, (3)}. - M. F. Hasler, Nov 20 2019

			a(4)=2 because 4=sigma(3), but also sigma(sigma(2)) with 2 iterations.
a(7)=3 because 7=sigma(4), but also sigma(sigma(3)), and sigma(sigma(sigma(2))), with 3 iterations.

Robert Israel, Table of n, a(n) for n = 1..10000
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005

Cf. A000203, A002191, A007369, A007497, A257670.

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
for n from 1 to N do
  s:= numtheory:-sigma(n);
  if s <= N then V[s]:= max(V[s],V[n]+1) fi
od:
convert(V,list); # Robert Israel, May 01 2017

a(n) = {if (n==1, return(1)); vn = vector(n-1, k, k+1); nb = 0; knb = 0; ok = 1; while(ok, nb++; vn = vector(#vn, k, sigma(vn[k])); svn = Set(vn); if (#select(x->x==n, svn), knb = nb); if (!#select(x->x<=n, svn), ok = 0);); knb;}

apply( A286011(n)=if(n<3,2-n, n=invsigma(n), vecmax(apply(self,n))+1), [1..99]) \\ See Alekseyev-link for invsigma(). - M. F. Hasler, Nov 20 2019

A309727 a(n) is the least integer k such that for some iteration of sigma applied to k, one gets the n-th term of A002191, the list of possible values for the function sum of divisors.

Original entry on oeis.org

1, 2, 2, 5, 2, 2, 5, 9, 9, 2, 10, 19, 2, 5, 29, 16, 16, 22, 37, 10, 27, 19, 43, 33, 34, 5, 49, 2, 61, 16, 67, 29, 73, 45, 49, 43, 27, 22, 50, 19, 52, 101, 16, 85, 109, 22, 73, 5, 81, 33, 67, 64, 50, 86, 81, 137, 76, 66, 149, 111, 99, 157, 81, 106, 163, 2, 52, 173, 129

Offset: 1

Michel Marcus, Oct 14 2019

nonn

The set union of this sequence is 1 U A007369.

			For n = 5, A002191(5) is 7, and 4 iterations of sigma applied to 2 give 7, and no integer less than 2 will give 7, so a(5)=2.

Michel Marcus, Table of n, a(n) for n = 1..2503

Cf. A000203, A002191, A007369, A007497, A051572, A257349.

A257670 is a better version for this sequence.

list(lim) = select(n->n<=lim, Set(vector(lim\=1, n, sigma(n))));
lista(nn) = {my(vs = list(nn), v = vector(#vs)); v[1] = 1; for (n=2, #vs, for (k=2, vs[n], my(kk=k); while (sigma(kk) <= vs[n], kk=sigma(kk)); if (kk == vs[n], v[n] = k; break););); v;}

a(n) = 2 when A002191(n) is in A007497.
a(n) = 5 when A002191(n) is in A051572.
a(n) = 16 when A002191(n) is in A257349.

A337436 6a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2A007494(n).

Original entry on oeis.org

1, 5, 23, 33, 322, 87, 325, 278, 495, 1293, 2027, 4725, 3468, 2690, 27177, 14438, 4245, 6773, 13283, 24938, 104283, 92067, 28893, 60015, 119362, 46905, 44270, 106323, 90713, 67475, 266618, 207107, 139708, 1496910, 716182, 598867, 439633, 688518, 224922, 315893

Offset: 1

Hugo Pfoertner, Sep 02 2020

nonn

Apart from the atypical case [3, 5, 7], prime gaps nextprime(p+1)-p following a pair of twin primes p-2, p can only have the sizes 4, 6, 10, 12, 16, 18, ..., i.e., numbers k of the form 2*(k == 0 or 2 mod 3) = 2*A007494(n).

			a(1) = 1: The first occurrence of 3 consecutive primes [p-2, p, p+4] is at p = 6*a(1) + 1 = 7 -> [5, 7, 11],
a(2) = 5: consecutive primes [p-2, p, p+6] first occur at p = 6*a(2) * 1 = 31 -> [29, 31, 37],
a(3) = 23: consecutive primes [p-2, p, p+10] first occur at p = 6*a(3) + 1 = 139 -> [137, 139, 149].

Cf. A007497, A329160, A329161, A337435.

A285634 a(1) = 4, a(n) = Product_{d|a(n-1)} d.

Original entry on oeis.org

4, 8, 64, 2097152, 3450873173395281893717377931138512726225554486085193277581262111899648

Offset: 1

Ilya Gutkovskiy, Apr 23 2017

Iterating the product-of-divisors function.
The next term is too large to include.
Let a(n) = Product_{d|a(n-1)} d, with a(1) = p^k, p is a prime, k >= 0 and b(n) = b(n-1)*(b(n-1) + 1)/2, with b(1) = k, then a(n) = p^b(n).
The next term has 8067 digits. - Harvey P. Dale, Apr 18 2019

			a(1) = 4;
a(2) = 8 because 4 has 3 divisors {1, 2, 4} and 1*2*4 = 8;
a(3) = 64 because 64 has 7 divisors {1, 2, 4, 8, 16, 32, 64} and 1*2*4*8*16*32*64 = 2097152, etc.
...
a(6) = 2^26796;
a(7) = 2^359026206;
a(8) = 2^64449908476890321;
a(9) = 2^2076895351339769460477611370186681, etc.

Eric Weisstein's World of Mathematics, Divisor Product

Cf. A000005, A000079, A007497, A007501, A007955.

RecurrenceTable[{a[1] == 4, a[n] == Sqrt[a[n - 1]]^DivisorSigma[0, a[n - 1]]}, a, {n, 5}]
NestList[Times@@Divisors[#]&,4,4] (* Harvey P. Dale, Apr 18 2019 *)

a(1) = 4, a(n) = a(n-1)^(A000005(a(n-1))/2).

a(n) = 2^A007501(n-1).

Previous Showing 11-20 of 20 results.

cp's OEIS Frontend

A120611 Sum of previous term and preceding divisors.

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A283166 a(0) = 0; a(1) = 1; a(2*n) = sigma(a(n)), a(2*n+1) = sigma(a(n)) + sigma(a(n+1)).

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A059460 Iteration of unitary-sigma function: a(1) = 2, a(n) = usigma(a(n-1)).

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A126850 a(n) = OrdinaryUnitarySigma(a(n-1)).

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A283167 a(0) = 1; a(2*n) = 2*a(n), a(2*n+1) = sigma(a(n)).

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A283764 a(0) = 0; a(1) = 1; a(2*n) = sigma(a(n)), a(2*n+1) = sigma(a(n)+a(n+1)).

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A286011 a(1)=1, and for n>1, a(n) is the maximum number of iterations of sigma resulting in n, starting at some integer k; or 0 if n cannot be reached from any k.

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A309727 a(n) is the least integer k such that for some iteration of sigma applied to k, one gets the n-th term of A002191, the list of possible values for the function sum of divisors.

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A283166 a(0) = 0; a(1) = 1; a(2n) = sigma(a(n)), a(2n+1) = sigma(a(n)) + sigma(a(n+1)).

A283167 a(0) = 1; a(2n) = 2a(n), a(2*n+1) = sigma(a(n)).

A283764 a(0) = 0; a(1) = 1; a(2n) = sigma(a(n)), a(2n+1) = sigma(a(n)+a(n+1)).

A337436 6a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2A007494(n).