Labos Elemer - cp's OEIS frontend

A318883 Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.

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

Offset: 1

Antti Karttunen, Sep 22 2018, after Labos Elemer's A097033

nonn

This sequence implements the original definition given for A097033.

			For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle (of length 1 in this case), thus there are no transient part, and a(1) = 0.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle after a transient part of length 1, thus a(2) = 1.
For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(54) = 30, thus a(30) = a(42) = a(54) = 0, as 30, 42 and 54 are all contained in their own terminal cycle, without a preceding transient part.
For n = 1506, the iteration-list is {1506, 1518, 1938, 2382, 2394, 2406, [2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582, 2418, ..., ad infinitum]}. After a transient of length 6 the iteration ends in a cycle of length 14, thus a(1506) = 6.
If a(n) = 0, then n is a term in an attractor set like A002827, A063991, A097024, A097030.

Antti Karttunen, Table of n, a(n) for n = 1..87360

Cf. A003062, A063919, A097031, A097033, A318880, A318882.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
transient[k_] := Module[{iter=NestWhileList[a063919, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]]-1
a318883[n_] := Map[transient, Range[n]]
a318883[105] (* Hartmut F. W. Hoft, Jan 25 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n));
A318883(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(mapget(visited, n)-1), mapput(visited, n, j)); n = A063919(n)); };
\\ Or by using lists:
pil(item,lista) = { for(i=1,#lista,if(lista[i]==item,return(i))); (0); };
A318883(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(k-1)); listput(visited, n); n = A063919(n)); };

a(n) = A318882(n) - A097031(n).

a(n) = A097033(n) + A318880(n) - 1.

A318882 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n. Number of distinct terms in iteration list.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 22 2018, after Labos Elemer's A097032

nonn

This sequence implements the original definition given for A097032.

			For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle of length 1 without a preceding transient part, thus a(1) = 0+1 = 1.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle of length 1, after a transient part of length 1, thus a(2) = 1+1 = 2.
For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(54) = 30, thus a(30) = a(42) = a(54) = 0+3 = 3, as 30, 42 and 54 are all contained in their own terminal cycle of length 3, without a preceding transient part.
For n = 1506, the iteration-list is {1506, 1518, 1938, 2382, 2394, 2406, [2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582, 2418, ..., ad infinitum]}. After a transient of length 6 the iteration ends in a cycle of length 14, thus a(1506) = 6+14 = 20.

Antti Karttunen, Table of n, a(n) for n = 1..87360

Cf. A063919, A097031, A097032, A318883.

Cf. A002827 (the positions of ones after the initial 1).

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
a318882[n_] := Map[Length[NestWhileList[a063919, #, UnsameQ, All]]-1&, Range[n]]
a318882[105] (* Hartmut F. W. Hoft, Jan 25 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n));
A318882(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-1), mapput(visited, n, j)); n = A063919(n)); };
\\ Or by using lists:
pil(item,lista) = { for(i=1,#lista,if(lista[i]==item,return(i))); (0); };
A318882(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(j-1)); listput(visited, n); n = A063919(n)); };

a(n) = A097031(n) + A318883(n).

a(n) = A097032(n) + A318880(n) - 1.

A318459 a(n) = gcd(n, tau(n), phi(n)), where tau = A000005 and phi = A000010.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 1, 1, 2, 1, 8, 1

Offset: 1

Antti Karttunen, Sep 07 2018, after Labos Elemer's A074389

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A000010, A009191, A009195, A009213.

Cf. also A074389.

Table[GCD[n,DivisorSigma[0,n],EulerPhi[n]],{n,110}] (* Harvey P. Dale, Jul 30 2019 *)

A318459(n) = gcd([n, numdiv(n), eulerphi(n)]);

a(n) = gcd(n, A000005(n), A000010(n)).

a(n) = gcd(n,A009213(n)) = gcd(A000005(n),A009195(n)) = gcd(A000010(n),A009191(n)).

A118860 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1 and 4k+1 are all primes.

Original entry on oeis.org

21968100, 37674210, 81875220, 356467230, 416172330, 750662640, 1007393730, 1150070040, 1586271960, 1963954650, 3127171320, 3669568560, 4377895410, 4383541050, 5575083360, 5686935870, 5708418870, 7365234450, 7478474430, 7681046100, 8453862690, 8898688680

Offset: 1

Labos Elemer, May 03 2006

nonn

All terms are multiples of 210, hence simpler code is possible.

			21968100 is a term because 21968099, 21968101, 43936199, 43936201, 65904299, 65904301, 87872399, 87872401 are all prime.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A014574, A066388 and A118859.
Subsequence: A349321.
Cf. A174293, A348348.

tb={};Do[If[(PrimeQ[n-1]&&PrimeQ[n+1])&& (PrimeQ[2*n-1]&&PrimeQ[2*n+1])&& (PrimeQ[3*n-1]&&PrimeQ[3*n+1])&& (PrimeQ[4*n-1]&&PrimeQ[4*n+1]), Print[n];AppendTo[tb,n]], {n,21968100,10^8,210}];tb
Select[210*Range[424*10^5],AllTrue[{#-1,#+1,2#-1,2#+1,3#-1,3#+1,4#-1,4#+1},PrimeQ]&] (* Harvey P. Dale, Jul 23 2024 *)

isok(k) = if(k % 210, 0, for(i = 1, 4, forstep(j = -1, 1, 2, if(!isprime(i*k-j), return(0)))); 1); \\ Amiram Eldar, Mar 13 2025

a(n) = 210*A174293(n).

Edited by Don Reble, May 16 2006

a(20)-a(22) from Pontus von Brömssen, Oct 14 2021

A118859 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1 and 3k+1 are primes.

Original entry on oeis.org

6, 53550, 420420, 422310, 1624350, 2130240, 3399900, 5199810, 5246010, 6549270, 7384440, 7775880, 9516570, 9565710, 10430280, 11845260, 13207950, 14562870, 14619990, 18747960, 20099940, 21596820, 21968100, 24358950, 24610740, 26916120, 28359240, 30838080

Offset: 1

Labos Elemer, May 03 2006

nonn

			6 is a term because 5, 7, 11, 13, 17, 19 are all prime.

Amiram Eldar, Table of n, a(n) for n = 1..4000

Subsequence of A014574 and A066388.
Subsequences: A118860, A349321.
Cf. A014574, A290811, A348348.

Select[Range[25*10^6],AllTrue[Flatten[{#+{1,-1},2#+{1,-1},3#+{1,-1}}], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 13 2016 *)

isok(k) = isprime(k-1) && isprime(k+1) && isprime(2*k-1) && isprime(2*k+1) && isprime(3*k-1) && isprime(3*k+1); \\ Amiram Eldar, Mar 13 2025

a(n) = 6*A290811(n).

Edited by Don Reble, May 16 2006

a(26)-a(28) from Jon E. Schoenfield, Dec 07 2021

A118114 a(n) = binomial(3n,n) mod((n+1)(n+2)).

Original entry on oeis.org

3, 3, 4, 15, 21, 28, 0, 81, 55, 99, 0, 0, 84, 120, 0, 153, 171, 285, 0, 231, 253, 0, 360, 0, 0, 0, 0, 522, 0, 496, 0, 561, 833, 945, 0, 703, 741, 156, 0, 861, 903, 1419, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2016, 1664, 2145, 2211, 3417, 0, 2415, 2485, 2556, 0

Offset: 1

Labos Elemer, Apr 13 2006

nonn

Compared with A118112: larger nonzero value more often and in non-monotonic order.

			For n=5, binomial(15,5) = 3003 = (5+1)*(5+2)*71 + 21, a(5) = 21, the residue.
Interestingly, a very large zone of zeros occurs between about n=5460 and n=7800, uninterrupted by nonzero residues.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000108, A118112, A118113.

seq(binomial(3*n,n) mod((n+1)*(n+2)),n=1..71); # Emeric Deutsch, Apr 15 2006

Table[Mod[Binomial[3*k, k], (k + 1)*(k + 2)], {k, 1, 1000}]

a(n) = binomial(3*n,n) % ((n+1)*(n+2)); \\ Michel Marcus, Jan 05 2017

A118112 a(n) = binomial(3n,n) mod (n+1).

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 19, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 35, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Apr 13 2006

nonn

These divisibilities are analogous to those of Catalan numbers. For rather long sequences of consecutive integers, a(n)=0. For the first 10000 integers 9678 residues equals zero. See A118113.

If n+1 is in A061345, a(n)=0. This follows from Kummer's theorem. - Robert Israel, May 09 2018

			For n=9, binomial(27,7) = 4686825; 4686825 mod 10 = 5.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Kummer's theorem

Cf. A000108, A061345, A118113.

seq(binomial(3*n,n) mod (n+1), n=1..200); # Robert Israel, May 09 2018

Mathematica
```
Table[Mod[Binomial[3*k,k],k+1],{k,500}]
```

a(n) = binomial(3*n, n) % (n+1); \\ Michel Marcus, May 10 2018

a(n) = binomial(3n,n) mod (n+1).

Mathematica program corrected by Harvey P. Dale, Dec 28 2012

A115560 Twin prime pairs k-1 and k+1 such that the squares of both are present in A115557.

Original entry on oeis.org

11, 13, 29, 31, 197, 199, 239, 241, 419, 421, 659, 661, 881, 883, 1019, 1021, 1061, 1063, 1481, 1483, 1877, 1879, 3167, 3169, 3821, 3823, 4019, 4021, 4049, 4051, 4787, 4789, 6359, 6361, 7589, 7591, 9437, 9439, 13691, 13693, 14447, 14449, 14627, 14629, 16451, 16453

Offset: 1

Labos Elemer, Jan 25 2006

nonn

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

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115558, A115559.

ta={{0}};tb={{0}}; Do[s=DivisorSigma[1,DivisorSigma[0,n]]; s1=DivisorSigma[0,DivisorSigma[1,n]]; If[Equal[s-s1,0]&&IntegerQ[Sqrt[n]&&PrimeQ[Sqrt[n]]],Print[n]; ta=Append[ta,n];tb=Append[tb,Sqrt[n]]],{n,1,100000000}] ta=Delete[ta,1];tb=Delete[tb,1];ni=Intersection[tb,2+tb]; Union[ni,ni-2]

isok(n) = issquare(n) && (sigma(numdiv(n)) == numdiv(sigma(n))); \\ A115557
lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isok(p^2) && isok((p+2)^2), print1(p, ", ", p+2, ", ")););} \\ Michel Marcus, Jul 17 2019

The commutator [sigma, tau] is zero and a(n) is the square root of special prime solutions. These solutions are twin primes. Both twins are displayed.

More terms from Amiram Eldar, Jul 17 2019

A115558 a(n) is the square root of A115557(n).

Original entry on oeis.org

1, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 73, 83, 97, 103, 113, 127, 157, 179, 197, 199, 223, 227, 233, 239, 241, 251, 257, 271, 281, 311, 316, 317, 333, 353, 389, 401, 409, 419, 421, 443, 449, 461, 467, 479, 491, 503, 509, 549, 563, 587, 593, 599, 617, 641

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115559, A115560.

Select[Range[1000], DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)

isok(k) = numdiv(sigma(k^2)) == sigma(numdiv(k^2)); \\ Amiram Eldar, Jan 28 2025

The commutator [sigma, tau] is zero and a(n) is the square root of solutions. Both prime and composite numbers.

A115559 Nonprime terms of A115558.

Original entry on oeis.org

1, 316, 333, 549, 844, 963, 981, 1052, 1233, 1251, 1304, 1341, 1359, 1474, 1629, 1688, 1737, 1738, 1996, 2061, 2144, 2216, 2421, 2528, 2547, 2763, 2979, 3033, 3082, 3123, 3141, 3148, 3231, 3244, 3283, 3303, 3411, 3573, 3634, 3871, 3879, 3897, 3988, 4113

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Intersection of A018252 and A115558.

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115560.

Select[Range[5000], !PrimeQ[#] && DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)

isok(n)= !isprime(n) && (sigma(numdiv(n^2)) == numdiv(sigma(n^2))); \\ Michel Marcus, Dec 20 2013

The commutator [sigma, tau] is zero and a(n) is the square root of solutions. Moreover this square root is a nonprime number.

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User: Labos Elemer

A318883 Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.

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A318882 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n. Number of distinct terms in iteration list.

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A318459 a(n) = gcd(n, tau(n), phi(n)), where tau = A000005 and phi = A000010.

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A118860 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1 and 4k+1 are all primes.

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A118859 Numbers k such that k-1, k+1, 2*k-1, 2*k+1, 3*k-1 and 3*k+1 are primes.

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A118114 a(n) = binomial(3n,n) mod((n+1)(n+2)).

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A118112 a(n) = binomial(3n,n) mod (n+1).

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A118859 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1 and 3k+1 are primes.