A173557 -id:A173557 - cp's OEIS frontend

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 6, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 2, 8, 2, 2, 2, 2, 2, 8, 6, 2, 2, 8, 6, 4, 2, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 4, 24, 2, 2, 2, 6, 8, 6, 12, 24, 2, 2, 2, 2, 2, 12, 4, 6, 8, 2, 2, 8, 8, 2, 4, 2, 24, 2, 2, 6, 4

Offset: 1

Labos Elemer, Dec 04 2001

nonn

Frequently equal, but not identical, to A009223 (i.e. GCD of sigma and phi of n).

Antti Karttunen, Table of n, a(n) for n = 1..23374
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A048250, A173557, A023900, A000203, A007947, A000010, A009223, A066087, A322359, A322360, A322362.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}]
a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

a(n)=my(f=factor(n)[,1]);gcd(prod(i=1,#f,f[i]+1),prod(i=1,#f,f[i]-1)) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = gcd(A048250(n), A023900(n)) = gcd(A000203(A007947(n)), A000010(A007947(n))).

a(n) = A322360(n) / A322359(n). - Antti Karttunen, Dec 04 2018

Name edited, part of the old name transferred to the formula section by Antti Karttunen, Dec 04 2018

A295887 Filter sequence combining A003557(n) and A173557(n); the restricted growth sequence transform of A291756.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 9, 10, 5, 11, 12, 13, 7, 14, 15, 10, 8, 16, 17, 18, 10, 19, 20, 21, 11, 22, 23, 24, 13, 25, 26, 27, 14, 25, 28, 29, 10, 30, 31, 32, 16, 33, 34, 35, 18, 36, 37, 38, 19, 29, 39, 27, 21, 40, 41, 42, 22, 43, 44, 45, 24, 46, 47, 48, 25, 49, 50, 51, 27, 52, 53, 42, 25, 54, 55, 56, 29, 57, 37, 58, 30, 59, 60, 61, 32, 51, 62, 42, 33, 51

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

First define function f(n) = (1/2)*(2 + ((A003557(n) + A173557(n))^2) - A003557(n) - 3*A173557(n)), or in short, f(n) = P(A003557(n),  A173557(n)), where P(n,k) is triangular table sequence A000027 used as an injective N x N -> N pairing function. Then apply the restricted growth sequence transform to the sequence f(1), f(2), f(3), ... See the example-section.
This is also the restricted growth sequence transform of sequence A291756, as A291756(n) = P(A003557(n), A000010(n)), where again P(n,k) is sequence A000027 used as a pairing function. Given either an ordered pair (A003557(n),A000010(n)) or (A003557(n),A173557(n)), the other one can be computed because A000010(n) = A003557(n)*A173557(n).
Note that the exact pairing function P used is not important, as long as it provides an injective mapping N x N -> N. So instead of Cantor's mapping we could as well used bit-interleaving A054238 (Morton code) to pack together A003557(n) and A173557(n), or equally, A000010(n) and A003557(n).

			The first ten terms of the sequence f(n) = (1/2)*(2 + ((A003557(n) + A173557(n))^2) - A003557(n) - 3*A173557(n)) are 1, 1, 2, 3, 7, 2, 16, 10, 9, 7. When we assign to each newly occurring term the least unused number k so far (starting by giving k=1 for the initial term, this k increases by one for each new distinct term produced by f(n) when n grows), and for each repeated term the same number it was given the previous time (equal to the number it was given for the first time), we obtain 1, 1, 2, 3, 4, 2, 5, 6, 7, 4, the first 10 terms of this sequence. Note how f(10) = 7 gets 4 because when seven occurred for the first time at f(5), it was the 4th distinct new number in that sequence.
This is also true for the sequence A291756 although there the terms are different: 1, 1, 2, 5, 7, 2, 16, 25, 31, 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pairing Function
Wikipedia, Pairing Function

Cf. A000010, A003557, A054238, A173557, A291750, A291756, A295300, A295886.

allocatemem(2^30);
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted7(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted7(n))),"b295887.txt");

A322320 a(n) = gcd(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A003557, A173557, A322321, A322351, A322352.

Cf. also A066086, A322318.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322320(n) = gcd(A173557(n), A003557(n));

a(n) = gcd(A003557(n), A173557(n)) = gcd(A322351(n), A322352(n)).

a(n) = A000010(n) / A322321(n).

A322321 a(n) = lcm(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, 6, 8, 8, 16, 6, 18, 4, 12, 10, 22, 4, 20, 12, 18, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 24, 22, 46, 8, 42, 20, 32, 12, 52, 18, 40, 12, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 40, 18, 60, 24, 78, 8, 54, 40, 82, 12, 64, 42, 56, 20

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A003557, A173557, A322320, A322351, A322352.

Cf. also A322319, A322359.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322321(n) = lcm(A003557(n), A173557(n));

a(n) = lcm(A003557(n), A173557(n)) = lcm(A322351(n), A322352(n)).

a(n) = A000010(n) / A322320(n).

A342416 a(n) = gcd(A173557(n), A342001(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 8, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 12, 2, 1, 3, 8, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 1, 2, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 4, 2, 1, 1, 1, 1, 3, 1, 2, 6, 1, 1, 2, 2, 1, 1, 2, 2, 3, 8, 5, 1, 1, 4, 2, 2, 1, 24, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

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

Cf. A000010, A003415, A003557, A173557, A342001, A342413.

Array[GCD[#2, #1/#3] & @@ {If[#1 < 2, 0, #1 Total[#2/#1 & @@@ #2]], If[#1 == 1, 1, Times @@ Map[# - 1 &, #2[[All, 1]] ]], #1/Times @@ #2[[All, 1]]} & @@ {Abs[#], FactorInteger[#]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342413(n) = gcd(eulerphi(n), A003415(n));
A342416(n) = (A342413(n)/A003557(n));

a(n) = A342413(n) / A003557(n) = gcd(A173557(n), A342001(n)).

A322351 a(n) = min(A003557(n), A173557(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003557, A173557, A322320, A322321, A322352, A322355.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, Min[ Times@@ (First[#]^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322351(n) = min(A003557(n), A173557(n));

a(n) = min(A003557(n), A173557(n)).

a(n) = A000010(n) / A322352(n).

A322352 a(n) = max(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 3, 4, 10, 2, 12, 6, 8, 8, 16, 3, 18, 4, 12, 10, 22, 4, 5, 12, 9, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 8, 7, 5, 32, 12, 52, 9, 40, 6, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 8, 18, 60, 24, 78, 8, 27, 40, 82, 12, 64, 42, 56, 10, 88, 8

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003557, A173557, A322320, A322321, A322351, A322355.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, Max[ Times @@ (First[#]^ (Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322352(n) = max(A003557(n), A173557(n));

a(n) = max(A003557(n), A173557(n)).

a(n) = A000010(n) / A322351(n).

A345049 a(n) = A173557(n) * A345001(n).

Original entry on oeis.org

-1, 0, -2, 3, -12, 10, -30, 11, 2, 20, -90, 40, -132, 30, 16, 31, -240, 48, -306, 104, 0, 50, -462, 112, -36, 60, 26, 192, -756, 344, -870, 79, -80, 80, -240, 158, -1260, 90, -144, 312, -1560, 636, -1722, 440, 216, 110, -2070, 280, -162, 152, -320, 600, -2652, 186, -880, 600, -432, 140, -3306, 1120, -3540, 150, 348, 191

Offset: 1

Antti Karttunen, Jun 06 2021

sign

Cf. A173557, A345001, A345048, A345050, A345051.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
A345049(n) = (A173557(n) * A345001(n));

a(n) = A173557(n) * A345001(n).

a(n) = A345048(n) - A345050(n).

A318841 a(n) = n - A173557(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 7, 6, 1, 10, 1, 8, 7, 15, 1, 16, 1, 16, 9, 12, 1, 22, 21, 14, 25, 22, 1, 22, 1, 31, 13, 18, 11, 34, 1, 20, 15, 36, 1, 30, 1, 34, 37, 24, 1, 46, 43, 46, 19, 40, 1, 52, 15, 50, 21, 30, 1, 52, 1, 32, 51, 63, 17, 46, 1, 52, 25, 46, 1, 70, 1, 38, 67, 58, 17, 54, 1, 76, 79, 42, 1, 72, 21, 44, 31, 78, 1, 82, 19, 70, 33, 48

Offset: 1

Antti Karttunen, Sep 16 2018

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

Cf. A173557, A307868, A318833.

a[n_] := n - Times @@ (FactorInteger[n][[;;, 1]] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 16 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A318841(n) = (n-A173557(n));

a(n) = n - A173557(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A307868 = 0.528319... . - Amiram Eldar, Dec 16 2023

A387157 a(n) = A173557(sigma(n)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 6, 2, 2, 1, 8, 12, 2, 2, 6, 6, 2, 2, 30, 2, 24, 4, 12, 1, 2, 2, 8, 30, 12, 4, 6, 8, 2, 1, 12, 2, 2, 2, 72, 18, 8, 6, 8, 12, 2, 10, 12, 24, 2, 2, 30, 36, 60, 2, 6, 2, 8, 2, 8, 4, 8, 8, 12, 30, 2, 12, 126, 12, 2, 16, 12, 2, 2, 2, 96, 36, 36, 30, 24, 2, 12, 4, 60, 10, 12, 12, 6, 2, 20, 8, 8, 8, 24, 6, 12, 1, 2, 8

Offset: 1

Antti Karttunen, Aug 19 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A003958, A080398, A173557, A387158 (positions where equal to A173557(n)).

Cf. also A351442.

A387157[n_] := If[n == 1, 1, Times @@ (FactorInteger[DivisorSigma[1, n]][[All, 1]] - 1)];
Array[A387157, 100] (* Paolo Xausa, Aug 20 2025 *)

A387157(n) = factorback(apply(p -> p-1,factor(sigma(n))[,1]));

a(n) = A003958(A080398(n)).

cp's OEIS Frontend

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

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A295887 Filter sequence combining A003557(n) and A173557(n); the restricted growth sequence transform of A291756.

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A322320 a(n) = gcd(A003557(n), A173557(n)).

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A322321 a(n) = lcm(A003557(n), A173557(n)).

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A342416 a(n) = gcd(A173557(n), A342001(n)).

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A322351 a(n) = min(A003557(n), A173557(n)).

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A322352 a(n) = max(A003557(n), A173557(n)).

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A345049 a(n) = A173557(n) * A345001(n).

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