A344695 -id:A344695 - cp's OEIS frontend

A348733 a(n) = gcd(A003959(n), A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

1, 3, 4, 1, 6, 12, 8, 9, 2, 18, 12, 4, 14, 24, 24, 1, 18, 6, 20, 6, 32, 36, 24, 36, 2, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 2, 38, 60, 56, 54, 42, 96, 44, 12, 12, 72, 48, 4, 2, 6, 72, 14, 54, 12, 72, 72, 80, 90, 60, 24, 62, 96, 16, 1, 84, 144, 68, 18, 96, 144, 72, 18, 74, 114, 8, 20, 96, 168, 80, 6, 2, 126, 84, 32

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 1444 = 2^2 * 19^2, where a(1444) = 10 != 1*2 = a(4)*a(361). See A348740 for the list of such positions.

Antti Karttunen, Table of n, a(n) for n = 1..21125

Cf. A003959, A034448, A348732, A348734, A348735, A348740.

Cf. also A344695, A348047, A348503, A348946 for similar, almost multiplicative sequences.

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348733(n) = gcd(A003959(n), A034448(n));

a(n) = gcd(A003959(n), A034448(n)).
a(n) = gcd(A003959(n), A348732(n)) = gcd(A034448(n), A348732(n)).
a(n) = A003959(n) / A348734(n) = A034448(n) / A348735(n).

A348047 a(n) = gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 8, 8, 30, 72, 32, 9, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 24, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 196 = 4*49, where a(196) = 3, although a(4) = 1 and a(49) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A348029, A348048, A348049.

Cf. also A344695.

f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[Times @@ f @@@ FactorInteger[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348047(n) = gcd(sigma(n), A003959(n));

a(n) = gcd(A000203(n), A003959(n)).
a(n) = gcd(A000203(n), A348029(n)) = gcd(A003959(n), A348029(n)).
a(n) = A000203(n)/ A348048(n) = A003959(n) / A348049(n).

A348503 a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 15, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 15 != 3*1 = a(8)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A034448, A048146, A348504, A348505.
Differs from A344695 for the first time at n=72, where a(72) = 15, while A344695(72) = 3.
Differs from A348047 for the first time at n=27, where a(27) = 4, while A348047(27) = 8.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (fct = FactorInteger[n]), Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348503(n) = gcd(sigma(n), A034448(n));

a(n) = gcd(A000203(n), A034448(n)).
a(n) = gcd(A000203(n), A048146(n)) = gcd(A034448(n), A048146(n)).
a(n) = A000203(n) / A348504(n) = A034448(n) / A348505(n).

A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 2, 56, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 6, 72, 24, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96, 144, 72, 3, 74, 114, 124, 140, 96, 168, 80, 6, 1, 126

Offset: 1

Antti Karttunen, Nov 05 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 1 <> 91 = 7*13 = a(4)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A034448, A348944, A348945, A348947, A348948.

Cf. also A344695, A348047, A348503, A348733.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), (Times @@ f2 @@@ f + Times @@ f3 @@@ f)/2]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348946(n) = gcd(sigma(n), ((1/2)*(A003959(n)+A034448(n))));

a(n) = gcd(A000203(n), A348944(n)).
a(n) = gcd(A000203(n), A348945(n)) = gcd(A348944(n), A348945(n));
a(n) = A348944(n) / A348947(n) = A000203(n) / A348948(n).

A367991 The sum of the divisors of the squarefree part of n.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72

Offset: 1

Amiram Eldar, Dec 07 2023

First differs from A348503 at n = 72 and from A344695 at n = 108.

The sum of the infinitary divisors (A077609) of n that are squarefree (A005117). - Amiram Eldar, Jun 03 2025

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

Cf. A000203, A000290, A005117, A007913, A048250, A077609, A367990.
Cf. A002117, A013662.
Cf. A344695, A348503.

f[p_, e_] := If[OddQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]+1, 1));}

Multiplicative with a(p^e) = p + 1 if e is odd and 1 otherwise.
a(n) = A000203(A007913(n)) = A048250(A007913(n)).
a(n) = A048250(n)/A367990(n).
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)/zeta(3) = 0.900392677639... .

A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

14, 21, 26, 28, 35, 38, 39, 42, 52, 56, 57, 62, 63, 65, 70, 74, 76, 77, 78, 82, 84, 86, 93, 95, 98, 99, 104, 105, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 134, 140, 143, 146, 148, 152, 154, 155, 156, 158, 161, 166, 168, 171, 172, 175, 182, 183, 185, 186, 189, 190, 194, 195, 198, 201, 203, 206, 208, 209

Offset: 1

Antti Karttunen and Peter Munn, May 27 2021

nonn

Numbers k for which A344695(k^2) > 1.
It can be shown that sigma(m) and psi(m) share a factor if m is nonsquare. (See A344695 for more detail.) So here we consider only square numbers, m = k^2.
For prime p, sigma(p^2) and psi(p^2) are coprime, since sigma(p^2) = p^2 + p + 1 = psi(p^2) + 1. So all terms are composite. We can say more, since for prime p and positive integer e, psi(p^(2*e)) = p^(2*e-1) * (p+1), whereas sigma(p^(2*e)) can be shown to be congruent to 1 modulo p and to 1 modulo p+1, so shares no factors with p^(2*e-1) * (p+1). So all terms are divisible by more than one prime.
If k is in the sequence, m*k is also present for any positive integer m coprime to k.

			Sigma (A000203) and the Dedekind psi function (A001615) are both multiplicative, so we gain insight by showing the values of these functions using their multiplicative properties:-
sigma(14^2) = sigma(2^2) * sigma(7^2) = 7 * 57 = 7 * (3*19).
psi(14^2) = psi(2^2) * psi(7^2) = 6 * 56 = (2*3) * (2^3*7).
So sigma(14^2) and psi(14^2) share factors 3 and 7, so 14 is in the sequence.
Looking in particular at the shared factor 3, we see it is present in sigma(7^2) and psi(2^2). For prime p, sigma(p^2) and psi(p^2) equate to polynomials in p, so we deduce 3 divides sigma(p^2) for p congruent to 7 modulo 3, and divides psi(p^2) for p congruent to 2 modulo 3. From this we see all products of a prime of the form 3m+1 and a prime of the form 3m+2 are in the sequence; for instance (3*4+1) * (3*1+2) = 13 * 5 = 65.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A000290, A001615, A344695.

Subsequences: A344872.

filter:= proc(k) local n,F, sig, psi, t;
   n:= k^2;
   F:= map(t -> [t[1],2*t[2]], ifactors(k)[2]);
   sig:= mul((t[1]^(1+t[2])-1)/(t[1]-1),t=F);
   psi:= n*mul(1+1/t[1],t=F);
   igcd(sig,psi) > 1
end proc:
select(filter, [$1..300]); # Robert Israel, Jan 06 2024

filter[k_] := Module[{n, F, sig, psi},
   n = k^2;
   F = {#[[1]], 2 #[[2]]}& /@ FactorInteger[k];
   sig = Product[(t[[1]]^(1 + t[[2]]) - 1)/(t[[1]] - 1), {t, F}];
   psi = n*Product[1 + 1/t[[1]], {t, F}];
   GCD[sig, psi] > 1];
Select[Range[300], filter] // Quiet (* Jean-François Alcover, May 23 2024, after Robert Israel *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
isA344703(n) = (A344695(n^2)>1);

A366796 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366795(i) = A366795(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 1, 2, 6, 7, 8, 3, 1, 2, 3, 1, 5, 8, 9, 4, 10, 11, 4, 5, 1, 2, 3, 1, 4, 5, 1, 2, 12, 13, 10, 6, 11, 14, 6, 7, 14, 15, 16, 8, 6, 7, 8, 3, 1, 2, 3, 1, 4, 5, 1, 2, 6, 7, 8, 6, 1, 2, 3, 1, 7, 17, 18, 5, 19, 15, 5, 8, 20, 21, 22, 9, 5, 8, 9, 4, 23, 22, 24, 10, 25, 25, 10, 11, 5, 8, 9, 4, 10, 11, 4, 5, 1

Offset: 0

Antti Karttunen, Oct 26 2023

nonn

Restricted growth sequence transform of A366795.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A005940, A344695, A366795.

Cf. also A366802, A366805.

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; };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344695(n) = gcd(sigma(n), A001615(n));
A366795(n) = A344695(A005940(1+n));
v366796 = rgs_transform(vector(1+up_to,n,A366795(n-1)));
A366796(n) = v366796[1+n];

A349574 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1, 5, 1, 4, 1, 2, 1, 1, 1, 3, 6, 1, 7, 2, 1, 1, 1, 8, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 5, 10, 6, 1, 2, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 4, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 6, 2, 1, 1, 1, 5, 13, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 8, 1, 10, 4, 14, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Nov 22 2021

nonn

Restricted growth sequence transform of the ordered pair [A344696(n), A344697(n)].
For all i, j, A003557(i) = A003557(j) => a(i) = a(j); in other words, this sequence is a function of A003557. This follows because A344696(n) = A344696(A057521(n)), A344697(n) = A344696(A057521(n)), and A057521(n) = A064549(A003557(n)).
Apparently, A081770 gives the positions of 2's, which occur on those n where A344696(n) = 7 and A344697(n) = 6.

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

Cf. A000203, A001615, A003557, A005117 (positions of 1's), A057521, A064549, A081770, A280292, A344695, A344696, A344697.

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; };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
Aux349574(n) = { my(s=sigma(n),u=A001615(n),g=gcd(u,s)); [s/g, u/g]; };
v349574 = rgs_transform(vector(up_to, n, Aux349574(n)));
A349574(n) = v349574[n];

For all n >= 1, a(n) = a(A057521(n)). [See comments]

cp's OEIS Frontend

A348733 a(n) = gcd(A003959(n), A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

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A348047 a(n) = gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

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A348503 a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

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A367991 The sum of the divisors of the squarefree part of n.

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A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.

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A366796 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366795(i) = A366795(j) for all i, j >= 0.

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A349574 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

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