A080398 -id:A080398 - cp's OEIS frontend

A353785 a(n) = A353783(n) / A080398(n).

1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 1, 4, 2, 1, 3, 1, 2, 1, 4, 2, 4, 2, 1, 1, 4, 4, 1, 2, 16, 3, 2, 3, 4, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 4, 8, 2, 1, 1, 6, 1, 9, 4, 2, 4, 2, 1, 2, 2, 1, 16, 4, 1, 1, 2, 2, 3, 4, 4, 12, 1, 1, 1, 2, 2, 4, 2, 8, 1, 11, 1, 2, 4, 3, 2, 2, 2, 3, 1, 4, 4, 16, 8, 2, 6, 7, 1, 2, 1, 1, 6, 4, 1, 4

Offset: 1

Antti Karttunen, May 08 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A080398, A353783.

Array[Apply[LCM, DivisorSigma[1, Power @@@ FactorInteger[#]]]/Apply[Times, FactorInteger[DivisorSigma[1, #]][[All, 1]]] &, 105] (* Michael De Vlieger, May 08 2022 *)

A080398(n) = factorback(factor(sigma(n))[, 1]);
A353783(n) = { my(f=factor(n)~); lcm(vector(#f, i, sigma(f[1, i]^f[2, i]))); };
A353785(n) = (A353783(n) / A080398(n));

a(n) = A353783(n) / A080398(n).

A353783 a(n) = LCM_{p^e||n} sigma(p^e), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 6, 12, 28, 14, 24, 12, 31, 18, 39, 20, 42, 8, 12, 24, 60, 31, 42, 40, 56, 30, 12, 32, 63, 12, 18, 24, 91, 38, 60, 28, 30, 42, 24, 44, 84, 78, 24, 48, 124, 57, 93, 36, 14, 54, 120, 12, 120, 20, 30, 60, 84, 62, 96, 104, 127, 42, 12, 68, 126, 24, 24, 72, 195, 74, 114, 124, 140, 24, 84, 80

Offset: 1

Antti Karttunen, May 08 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A080398, A087207, A351560, A353784, A353785.
Cf. also A345044, A345046.
Cf. A336547 (positions where equal to sigma).

Array[LCM @@ DivisorSigma[1, Power @@@ FactorInteger[#]] &, 79] (* Michael De Vlieger, May 08 2022 *)

A353783(n) = { my(f=factor(n)~); lcm(vector(#f, i, sigma(f[1, i]^f[2, i]))); };

a(n) = A000203(n) / A353784(n).
a(n) = A353785(n) * A080398(n).
For all n >= 1, A087207(a(n)) = A351560(n).

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

Original entry on oeis.org

1, 1456, 15480, 114660, 2244600, 3894768, 25108200, 27052704, 65021040, 112402080, 1973921400

Offset: 1

Antti Karttunen, Feb 19 2022

Numbers k such that their abundancy index [sigma(k)/k] is equal to A351557(k)/A351556(k).
Question: If the above ratio is neither 1 nor 2, must it then be > 2? Are all even terms abundant?
a(12) > 2281701376 if it exists.

Cf. A000203, A005101, A007947, A019565, A080398, A351556, A351557, A351558, A351559.
Subsequence of A386424.
Cf. also A351458, A351548.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA351549(n) = { my(s=sigma(n), z=A019565(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

Original entry on oeis.org

0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 3, 8, 9, 2, 3, 0, 1, 2, 1, 0, 1, 2, 3, 0, 0, 10, 1, 8, 5, 2, 1, 0, 1, 2, 3, 32, 1, 6, 1, 0, 9, 2, 1, 8, 33, 2, 3, 0, 0, 2, 3, 0, 1, 6, 3, 0, 1, 2, 3, 8, 1, 2, 33, 0, 1, 2, 65, 0, 1, 2, 3, 0, 1, 2, 1, 12, 1, 10, 5, 0, 16, 2, 3, 0, 1, 18, 7, 0, 1, 2, 9, 8, 1, 2, 7, 0, 1, 2, 35, 0, 65, 2, 33

Offset: 1

Antti Karttunen, Feb 19 2022

Cf. A000203, A004198, A007947, A019565, A080398, A351557, A351560.

Cf. also A324644, A351558.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 103}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351559(n) = A048675(gcd(sigma(n), A019565(n)));

a(n) = A048675(A351557(n)) = A048675(gcd(sigma(n), A019565(n))).

a(n) = n AND A351560(n), where AND is bitwise-and, A004198.

A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

Original entry on oeis.org

6, 28, 120, 135, 270, 496, 672, 891, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 200655, 273000, 293760, 307125, 401310, 441936, 446880, 502740, 523776, 544635, 614250, 707616, 802620, 819000, 884520

Offset: 1

Antti Karttunen, Feb 07 2020

nonn

Numbers k such that sigma(k) has the same set of distinct prime factors as 2*k.
Numbers k such that A007947(sigma(k)) is equal to A007947(2*k), or equally, that A087207(sigma(k)) is equal to A087207(2*k).
Of the first 256 terms 44 are odd, and none occurs in A228058. Compare also to A331752.

Antti Karttunen, Table of n, a(n) for n = 1..256
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A007947, A080398, A087207, A228058, A331751, A331752.

Subsequences: A000396, A027687.

Select[Range[10^6], SameQ @@ Map[Times @@ FactorInteger[#][[All, 1]] &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020 *)

A007947(n) = factorback(factorint(n)[, 1]);
isA332208(n) = (A007947(sigma(n)) == A007947(2*n));

{n: A080398(n) == A007947(2n)}.

A351557 a(n) = gcd(sigma(n), A019565(n)).

Original entry on oeis.org

1, 3, 2, 1, 2, 3, 2, 1, 1, 3, 6, 7, 14, 3, 6, 1, 2, 3, 2, 1, 2, 3, 6, 1, 1, 21, 2, 7, 10, 3, 2, 1, 2, 3, 6, 13, 2, 15, 2, 1, 14, 3, 2, 7, 26, 3, 6, 1, 1, 3, 6, 1, 2, 15, 6, 1, 2, 3, 6, 7, 2, 3, 26, 1, 2, 3, 34, 1, 2, 3, 6, 1, 2, 3, 2, 35, 2, 21, 10, 1, 11, 3, 6, 1, 2, 33, 30, 1, 2, 3, 14, 7, 2, 3, 30, 1, 2, 3, 78

Offset: 1

Antti Karttunen, Feb 19 2022

nonn

Cf. A000203, A007947, A019565, A080398.

Cf. also A324644, A351549, A351556, A351559.

Table[GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 99}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A351557(n) = gcd(sigma(n), A019565(n));

a(n) = gcd(A000203(n), A019565(n)) = gcd(A080398(n), A019565(n)).
a(n) = A007947(a(n)).
a(n) = A019565(A351559(n)).

A351560 a(n) is a binary representation of the primes that divide sigma(n) [the sum of divisors of n function], shown in decimal.

Original entry on oeis.org

0, 2, 1, 8, 3, 3, 1, 6, 32, 3, 3, 9, 9, 3, 3, 1024, 3, 34, 5, 11, 1, 3, 3, 7, 1024, 11, 5, 9, 7, 3, 1, 10, 3, 3, 3, 40, 129, 7, 9, 7, 11, 3, 17, 11, 35, 3, 3, 1025, 130, 1026, 3, 9, 3, 7, 3, 7, 5, 7, 7, 11, 1025, 3, 33, 1073741824, 11, 3, 65, 11, 3, 3, 3, 38, 2049, 131, 1025, 13, 3, 11, 5, 1027, 16, 11, 11, 9, 3, 19

Offset: 1

Antti Karttunen, Feb 19 2022

nonn

This is not additive sequence, but "oritive": For all coprime x, y (with gcd(x,y)=1), a(x*y) = a(x) OR a(y), where OR is bitwise-or (A003986). Compare also with A080398.

Cf. A000203, A003986, A007947, A048675, A080398, A087207, A351559 [= n AND a(n)].

{0}~Join~Array[Total[2^(PrimePi[#] - 1) & /@ FactorInteger[DivisorSigma[1, #]][[All, 1]]] &, 85, 2] (* Michael De Vlieger, Feb 20 2022 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351560(n) = A048675(A007947(sigma(n)));

a(n) = A087207(A000203(n)) = A048675(A080398(n)) = A048675(A007947(A000203(n))).

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

nonn

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)).

A355928 Squarefree part of the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 7, 6, 3, 2, 15, 13, 2, 3, 7, 14, 6, 6, 31, 2, 39, 5, 42, 2, 1, 6, 15, 31, 42, 10, 14, 30, 2, 2, 7, 3, 6, 3, 91, 38, 15, 14, 10, 42, 6, 11, 21, 78, 2, 3, 31, 57, 93, 2, 2, 6, 30, 2, 30, 5, 10, 15, 42, 62, 6, 26, 127, 21, 1, 17, 14, 6, 1, 2, 195, 74, 114, 31, 35, 6, 42, 5, 186, 1, 14, 21, 14, 3, 33, 30, 5, 10

Offset: 1

Antti Karttunen, Jul 24 2022

nonn

Not multiplicative.

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

Cf. A000203, A007913, A006532 (positions of 1's), A355929.

Cf. also A080398.

PARI
```
A355928(n) = core(sigma(n));
```

from sympy.ntheory.factor_ import core, divisor_sigma
def A355928(n): return core(divisor_sigma(n)) # Chai Wah Wu, Jul 28 2022

a(n) = A007913(A000203(n)).

a(n) = A355929(n) + A007913(n).

A387156 a(n) = A003557(sigma(n)), where A003557(n) is multiplicative with a(p^e) = p^(e-1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 2, 2, 1, 4, 4, 1, 3, 1, 2, 1, 16, 6, 4, 2, 1, 1, 4, 4, 1, 12, 16, 3, 8, 9, 8, 1, 1, 2, 4, 3, 1, 16, 2, 2, 1, 12, 8, 2, 1, 1, 12, 7, 9, 4, 12, 4, 8, 3, 2, 4, 1, 16, 4, 1, 2, 24, 2, 3, 16, 24, 12, 1, 1, 1, 2, 2, 16, 4, 8, 1, 11, 3, 2, 16, 18, 2, 4, 6, 3, 3, 8, 4, 64, 24, 4, 6, 7, 3, 2

Offset: 1

Antti Karttunen, Aug 19 2025

nonn

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

Cf. A000203, A003557, A062401, A080398, A387157.

Cf. also A386424, A386425.

A387156[n_] := # / Times @@ FactorInteger[#][[All, 1]] & [DivisorSigma[1, n]];
Array[A387156, 100] (* Paolo Xausa, Aug 20 2025 *)

A387156(n) = { my(s=sigma(n)); s/factorback(factor(s)[,1]); };

a(n) = A000203(n) / A080398(n).

a(n) = A062401(n) / A387157(n).

cp's OEIS Frontend

A353785 a(n) = A353783(n) / A080398(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A353783 a(n) = LCM_{p^e||n} sigma(p^e), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351557 a(n) = gcd(sigma(n), A019565(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351560 a(n) is a binary representation of the primes that divide sigma(n) [the sum of divisors of n function], shown in decimal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica