A344697 -id:A344697

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A344699

a(n) = A344697(A108951(n)).

+20
5

1, 1, 1, 6, 1, 6, 1, 4, 72, 6, 1, 4, 1, 6, 72, 24, 1, 48, 1, 4, 72, 6, 1, 24, 2160, 6, 18, 4, 1, 48, 1, 16, 72, 6, 2160, 288, 1, 6, 72, 24, 1, 48, 1, 4, 18, 6, 1, 16, 5760, 288, 72, 4, 1, 108, 2160, 24, 72, 6, 1, 288, 1, 6, 18, 96, 2160, 48, 1, 4, 72, 288, 1, 64, 1, 6, 108, 4, 5760, 48, 1, 16, 2592, 6, 1, 288, 2160

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10201

Index entries for sequences related to primorial numbers

Index entries for sequences related to sigma(n)

PROG

(PARI)

A034386(n) = prod(i=1, primepi(n), prime(i));

A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };

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

A344697(n) = { my(u=A001615(n)); (u/gcd(u, sigma(n))); };

A344699(n) = A344697(A108951(n));

CROSSREFS

Cf. A000203, A001615, A108951, A337203, A344697, A344698, A344701 (apparently positions of records).

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 26 2021

STATUS

approved

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.

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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.

LINKS

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

FORMULA

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

PROG

(PARI)

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];

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 22 2021

STATUS

approved

A344695

a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

+10
19

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, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702.

A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative.

If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703.

Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = gcd(A000203(n), A001615(n)).

For prime p, a(p^e) = (p+1)^(e mod 2).

For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).

a(A007913(n)) | a(n).

a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)).

a(n) = A000203(n) / A344696(n).

a(n) = A001615(n) / A344697(n).

MATHEMATICA

Table[GCD[DivisorSigma[1, n], DivisorSum[n, MoebiusMu[n/#]^2*#&]], {n, 100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

PROG

(PARI)

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));

(Python 3.8+)

from math import prod, gcd

from sympy import primefactors, divisor_sigma

def A001615(n):

plist = primefactors(n)

return n*prod(p+1 for p in plist)//prod(plist)

def A344695(n): return gcd(A001615(n), divisor_sigma(n)) # Chai Wah Wu, Jun 03 2021

CROSSREFS

Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1).

Cf. also A048250, A291750, A291751, A340070, A323363.

Subsets of range: A008864, A065091 (conjectured).

KEYWORD

nonn

AUTHOR

Antti Karttunen and Peter Munn, May 26 2021

STATUS

approved

A344696

a(n) = sigma(n) / gcd(sigma(n), A001615(n)).

+10
9

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 10, 7, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 10, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 21, 1, 57, 13

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 35, although a(4) = 7 and a(27) = 10. See A344702.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10404

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A000203(n) / A344695(n).

PROG

(PARI)

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

A344696(n) = { my(u=sigma(n)); (u/gcd(u, A001615(n))); };

CROSSREFS

Cf. A000203, A001615, A005117 (positions of ones), A344695, A344697, A344698, A344702.

Cf. also A344756.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 26 2021

STATUS

approved

A344702

Positions k where A344695(k) is not multiplicative.

+10
4

108, 196, 200, 216, 288, 432, 441, 500, 540, 588, 600, 675, 676, 756, 784, 800, 864, 882, 980, 1000, 1080, 1125, 1188, 1225, 1323, 1350, 1372, 1400, 1404, 1440, 1444, 1500, 1512, 1521, 1568, 1728, 1764, 1800, 1836, 2000, 2016, 2028, 2052, 2156, 2160, 2200, 2205, 2250, 2352, 2376, 2400, 2450, 2484, 2548, 2592, 2600

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k with a factorization into coprime x and k/x with A344695(x) * A344695(k/x) <> A344695(k). - Peter Munn, Jun 04 2021

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

For 108 = 4*27, A344695(108) = 8, although A344695(4) = 1 and A344695(27) = 4, and 1*4 != 8, therefore 108 is included in this sequence.

For 441 = 9*49, A344695(441) = 3, although A344695(9) = 1 and A344695(49) = 1, and 1*1 != 3, therefore 441 is included in this sequence.

PROG

(PARI)

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));

A344695mult(n) = { my(f = factor(n)); prod(k=1, #f~, A344695(f[k, 1]^f[k, 2])); };

isA344702(n) = (A344695(n)!=A344695mult(n));

CROSSREFS

Cf. A000203, A001615, A344695, A344696, A344697.

Subsequence of A013929 and of A024619.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 27 2021

STATUS

approved

A348049

a(n) = A003959(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.

+10
4

1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 9, 1, 1, 1, 81, 1, 16, 1, 9, 1, 1, 1, 9, 36, 1, 8, 9, 1, 1, 1, 27, 1, 1, 1, 144, 1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 81, 64, 36, 1, 9, 1, 8, 1, 9, 1, 1, 1, 9, 1, 1, 16, 729, 1, 1, 1, 9, 1, 1, 1, 144, 1, 1, 36, 9, 1, 1, 1, 81, 256, 1, 1, 9, 1, 1, 1, 9, 1, 16, 1, 9, 1, 1, 1, 27, 1, 64

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Not multiplicative. For example, a(196) = 192 != a(4) * a(49).

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A003959(n) / A348047(n) = A003959(n) / gcd(A000203(n), A003959(n)).

MATHEMATICA

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

PROG

(PARI)

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

A348049(n) = { my(u=A003959(n)); (u/gcd(u, sigma(n))); };

CROSSREFS

Cf. A000203, A003959, A005117 (positions of 1's), A348029, A348047, A348048.

Cf. also A344697.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 21 2021

STATUS

approved

A348505

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

+10
4

1, 1, 1, 5, 1, 1, 1, 3, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 3, 26, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 50, 1, 1, 1, 3, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1, 1, 5, 1, 1, 1, 3, 1, 10, 1, 5, 1, 1, 1, 11, 1, 50, 10, 130

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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) = 6 != 3*10 = a(8) * a(9).

LINKS

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

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A034448(n) / A348503(n) = A034448(n) / gcd(A000203(n), A034448(n)).

MATHEMATICA

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

PROG

(PARI)

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

A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };

CROSSREFS

Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).

Cf. also A344697, A348049.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 29 2021

STATUS

approved

A344757

a(n) = A069359(n) / gcd(A003415(n), A069359(n)).

+10
3

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 5, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 13, 8, 1, 1, 5, 1, 7, 1, 15, 1, 5, 1, 9, 1, 1, 1, 31, 1, 1, 10, 1, 1, 1, 1, 19, 1, 1, 1, 5, 1, 1, 8, 21, 1, 1, 1, 7, 1, 1, 1, 41, 1, 1, 1, 13, 1, 31, 1, 25, 1, 1, 1, 5, 1, 9, 14, 1, 1, 1, 1, 15

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,11

LINKS

Antti Karttunen, Table of n, a(n) for n = 2..12012

Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

FORMULA

a(n) = A069359(n) / A340070(n) = A069359(n) / gcd(A003415(n), A069359(n)).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359