A001615 -id:A001615 - cp's OEIS frontend

A291126 Psibonacci numbers: solutions n of the equation psi(n) = psi(n-1) + psi(n-2), where psi is the Dedekind psi function (A001615).

3, 6, 210, 88200, 101970, 193290, 289680, 993990, 11264550, 59068230, 72776970, 98746230, 122460690, 126500910, 132766770, 234150930, 514442214, 531391650, 638082390, 650428020, 790769790, 1249160790, 3727074450, 4775972850, 8299675650, 9530202210

Offset: 1

Amiram Eldar, Aug 19 2017

nonn

Analogous to phibonacci numbers (A065557) and other sequences (see crossrefs).

			psi(210) = 576 = 240 + 336 = psi(209) + psi(208), therefore 210 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..70 (terms < 4*10^12)

Cf. A001615, A065557, A065900, A076136, A076251, A145469.

psi[n_]:=If[n < 1, 0, n Sum[ MoebiusMu[ d]^2 / d, {d, Divisors @ n}]];
Select[Range[10^6], psi[#]==psi[#-1]+psi[#-2] &]

a(21)-a(26) from Giovanni Resta, Aug 26 2018

A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 9, 16, 36, 0, 12, 0, 48, 48, 27, 0, 24, 0, 18, 64, 72, 0, 60, 36, 84, 32, 24, 0, 0, 0, 45, 96, 108, 96, 84, 0, 120, 112, 90, 0, 0, 0, 36, 48, 144, 0, 84, 64, 72, 144, 42, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 64, 99, 168, 0, 0, 54, 192, 0, 0, 132, 0, 228, 96, 60, 192, 0, 0, 126, 112, 252, 0, 288, 216, 264, 240, 180, 0

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

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

Cf. A001615, A323363.

Cf. also A319340, A322581, A323365, A323372, A323399.

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));
A323364(n) = (A001615(n) + A323363(n));

A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 12, 6, 1, 1, 8, 1, 3, 8, 8, 1, 3, 1, 12, 2, 1, 1, 4, 10, 3, 9, 16, 1, 1, 1, 16, 2, 1, 12, 12, 1, 3, 8, 4, 1, 1, 1, 24, 3, 1, 1, 16, 14, 45, 4, 28, 1, 27, 8, 4, 2, 1, 1, 4, 1, 3, 3, 96, 6, 1, 1, 36, 2, 1, 1, 12, 1, 3, 5, 40, 6, 1, 1, 16, 108, 1, 1, 4, 2, 3, 8, 4, 1, 3, 4, 48, 2, 1, 24, 16, 1, 7, 3, 20

Offset: 1

Antti Karttunen, Mar 28 2021

nonn

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

Cf. A001615, A003415, A003557, A342459, A342919.
Cf. A301939 (gives the positions at which a(n) = A001615(n) = A003415(n)).
Cf. also A175732, A342413, A342915.

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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342458(n) = gcd(A001615(n), A003415(n));

a(n) = gcd(A001615(n), A003415(n)).
a(n) = A003557(n) * A342459(n).
a(n) = A003415(n) / A342919(n).

A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 1, 6, 1, 8, 3, 2, 1, 12, 1, 14, 3, 8, 1, 18, 1, 20, 3, 2, 1, 24, 1, 2, 3, 4, 1, 30, 1, 32, 3, 2, 1, 12, 1, 38, 3, 8, 1, 42, 1, 44, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 62, 3, 32, 1, 6, 1, 68, 3, 2, 1, 72, 1, 74, 3, 4, 1, 6, 1, 80, 9, 2, 1, 84, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 24, 1, 98, 3, 4, 1, 102

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

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

Cf. A001615, A342916, A342917.
Cf. also A049559, A342458.
After n=1 differs from A143771 for the first time at n=44, where a(44) = 9, while A143771(44) = 3.

psi[n_] := If[n==1, 1, Times @@ ((#1+1)*#1^(#2-1)& @@@ FactorInteger[n])];
a[n_] := GCD[n+1, psi[n]];
Array[a, 105] (* Jean-François Alcover, Dec 22 2021 *)

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
A342915(n) = gcd(1+n,A001615(n));

a(n) = gcd(1+n, A001615(n)).

a(n) = (1+n) / A342916(n) = A001615(n) / A342917(n).

A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

Original entry on oeis.org

0, 1, 1, 7, 1, 12, 1, 30, 10, 16, 1, 65, 1, 20, 18, 104, 1, 83, 1, 93, 22, 28, 1, 254, 16, 32, 63, 121, 1, 167, 1, 320, 30, 40, 26, 391, 1, 44, 34, 374, 1, 215, 1, 177, 143, 52, 1, 840, 22, 165, 42, 205, 1, 450, 34, 494, 46, 64, 1, 827, 1, 68, 183, 912, 38, 311, 1, 261, 54, 295, 1, 1430, 1, 80, 197, 289, 38, 359

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of Dedekind psi function (A001615) with the arithmetic derivative (A003415).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001615, A003415.

Cf. also A305809, A322577, A347131, A347133, A347135.

Table[DivisorSum[n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/#]*If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &], {n, 78}] (* Michael De Vlieger, Oct 19 2021, after Jan Mangaldan at A001615 *)

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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347132(n) = sumdiv(n,d,A001615(n/d)*A003415(d));

a(n) = Sum_{d|n} A001615(n/d) * A003415(d).

A348982 a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 6, 1, 11, 1, 22, 9, 15, 1, 52, 1, 19, 17, 66, 1, 69, 1, 76, 21, 27, 1, 176, 15, 31, 51, 100, 1, 145, 1, 178, 29, 39, 25, 288, 1, 43, 33, 264, 1, 189, 1, 148, 123, 51, 1, 508, 21, 145, 41, 172, 1, 339, 33, 352, 45, 63, 1, 632, 1, 67, 159, 450, 37, 277, 1, 220, 53, 265, 1, 924, 1, 79, 175, 244, 37, 321

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A001615 with A322582.

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

Cf. A001615, A003958, A322582, A327251, A348980, A348981, A348982, A348983, A349132.

Cf. also A347132, A349142.

f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

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
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348982(n) = sumdiv(n,d,A001615(n/d)*A322582(d));

a(n) = Sum_{d|n} A001615(n/d) * A322582(d).
For all n >= 1, a(n) <= A347132(n) <= A349142(n).
a(n) = A327251(n) - A349132(n). - Antti Karttunen, Nov 14 2021

A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 22, 24, 40, 22, 60, 26, 56, 60, 46, 34, 96, 38, 100, 84, 88, 46, 132, 70, 104, 84, 140, 58, 240, 62, 94, 132, 136, 140, 240, 74, 152, 156, 220, 82, 336, 86, 220, 240, 184, 94, 276, 140, 280, 204, 260, 106, 336, 220, 308, 228, 232, 118, 600, 122, 248, 336, 190, 260, 528, 134, 340, 276, 560

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with Dedekind psi function, A001615.

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

Cf. A001615, A003958, A327251, A348982, A349130, A349131, A349133, A349172.

f[p_, e_] := (p + 1)*p^e - p*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

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)));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349132(n) = sumdiv(n,d,A001615(d)*A003958(n/d));

a(n) = Sum_{d|n} A001615(d) * A003958(n/d).
a(n) = A327251(n) - A348982(n).
For all n >= 1, a(n) <= A349172(n).
Multiplicative with a(p^e) = (p+1)*p^e - p*(p-1)^e. - Amiram Eldar, Nov 09 2021

A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 8, 1, 13, 1, 40, 11, 17, 1, 80, 1, 21, 19, 164, 1, 99, 1, 112, 23, 29, 1, 364, 17, 33, 77, 144, 1, 191, 1, 604, 31, 41, 27, 528, 1, 45, 35, 524, 1, 243, 1, 208, 165, 53, 1, 1424, 23, 187, 43, 240, 1, 597, 35, 684, 47, 65, 1, 1072, 1, 69, 209, 2084, 39, 347, 1, 304, 55, 327, 1, 2244, 1, 81, 221, 336, 39, 399

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A001615 with A348507.

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

Cf. A001615, A003959, A327251, A348507, A349140, A349141, A349143, A349172.

Cf. also A347132, A348982.

f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #)*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

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
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349142(n) = sumdiv(n,d,A001615(d)*A348507(n/d));

a(n) = Sum_{d|n} A001615(n/d) * A348507(d).
For all n >= 1, a(n) >= A347132(n) >= A348982(n).
a(n) = A349172(n) - A327251(n). - Antti Karttunen, Nov 14 2021

A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.

Original entry on oeis.org

1, 6, 8, 24, 12, 48, 16, 84, 44, 72, 24, 192, 28, 96, 96, 276, 36, 264, 40, 288, 128, 144, 48, 672, 102, 168, 212, 384, 60, 576, 64, 876, 192, 216, 192, 1056, 76, 240, 224, 1008, 84, 768, 88, 576, 528, 288, 96, 2208, 184, 612, 288, 672, 108, 1272, 288, 1344, 320, 360, 120, 2304, 124, 384, 704, 2724, 336, 1152, 136

Offset: 1

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A001615 with A003959.

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

Cf. A001615, A003959, A327251, A349142, A349170, A349171, A349172, A349173.

f[p_, e_] := (p + 2)*(p + 1)^e - (p + 1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

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)));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349172(n) = sumdiv(n,d,A001615(d)*A003959(n/d));

a(n) = Sum_{d|n} A001615(d) * A003959(n/d).
a(n) = A327251(n) + A349142(n).
For all n >= 1, a(n) >= A349132(n).
Multiplicative with a(p^e) = (p+2)*(p+1)^e - (p+1)*p^e. - Amiram Eldar, Nov 09 2021

A295888 Filter combining prime signature of n (A101296) with Dedekind's psi (A001615).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 42, 47, 48, 49, 42, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 56, 60, 63, 64, 65, 66, 67, 67, 56, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 77

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

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

Cf. A001615, A046523, A101296, A291750, A295300, A295880, A295887, A296090.

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])); }
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
Anotsubmitted8(n) = (1/2)*(2 + ((A046523(n)+A001615(n))^2) - A046523(n) - 3*A001615(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted8(n))),"b295888.txt");

Restricted growth sequence transform of function f(n) = (1/2)*(2 + ((A046523(n) + A001615(n))^2) - A046523(n) - 3*A001615(n)), where values A046523(n) and A001615(n) are packed together to a(n) with the 2-argument form of A000027, also known as Cantor pairing-function.

cp's OEIS Frontend

A291126 Psibonacci numbers: solutions n of the equation psi(n) = psi(n-1) + psi(n-2), where psi is the Dedekind psi function (A001615).

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A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

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A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

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A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

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A348982 a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

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A349132 a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.

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A349142 a(n) = Sum_{d|n} psi(n/d) * A348507(d), where psi is Dedekind psi (A001615), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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