A003972 -id:A003972 - cp's OEIS frontend

A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

1, 1, 1, 3, 1, 4, 1, 9, 5, 3, 1, 6, 1, 5, 12, 27, 1, 20, 1, 18, 20, 12, 1, 36, 7, 16, 25, 30, 1, 6, 1, 81, 3, 9, 15, 15, 1, 11, 16, 27, 1, 20, 1, 18, 20, 28, 1, 54, 11, 42, 36, 12, 1, 100, 4, 45, 44, 15, 1, 72, 1, 36, 100, 243, 48, 48, 1, 54, 7, 12, 1, 180, 1, 40, 42, 66, 60, 64, 1, 162, 125, 21, 1, 120, 9, 23, 60, 108

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A160595.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A160595, A253885, A340071, A340073, A340075 (gives the odd part).

Cf. also A340082.

f:= proc(n) local F,x,p,t;
  F:= ifactors(n)[2];
  x:= mul(nextprime(t[1])^t[2],t=F);
  p:= numtheory:-phi(x);
  p/igcd(x-1,p)
end proc:
map(f,[$1..100]); # Robert Israel, Dec 28 2020

a[n_] := Module[{x, p, e, phi}, x = Product[{p, e} = pe; NextPrime[p]^e, {pe, FactorInteger[n]}]; phi = EulerPhi[x]; phi/GCD[x-1, phi]];
Array[a, 100] (* Jean-François Alcover, Jan 04 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };

a(n) = A160595(A003961(n)).

a(n) = A003972(n) / A340071(n).

A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

Original entry on oeis.org

1, 2, 3, 6, 4, 6, 6, 18, 15, 8, 6, 18, 6, 12, 12, 54, 6, 30, 6, 24, 18, 12, 10, 54, 28, 12, 75, 36, 8, 24, 8, 162, 18, 12, 24, 90, 10, 12, 18, 72, 6, 36, 6, 36, 60, 20, 10, 162, 66, 56, 18, 36, 12, 150, 24, 108, 18, 16, 8, 72, 8, 16, 90, 486, 24, 36, 10, 36, 30, 48, 6, 270, 8, 20, 84, 36, 36, 36, 10, 216, 375, 12

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because both A003961 and A349125 are.

Convolving this with A349127 gives A003972.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A064989, A349125, A349382 (Dirichlet inverse), A349383 (sum with it).

Cf. also A003972, A349127, and A349355, A349356 and A349384, A349385, and A349387.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349125(n) = (moebius(n)*A064989(n));
A349381(n) = sumdiv(n,d,A003961(n/d)*A349125(d));

a(n) = Sum_{d|n} A003961(n/d) * A349125(d).

a(n) = A349383(n) - A349382(n).

A332463 Möbius transform of A332223.

Original entry on oeis.org

1, 1, 3, 3, 7, 4, 15, 2, 21, 9, 31, 13, 63, 4, 10, 42, 127, -3, 255, 14, 21, 88, 511, 22, 117, 320, 24, 97, 1023, -22, 2047, -36, 190, 1444, 82, 34, 4095, -200, 120, 306, 8191, 14, 16383, -59, 13, 4180, 32767, 30, 609, -103, 15494, -303, 65535, 30, 141, -32, -6, 8920, 131071, 132, 262143, 506030, 564, 834, 658, -149

Offset: 1

Antti Karttunen, Feb 22 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed using Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A008683, A332223.

Cf. also A003972, A324201, A324543.

A332463(n) = sumdiv(n,d,moebius(n/d)*A332223(d)); \\ Needs also code from A332223.

a(n) = Sum_{d|n} A008683(n/d) * A332223(d).

A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 5, 9, 5, 3, 3, 3, 1, 5, 3, 27, 9, 5, 11, 9, 5, 3, 7, 9, 21, 1, 25, 15, 15, 3, 9, 81, 3, 9, 15, 15, 5, 11, 1, 27, 21, 5, 23, 9, 15, 7, 13, 27, 55, 21, 9, 3, 29, 25, 9, 45, 11, 15, 15, 9, 33, 9, 25, 243, 3, 3, 35, 27, 7, 15, 9, 45, 39, 5, 21, 33, 15, 1, 41, 81, 125, 21, 11, 15, 27, 23, 15, 27, 3, 15

Offset: 1

Antti Karttunen, Dec 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A000265, A003961, A003972, A005117, A053575, A151800, A339903, A340075.

A000265(n) = (n>>valuation(n,2));
A339904(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,my(q=nextprime(1+f[i,1])); A000265(q-1)*(q^(f[i,2]-1))));

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339903(n) = A000265(eulerphi(A003961(n)));

Multiplicative with a(p^e) = A000265(q-1) * q^(e-1), where q = A151800(p), the next prime larger than p.
a(n) = A000265(A003972(n)) = A053575(A003961(n)) = A000265(A000010(A003961(n))).
For all squarefree numbers k, a(k) = A339903(k).

A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.

Original entry on oeis.org

1, 5, 8, 19, 12, 40, 18, 65, 49, 60, 24, 152, 30, 90, 96, 211, 36, 245, 42, 228, 144, 120, 52, 520, 109, 150, 272, 342, 60, 480, 68, 665, 192, 180, 216, 931, 78, 210, 240, 780, 84, 720, 90, 456, 588, 260, 100, 1688, 247, 545, 288, 570, 112, 1360, 288, 1170, 336, 300, 120, 1824, 128, 340, 882, 2059, 360, 960, 138

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of the identity function (A000027) with the prime shifted identity (A003961). Multiplicative because both A000027 and A003961 are.
Dirichlet convolution of Euler phi (A000010) with the prime shifted sigma (A003973).
Dirichlet convolution of sigma (A000203) with the prime shifted phi (A003972).
Inverse Möbius transform of A347137.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A003972, A003973, A151800, A347121, A347137 (Möbius transform).

Cf. also A038040, A336845, A336475, A347236.

f[p_, e_] := ((np = NextPrime[p])^(e + 1) - p^(e + 1))/(np - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347136(n) = sumdiv(n,d,d*A003961(n/d));

a(n) = Sum_{d|n} d * A003961(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A003973(d).
a(n) = Sum_{d|n} A000203(n/d) * A003972(d).
a(n) = Sum_{d|n} A347137(d).
For all primes p, a(p) = p + A003961(p).
a(n) = A347121(n) + 2*n.
Multiplicative with a(p^e) = (A151800(p)^(e+1) - p^(e+1))/(A151800(p)-p). - Amiram Eldar, Aug 24 2021

A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

Original entry on oeis.org

1, 4, 12, 24, 30, 48, 70, 144, 180, 120, 132, 288, 208, 280, 360, 864, 306, 720, 418, 720, 840, 528, 644, 1728, 1050, 832, 2700, 1680, 870, 1440, 1116, 5184, 1584, 1224, 2100, 4320, 1480, 1672, 2496, 4320, 1722, 3360, 1978, 3168, 5400, 2576, 2444, 10368, 5390, 4200, 3672, 4992, 3074, 10800, 3960, 10080, 5016, 3480

Offset: 1

Antti Karttunen, May 10 2022

Question: Does a(n) divide A353790(n) only when n=1? Compare to A353764.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization.

Cf. A003961, A003972, A151800, A353749, A353764, A353790.

Cf. also A353793.

f[p_, e_] := ((q = NextPrime[p]) - 1) * q^(e - 1) * p^e; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 10 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A353789(n) = (n * eulerphi(A003961(n)));

from math import prod
from sympy import nextprime, factorint
def A353789(n): return prod((q:= nextprime(p))**(e-1)*p**e*(q-1) for p, e in factorint(n).items()) # Chai Wah Wu, May 10 2022

Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.
a(n) = A353749(A003961(n)) = n * A003972(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2-p+1)/(p^3 - p*q)) = 0.836506229..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022

A337544 a(n) = 2*phi(A003961(n)) - A003961(n).

Original entry on oeis.org

1, 1, 3, 3, 5, 1, 9, 9, 15, 3, 11, 3, 15, 7, 13, 27, 17, 5, 21, 9, 25, 9, 27, 9, 35, 13, 75, 21, 29, -9, 35, 81, 31, 15, 43, 15, 39, 19, 43, 27, 41, -5, 45, 27, 65, 25, 51, 27, 99, 21, 49, 39, 57, 25, 53, 63, 61, 27, 59, -27, 65, 33, 125, 243, 73, -3, 69, 45, 79, 9, 71, 45, 77, 37, 91, 57, 97, 1, 81, 81, 375, 39, 87, -15, 83

Offset: 1

Antti Karttunen, Sep 21 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A083254.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f);
A337544(n) = (2*eulerphi(A003961(n)) - A003961(n));

a(n) = A083254(A003961(n)).

a(n) = 2*A003972(n) - A003961(n).

A340073 a(n) = (x-1) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 13, 6, 5, 1, 11, 1, 8, 17, 40, 1, 37, 1, 31, 27, 19, 1, 67, 8, 25, 31, 49, 1, 13, 1, 121, 4, 14, 19, 28, 1, 17, 21, 47, 1, 41, 1, 29, 29, 43, 1, 101, 12, 73, 47, 19, 1, 187, 5, 74, 57, 23, 1, 157, 1, 55, 137, 364, 59, 97, 1, 85, 9, 23, 1, 337, 1, 61, 61, 103, 71, 127, 1, 283, 156, 32, 1, 247, 11

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A160596.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A160596, A253885, A340071, A340072.

Cf. also A340083.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340073(n) = { my(x=A003961(n)); (x-1)/gcd(x-1, eulerphi(x)); };

a(n) = A160596(A003961(n)).

a(n) = A253885(n-1) / A340071(n) = (A003961(n)-1) / A340071(n).

A347137 a(n) = Sum_{d|n} phi(d) * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes, and phi is Euler totient function.

Original entry on oeis.org

1, 4, 7, 14, 11, 28, 17, 46, 41, 44, 23, 98, 29, 68, 77, 146, 35, 164, 41, 154, 119, 92, 51, 322, 97, 116, 223, 238, 59, 308, 67, 454, 161, 140, 187, 574, 77, 164, 203, 506, 83, 476, 89, 322, 451, 204, 99, 1022, 229, 388, 245, 406, 111, 892, 253, 782, 287, 236, 119, 1078, 127, 268, 697, 1394, 319, 644, 137, 490

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of Euler phi (A000010) with the prime shift function (A003961). Multiplicative because both A000010 and A003961 are.
Dirichlet convolution of the identity function (A000027) with the prime shifted phi (A003972).
Möbius transform of A347136.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000027, A000040, A001043, A003961, A003972, A008683, A151800, A347122, A347136 (inverse Möbius transform).

Cf. also A018804, A347237.

f[p_, e_] := (q = NextPrime[p])^e + (p - 1)*(q^e - p^e)/(q - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347137(n) = sumdiv(n,d,eulerphi(n/d)*A003961(d));

a(n) = Sum_{d|n} A000010(n/d) * A003961(d).
a(n) = Sum_{d|n} d * A003972(n/d).
a(n) = Sum_{d|n} A008683(n/d) * A347136(d).
a(n) = A347122(n) + 2*A000010(n).
a(A000040(n)) = A001043(n) - 1.
Multiplicative with a(p^e) = q(p)^e + (p-1)*(q(p)^e - p^e)/(q(p) - p), where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Sep 16 2023

A347376 Möbius transform of A250469.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 10, 12, 20, 18, 12, 12, 16, 26, 24, 24, 18, 16, 22, 24, 40, 48, 28, 24, 42, 56, 40, 36, 30, 24, 36, 48, 68, 78, 60, 36, 40, 86, 74, 48, 42, 32, 46, 60, 60, 104, 52, 48, 110, 78, 102, 72, 58, 68, 72, 72, 118, 138, 60, 48, 66, 144, 80, 96, 96, 52, 70, 96, 142, 84, 72, 72, 78, 176, 108, 108, 120, 70

Offset: 1

Antti Karttunen, Sep 01 2021

nonn

Question: Are all terms positive?

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A003972, A008683, A020639, A078898, A250469, A347377.

Cf. also A346479.

up_to = 10000;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
A347376(n) = sumdiv(n,d,moebius(n/d)*A250469(d));

a(n) = Sum_{d|n} A008683(n/d) * A250469(d).

a(n) = A003972(n) - A347377(n).

cp's OEIS Frontend

A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

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A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

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A332463 Möbius transform of A332223.

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A339904 The odd part of {Euler totient function phi applied to the prime shifted n}: a(n) = A000265(A000010(A003961(n))).

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A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.

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A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

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A337544 a(n) = 2*phi(A003961(n)) - A003961(n).

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A340073 a(n) = (x-1) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

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