A325813 -id:A325813 - cp's OEIS frontend

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 4, 5, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 1, 4, 4, 1, 2, 1, 1, 2, 3, 4, 4, 1, 1, 2, 2, 1, 4, 5, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 4, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 4, 2, 5, 4, 1, 1, 2, 2, 3, 1, 2, 1, 4, 4, 1, 1, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Cf. A000120, A000203, A005187, A033879, A326131.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326140, A326144.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A326130(n) = gcd(hammingweight(n), A294898(n));

A326130(n) = gcd(hammingweight(n), sigma(n)-A005187(n));

a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), A000203(n)-A005187(n)).

A326140 a(n) = gcd(A318878(n), A318879(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 2, 12, 2, 6, 1, 16, 1, 18, 2, 10, 2, 22, 2, 19, 2, 14, 6, 28, 6, 30, 1, 18, 2, 22, 1, 36, 2, 22, 2, 40, 2, 42, 2, 12, 2, 46, 2, 41, 1, 30, 6, 52, 2, 38, 2, 34, 2, 58, 6, 60, 2, 22, 1, 46, 6, 66, 2, 42, 2, 70, 1, 72, 2, 26, 6, 58, 2, 78, 2, 41, 2, 82, 2, 62, 2, 54, 2, 88, 6, 70, 2, 58, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A033879, A083254, A318878, A318879, A326141.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326144.

A326140(n) = { my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); gcd(t,u); };

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);
A318879(n) = sumdiv(n,d,d=d-(2*eulerphi(d)); (d>0)*d);
A326140(n) = gcd(A318878(n), A318879(n));

A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

Original entry on oeis.org

1, 6, 12, 28, 56, 60, 108, 120, 132, 168, 264, 280, 312, 408, 420, 440, 456, 496, 528, 540, 552, 696, 700, 728, 744, 756, 760, 888, 984, 992, 1032, 1128, 1140, 1188, 1272, 1404, 1416, 1456, 1464, 1608, 1704, 1710, 1752, 1836, 1896, 1992, 2052, 2136, 2328, 2424, 2472, 2484, 2568, 2616, 2646, 2712, 3048, 3132, 3144, 3288, 3336, 3344

Offset: 1

Antti Karttunen, May 23 2019

nonn

Numbers k for which A325813(k) is equal to abs(A325814(k)).
Numbers k such that A325814(k) is not zero (not in A064591) and divides A034460(k).
Conjecture: after the initial one all other terms are even. If this holds then there are no odd perfect numbers.

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences where any odd perfect numbers must occur

Cf. A034448, A034460, A048146, A064591, A325813, A325814, A325822.

Cf. A000396 (a subsequence).

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));
isA325812(n) = (A325813(n)==abs(A325814(n)));
\\ Alternatively:
isA325812(n) = (A325814(n) && !(A034460(n)%A325814(n)));

A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 4, 12, 2, 3, 1, 16, 1, 18, 2, 1, 2, 22, 4, 19, 2, 14, 4, 28, 6, 30, 1, 3, 2, 1, 1, 36, 2, 1, 2, 40, 2, 42, 4, 3, 2, 46, 4, 41, 1, 3, 2, 52, 2, 1, 8, 1, 2, 58, 12, 60, 2, 1, 1, 1, 6, 66, 2, 3, 2, 70, 1, 72, 2, 2, 4, 1, 2, 78, 2, 41, 2, 82, 4, 1, 2, 3, 4, 88, 6, 7, 4, 1, 2, 5, 4, 96, 1, 3, 1, 100, 6, 102, 2, 3

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

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)

Cf. A000203, A032742, A326065, A326066, A326067, A326068.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326066(n) = (sigma(n) - sigma(A032742(n)));
A326067(n) = (A326066(n) - n);
A326065(n) = sigma(A032742(n));
A326068(n) = (n - A326065(n));
A326069(n) = gcd(A326067(n), A326068(n));

a(n) = gcd(A326067(n), A326068(n)) = gcd(A326066(n) - n, n - A326065(n)).

A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 1, 18, 2, 2, 4, 22, 2, 1, 2, 2, 26, 28, 4, 30, 1, 6, 2, 2, 1, 36, 4, 2, 2, 40, 4, 42, 2, 6, 4, 46, 2, 1, 1, 6, 2, 52, 4, 2, 2, 2, 2, 58, 2, 60, 4, 2, 1, 2, 4, 66, 2, 6, 4, 70, 1, 72, 2, 2, 2, 2, 4, 78, 26, 1, 2, 82, 2, 2, 4, 6, 2, 88, 2, 14, 2, 2, 4, 10, 2, 96, 1, 6, 1, 100, 4, 102, 2, 6

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

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

Cf. A000203, A020639, A046666, A326146, A326148.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326073, A326129, A326130, A326140, A326144.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326147(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n);

a(n) = gcd(n-A020639(n), A000203(n)-A020639(n)-n).

For n > 1, a(n) = gcd(A046666(n), A326146(n)).

A326073 a(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n (and 1 for 1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 7, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

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

Cf. A000203, A020639, A326074.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140, A326144, A326147.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326073(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n);

a(n) = gcd(1+n-A020639(n), 1+A000203(n)-A020639(n)-n).

A325810 Lexicographically earliest sequence such that a(i) = a(j) => A034460(i) = A034460(j) and A325814(i) = A325814(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 35, 2, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 45, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Antti Karttunen, May 23 2019

nonn

Restricted growth sequence transform of the ordered pair [A034460(n), A325814(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A033879(i) = A033879(j),
  a(i) = a(j) => A325811(i) = A325811(j) => A325813(i) = A325813(i).

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

Cf. A033879, A034460, A324400, A325811, A325813, A325814.

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; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
v325810 = rgs_transform(vector(up_to,n,[A034460(n), A325814(n)]));
A325810(n) = v325810[n];

A325811 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) where f(n) = [A034460(n), A325814(n)] for all other numbers, except f(p) = -1 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 23 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  A325810(i) = A325810(j) => a(i) = a(j) => A325813(i) = A325813(i).

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

Cf. A034460, A305801, A325810, A325813, A325814.

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; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
Aux325811(n) = if(isprime(n)&&(n%2),-(n%2),[A034460(n), A325814(n)]);
v325811 = rgs_transform(vector(up_to,n,Aux325811(n)));
A325811(n) = v325811[n];

cp's OEIS Frontend

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

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A326140 a(n) = gcd(A318878(n), A318879(n)).

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A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

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A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

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A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

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Formula

A326073 a(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n (and 1 for 1), and sigma is the sum of divisors of n.

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A325810 Lexicographically earliest sequence such that a(i) = a(j) => A034460(i) = A034460(j) and A325814(i) = A325814(j) for all i, j.

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A325811 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) where f(n) = [A034460(n), A325814(n)] for all other numbers, except f(p) = -1 for odd primes.

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