A325313 -id:A325313 - cp's OEIS frontend

A325979 Odd numbers k for which gcd(A325977(k), A325978(k)) is equal to abs(A325978(k)).

1, 3465, 72981, 78651, 80937, 152703, 199341, 201771, 241605, 253287, 492507, 631881, 880821, 933147, 985473, 1063755, 1209285, 1244133, 1292445, 1313235, 1327095, 1347885, 1360881, 1451835, 1521135, 1597365, 1620375, 1814373, 2015475, 2664585, 6058233, 6676371, 8186751, 11119761, 17496243, 18379935, 28695627

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977(k) and A325978(k) are never zero for the same k, these are odd numbers k such that A325978(k) is not zero and divides A325977(k).

Of the first 281 terms, only a(5) = 80937, a(51) = 86086881, a(175) = 43024468437, and a(262) = 564858541521 are in A228058. - Updated Jul 20 2025

Giovanni Resta, Table of n, a(n) for n = 1..281 (terms < 10^12; first 65 terms from Antti Karttunen)
Index entries for sequences where any odd perfect numbers must occur.

Cf. A228058, A325975, A325977, A325978, A325981.

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);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
isA325979(n) = ((n%2)&&(A325975(n)==abs(A325978(n))));
\\ Or alternatively as:
isA325979(n) = if(!(n%2),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(y&&!(x%y)));

A325315 Bitwise-XOR of absolute values of (n - A048250(n)) and (n - A162296(n)).

Original entry on oeis.org

1, 3, 2, 1, 4, 0, 6, 1, 5, 2, 10, 4, 12, 4, 6, 1, 16, 15, 18, 6, 30, 24, 22, 20, 19, 10, 30, 0, 28, 52, 30, 1, 46, 54, 46, 51, 36, 48, 54, 54, 40, 28, 42, 12, 28, 52, 46, 100, 41, 57, 38, 14, 52, 28, 38, 8, 46, 26, 58, 40, 60, 28, 22, 1, 82, 12, 66, 10, 94, 12, 70, 83, 72, 98, 42, 20, 94, 20, 78, 102, 105, 126, 82, 32, 66, 120, 118, 12

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

Cf. A000396, A003987, A028982 (positions of odd terms), A048250, A162296, A228058, A325310, A325313, A325314.

Array[BitXor @@ Abs[#1 - Map[Total, {#3, Complement[#2, #3]}]] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 88] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325315(n) = bitxor(abs(A325313(n)),abs(A325314(n)));

a(n) = A003987(abs(A325313(n)), abs(A325314(n))).

A325375 a(n) = gcd(A325319(n), A325320(n)).

Original entry on oeis.org

3, 1, 9, 1, 3, 1, 1, 3, 3, 1, 9, 1, 1, 7, 1, 5, 9, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 9, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 9, 1, 3, 5, 3, 3, 9, 1, 1, 1, 3, 1, 3, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 3, 3, 1, 25, 1, 1, 9, 1, 1, 9, 1, 3, 1, 27, 1, 1, 1, 1, 3, 9, 1, 49, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A228058, A325319, A325320, A325376.

up_to = 25000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n];
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325319(n) = -A325313(A228058(n));
A325320(n) = -A325314(A228058(n));
A325375(n) = gcd(A325319(n),A325320(n));

a(n) = gcd(A325319(n), A325320(n)).

A325376 Terms k of A228058 such that gcd(k - A048250(k), A162296(k) - k) = A162296(k) - k.

Original entry on oeis.org

153, 477, 801, 1773, 2097, 2421, 3725, 4041, 4689, 4753, 5013, 5337, 6309, 6957, 7281, 7929, 8577, 8725, 9549, 9873, 11225, 11493, 13437, 14357, 14409, 14733, 15381, 17001, 17973, 18621, 19269, 19917, 21213, 21537, 23481, 24777, 25101, 25749, 26073, 26225, 26721, 27369, 28989, 29161, 29313, 29961, 31225, 32229, 32553, 33849

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

Also, terms of this sequence are A228058(k) for all such k that A325375(k) = A325320(k).
In range 1 .. 2^27 there are no such terms k of A228058 that gcd(k-A048250(k), A162296(k)-k) = k - A048250(k).
If any odd perfect number exists, then it must occur in this sequence, but should also satisfy the other condition given above.

Antti Karttunen, Table of n, a(n) for n = 1..12532
Index entries for sequences where any odd perfect numbers must occur

Cf. A048250, A162296, A228058, A325313, A325314, A325320, A325375, A325822.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n) && (gcd(n-A048250(n),A162296(n)-n) == A162296(n)-n),k++; print1(n,", ")));

A325963 Numbers n for which A034448(n)-n is equal to n-A048250(n).

Original entry on oeis.org

1, 4, 24, 240, 349440

Offset: 1

Antti Karttunen, Jun 04 2019

No other terms below 536870912 (2^29).

a(6) > 10^12, if it exists. - Giovanni Resta, Jun 07 2019

Cf. A034448, A048250, A034460, A325313.

Positions of zeros in A325977.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
isA325963(n) = ((A034448(n)-n) == (n-A048250(n)));

A325310 a(n) = A001511(A325315(n)), except when A325315(n) = 0, then a(n) = 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 2, 1, 1, 2, 2, 3, 3, 3, 2, 1, 5, 1, 2, 2, 2, 4, 2, 3, 1, 2, 2, 0, 3, 3, 2, 1, 2, 2, 2, 1, 3, 5, 2, 2, 4, 3, 2, 3, 3, 3, 2, 3, 1, 1, 2, 2, 3, 3, 2, 4, 2, 2, 2, 4, 3, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 1, 4, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 6, 2, 4, 2, 3, 4, 2, 2, 5, 2, 3, 2, 3, 6, 1, 2, 1, 3, 3, 2, 2, 2

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

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

Cf. A000396, A001511, A028982 (gives the positions of 1's), A048250, A162296, A228058, A325313, A325314, A325315, A325378, A325379.

Array[If[# == 0, 0, IntegerExponent[2 #, 2]] &[BitXor @@ Abs[#1 - Map[Total, {#3, Complement[#2, #3]}]]] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 105] (* Michael De Vlieger, Apr 21 2019 *)

A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325315(n) = bitxor(abs(A325313(n)),abs(A325314(n)));
A325310(n) = A001511ext(A325315(n));

If A325315(n) = 0, then a(n) = 0, otherwise a(n) = A001511(A325315(n)).

a(A228058(n)) = A001511(abs(A325379(n))), assuming there are no odd perfect numbers, in which case a(A228058(n)) >= 3 for all n.

cp's OEIS Frontend

A325979 Odd numbers k for which gcd(A325977(k), A325978(k)) is equal to abs(A325978(k)).

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A325315 Bitwise-XOR of absolute values of (n - A048250(n)) and (n - A162296(n)).

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A325375 a(n) = gcd(A325319(n), A325320(n)).

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A325376 Terms k of A228058 such that gcd(k - A048250(k), A162296(k) - k) = A162296(k) - k.

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A325963 Numbers n for which A034448(n)-n is equal to n-A048250(n).

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A325310 a(n) = A001511(A325315(n)), except when A325315(n) = 0, then a(n) = 0.

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