A325314 -id:A325314 - 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)));

A357493 Numbers k such that s(k) = 3*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

480, 2688, 56304, 89400, 195216, 2095104, 9724032, 69441408, 1839272960, 5905219584

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 3-perfect numbers (A005820) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -2*k.
a(11) > 10^11, if it exists.
If k is one of the 6 known 3-perfect numbers, then 4*k is a term.

			480 is a term since A162296(480) = 1440 = 3*480.

Cf. A162296, A325314.
Subsequence of A013929 and A068403.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), this sequence (m=3), A357494 (m=4).
Similar sequence: A005820.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 3*n]; Select[Range[2, 10^7], q]

A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

902880, 1534680, 361674720, 767685600, 4530770640, 4941414720, 5405788800, 5517818880, 16993944000, 20429240832, 94820077440

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 4-perfect numbers (A027687) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -3*k.
There are no numbers k below 10^11 such that A162296(k) = m*k for integers m > 4.

			902880 is a term since A162296(902880) = 3611520 = 4*902880.

Cf. A162296, A325314.
Subsequence of A013929 and A023198.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), A357493 (m=3), this sequence (m=4).
Similar sequence: A027687.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 4*n]; Select[Range[2, 2*10^6], q]

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,", ")));

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.

A357495 Lesser of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

880, 10480, 20080, 24928, 42976, 69184, 110565, 252080, 267712, 489472, 566656, 569240, 603855, 626535, 631708, 687424, 705088, 741472, 786896, 904365, 1100385, 1234480, 1280790, 1425632, 1749824, 1993750, 2012224, 2401568, 2439712, 2496736, 2542496, 2573344, 2671856

Offset: 1

Amiram Eldar, Oct 01 2022

nonn

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The larger counterparts are in A357496.
Both members of each pair are necessarily nonsquarefree numbers.

			880 is a term since s(880) = 1136 and s(1136) = 880.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A162296, A325314, A322609, A357493, A357494, A357496.
Subsequence of A013929.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 2, 3*10^6}]; seq

A357496 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

1136, 11696, 22256, 25472, 43424, 73664, 131355, 304336, 267968, 492608, 612704, 674920, 640305, 788697, 691292, 705344, 723392, 813728, 809776, 1117395, 1258335, 1559696, 1518570, 1598368, 1821376, 2218250, 2058944, 2678752, 2744288, 2765024, 2848864, 2610656, 3134224

Offset: 1

Amiram Eldar, Oct 01 2022

nonn

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The terms are ordered according to their lesser counterparts (A357495).
Both members of each pair are necessarily nonsquarefree numbers.

			1136 is a term since s(1136) = 880 and s(880) = 1136.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A162296, A325314, A322609, A357493, A357494, A357495.
Subsequence of A013929.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 2, 3*10^6}]; seq

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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A357493 Numbers k such that s(k) = 3*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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

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A357495 Lesser of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

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A357496 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

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