A048250 -id:A048250 - cp's OEIS frontend

A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

30, 70, 90, 246, 150, 266, 190, 210, 678, 342, 270, 310, 654, 574, 370, 570, 450, 738, 930, 490, 510, 722, 550, 570, 798, 1582, 690, 1026, 750, 2034, 790, 1230, 1626, 1178, 870, 1526, 910, 970, 990, 2046, 1558, 1406, 1722, 1962, 1150, 1170, 1210, 4062, 1710, 1290, 3390, 1350, 1862, 1390, 1938, 1410, 2214, 1470, 3030

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

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

Cf. A048250, A162296, A228058, A325319, A325320, A325379.

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), k++; print1(A162296(n) - A048250(n),", ")));

a(n) = A162296(A228058(n)) - A048250(A228058(n)).

a(n) = A325319(n) + A325320(n).

A344999 a(n) = A048250(n) * A345001(n).

Original entry on oeis.org

-1, 0, -4, 9, -18, 60, -40, 33, 4, 90, -108, 240, -154, 120, 48, 93, -270, 288, -340, 468, 0, 180, -504, 672, -54, 210, 52, 768, -810, 3096, -928, 237, -192, 270, -480, 948, -1330, 300, -336, 1404, -1638, 5088, -1804, 1584, 648, 360, -2160, 1680, -216, 684, -720, 2100, -2754, 1116, -1584, 2400, -960, 450, -3420, 10080

Offset: 1

Antti Karttunen, Jun 05 2021

sign

Cf. A000203, A003415, A048250, A345001.

Cf. A345003 [gives k for which a(k) = A344998(k)], A345004, A345005.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
A344999(n) = (A048250(n) * A345001(n));

a(n) = A048250(n) * A345001(n).

a(n) = A344998(n) - A345043(n).

A181797 a(n) = n multiplied by the sum of its squarefree divisors (A048250(n)).

Original entry on oeis.org

1, 6, 12, 12, 30, 72, 56, 24, 36, 180, 132, 144, 182, 336, 360, 48, 306, 216, 380, 360, 672, 792, 552, 288, 150, 1092, 108, 672, 870, 2160, 992, 96, 1584, 1836, 1680, 432, 1406, 2280, 2184, 720, 1722, 4032, 1892, 1584, 1080, 3312, 2256, 576, 392, 900

Offset: 1

Matthew Vandermast, Dec 05 2010

Sum of reciprocals converges to Pi^2/6. The natural density of positive integers m such that A003557(m) = n equals 6/(a(n)*Pi^2).
If m is coprime to 6, a(3m) = a(4m).
Apparently the absolute values of the Dirichlet inverse of A000082. - R. J. Mathar, Mar 14 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Natural density

Cf. A181798, A181799.

A181797 := proc(n) local f; f := ifactors(n)[2] ;  mul( op(1,d)^op(2,d)*( op(1,d)+1),d=f) ; end proc: # R. J. Mathar, Dec 05 2010

Table[n*Sum[d*MoebiusMu[d]^2, {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 02 2019 *)

PARI
```
a(n)=n*sumdiv(n,d,d*moebius(d)^2)
```

A181797 = lambda n: n * sum(d for d in divisors(n) if is_squarefree(d)) # D. S. McNeil, Dec 05 2010

a(n) = n*A048250(n). Multiplicative with a(p^e) = (p+1)*p^e.
Dirichlet g.f. zeta(s-1)*zeta(s-2)/zeta(2*s-4). - R. J. Mathar, Mar 14 2011
G.f.: x*f'(x), where f(x) = Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017
Sum_{k=1..n} a(k) ~ n^3 / 3. - Vaclav Kotesovec, Feb 02 2019
Sum_{k>=1} 1/a(k) = Pi^2/6. - Vaclav Kotesovec, Sep 19 2020

A322021 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of A291758, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291758(i) = A291758(j) <=> A046523(i) = A046523(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A304411(i) = A304411(j),
  a(i) = a(j) => A304412(i) = A304412(j).

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

Cf. A046523, A048250, A291751, A291758, A294877, A295300, A322022.

Cf. also A304411, A304412.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
v322021 = rgs_transform(vector(up_to, n, [A046523(n), A048250(n)]));
A322021(n) = v322021[n];

A322318 a(n) = gcd(A003557(n), A048250(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

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

Cf. A001615, A003557, A048250, A322319.

Cf. also A066086, A322320.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A322318(n) = gcd(A048250(n), A003557(n));

a(n) = gcd(A003557(n), A048250(n)).

a(n) = A001615(n) / A322319(n).

A344754 Numbers k such that A344753(k) is a multiple of A048250(k).

Original entry on oeis.org

1, 6, 24, 28, 54, 96, 112, 150, 153, 216, 294, 384, 448, 486, 496, 528, 672, 726, 864, 1014, 1080, 1377, 1500, 1536, 1734, 1792, 1944, 1980, 1984, 2112, 2166, 2250, 2376, 2688, 3174, 3456, 3672, 3750, 4320, 4374, 4560, 4753, 5046, 5292, 5766, 6000, 6048, 6144, 6720, 7168, 7776, 7936, 8128, 8214, 8448, 8700, 8736, 9024

Offset: 1

Antti Karttunen, May 29 2021

nonn

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A001065, A001615, A306927, A048250, A344753.
Subsequences: A000396, A344755.
Cf. also A344700.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA344754(n) = (0==(A344753(n)%A048250(n)));

A351450 a(n) = A064989(A048250(A003961(n))).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 5, 2, 4, 2, 2, 1, 3, 2, 2, 1, 4, 5, 6, 2, 1, 4, 2, 2, 1, 2, 17, 1, 10, 3, 2, 2, 10, 2, 8, 1, 7, 4, 2, 5, 2, 6, 8, 2, 2, 1, 6, 4, 6, 2, 5, 2, 4, 1, 29, 2, 13, 17, 4, 1, 4, 10, 4, 3, 12, 2, 31, 2, 3, 10, 2, 2, 10, 8, 10, 1, 2, 7, 12, 4, 3, 2, 2, 5, 25, 2, 8, 6, 34, 8, 2

Offset: 1

Antti Karttunen, Feb 11 2022

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

Cf. A003961, A048250, A064989, A151800, A351449, A351451 [= a(A055231(n))].

Cf. also A351441.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A351450(n) = A064989(A048250(A003961(n)));

Multiplicative with a(p^e) = A064989(q+1), where q = nextPrime(p) = A151800(p).

a(n) = A064989(A048250(A003961(n))).

A291758 Compound filter (prime signature of n & sum of squarefree divisors of n): a(n) = P(A046523(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 12, 19, 23, 142, 38, 53, 25, 259, 80, 265, 107, 412, 412, 169, 173, 265, 212, 418, 672, 826, 302, 619, 40, 1087, 63, 607, 467, 5080, 530, 593, 1384, 1717, 1384, 1117, 743, 2086, 1836, 844, 905, 7780, 992, 1093, 607, 2932, 1178, 1759, 59, 418, 2932, 1390, 1487, 619, 2932, 1105, 3576, 4471, 1832, 8575, 1955, 5056, 915, 2209, 3922, 14908, 2348, 2092, 5056

Offset: 1

Antti Karttunen, Sep 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000027, A046523, A048250, A291750, A291752, A291757.

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
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291758(n) = (1/2)*(2 + ((A046523(n)+A048250(n))^2) - A046523(n) - 3*A048250(n));

a(n) = (1/2)*(2 + ((A046523(n)+A048250(n))^2) - A046523(n) - 3*A048250(n)).

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))).

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,", ")));

cp's OEIS Frontend

A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

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A344999 a(n) = A048250(n) * A345001(n).

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A181797 a(n) = n multiplied by the sum of its squarefree divisors (A048250(n)).

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A322021 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

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A322318 a(n) = gcd(A003557(n), A048250(n)).

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A344754 Numbers k such that A344753(k) is a multiple of A048250(k).

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A351450 a(n) = A064989(A048250(A003961(n))).

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A291758 Compound filter (prime signature of n & sum of squarefree divisors of n): a(n) = P(A046523(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

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

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