cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Antti Karttunen, Jun 04 2019

Keywords

Comments

No other terms below 536870912 (2^29).
a(6) > 10^12, if it exists. - Giovanni Resta, Jun 07 2019

Crossrefs

Cf. A034448, A048250, A034460, A325313.
Positions of zeros in A325977.

Programs

  • PARI
    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)));

A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 13 2019

Keywords

Comments

For all i, j:
A295300(i) = A295300(j) => a(i) = a(j),
A305801(i) = A305801(j) => a(i) = a(j),
a(i) = a(j) => A294877(i) = A294877(j).

Links

Crossrefs

Cf. A003557, A046523, A048250, A294877, A295300, A305801.
Differs from A323401 for the first time at n = 382 where a(382) = 253, while A323401(382) = 140.

Programs

  • PARI
    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; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
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)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux326199(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v326199 = rgs_transform(vector(up_to,n,Aux326199(n)));
A326199(n) = v326199[n];

A342459 a(n) = gcd(A048250(n), A342001(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 1, 3, 8, 1, 1, 1, 1, 6, 2, 1, 1, 1, 2, 3, 1, 8, 1, 1, 1, 1, 2, 1, 12, 2, 1, 3, 8, 1, 1, 1, 1, 12, 1, 1, 1, 2, 2, 9, 4, 14, 1, 3, 8, 1, 2, 1, 1, 2, 1, 3, 1, 3, 6, 1, 1, 18, 2, 1, 1, 1, 1, 3, 1, 20, 6, 1, 1, 2, 4, 1, 1, 2, 2, 3, 8, 1, 1, 1, 4, 24, 2, 1, 24, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 28 2021

Keywords

Links

Crossrefs

Cf. A001615, A003415, A003557, A048250, A342001, A342458, A342919.
Cf. also A066086, A342416.

Programs

Formula

a(n) = A342458(n) / A003557(n) = gcd(A048250(n), A342001(n)).
a(n) = A342001(n) / A342919(n).

A345004 Numbers k such that A345001(k)*A048250(k) is equal to A342001(k)*A344753(k) and A345001(n)/A342001(n) is an integer.

Original entry on oeis.org

6, 28, 496, 5292, 8128, 33550336
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2021

Keywords

Comments

Equally, we may require that A344753(k)/A048250(k) is an integer, thus this is a subsequence of A344754.

Links

Crossrefs

Cf. A000396 (subsequence), A003415, A003557, A048250, A342001, A344753, A345001.
Intersection of A344754 and A345003.

Programs

A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 639, 1023, 2047, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 126975, 131071, 204799, 229375, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095
Offset: 1

Views

Author

Antti Karttunen, Sep 01 2025

Keywords

Comments

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Crossrefs

Cf. A000225 (subsequence), A000265, A004767, A048250.
For similar sequences, see A336700, A387411, A387415, A387418, A387419.

Programs

A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

Original entry on oeis.org

15265, 27962, 30217, 30530, 45795, 50541, 54379, 54905, 57598, 60434, 61060, 64255, 66526, 72357, 72713, 89585, 90651, 91590, 101082, 101949, 108758, 109810, 120868, 122120, 128510, 136555, 137385, 137883, 138761, 144714, 145426, 149739, 151085, 152633, 161386, 163137, 164715, 166315, 179170, 181302, 181543, 182942
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2017

Keywords

Comments

Because Euler phi(n) = A000010(n) = A003557(n) * A173557(n), Dedekind psi(n) = A001615(n) = A003557(n) * A048250(n), and because also sigma(n) (A000203) can be computed from those three elements (see A291750), these numbers form also a subset of the positions of such duplicated occurrences of values computed for those functions. See for example A069822 and A296214.
a(11) = 61060 is the first term that is not squarefree.

Examples

			15265 is a term because A003557(15265) = 1 = A003557(15169), A048250(15265) = 19008 = A048250(15169), A173557(15265) = 11760 = A173557(15169).
27962 is a term because A003557(27962) = 1 = A003557(26355), A048250(27962) = 48384 = A048250(26355), A173557(27962) = 12000 = A173557(26355).

Links

Crossrefs

Cf. A000010, A001615, A003557, A048250, A173557.
Subsequence of A069822 and of A296214.

Programs

  • PARI
    search_up_to = (2^23);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted1(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
Akaikki3(n) = (1/2)*(2 + ((A048250(n)+Anotsubmitted1(n))^2) - A048250(n) - 3*Anotsubmitted1(n));
om = Map(); m = 0; i=0; for(n = 1, search_up_to, k = Akaikki3(n); if(!mapisdefined(om,k), mapput(om,k,n), i++; write("b296087.txt", i, " ", n)));

A322319 a(n) = lcm(A003557(n), A048250(n)).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 12, 14, 24, 24, 24, 18, 12, 20, 18, 32, 36, 24, 12, 30, 42, 36, 24, 30, 72, 32, 48, 48, 54, 48, 12, 38, 60, 56, 36, 42, 96, 44, 36, 24, 72, 48, 24, 56, 90, 72, 42, 54, 36, 72, 24, 80, 90, 60, 72, 62, 96, 96, 96, 84, 144, 68, 54, 96, 144, 72, 12, 74, 114, 120, 60, 96, 168, 80, 72
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A001615, A003557, A048250, A322318.
Cf. also A322321, A322359.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    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]));
A322319(n) = lcm(A048250(n), A003557(n));

Formula

a(n) = lcm(A003557(n), A048250(n)).
a(n) = A001615(n) / A322318(n).

A323159 Greatest common divisor of product (1+(p^e)) and product (1+p), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A048250(n)).

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 2, 18, 12, 4, 14, 24, 24, 1, 18, 6, 20, 6, 32, 36, 24, 12, 2, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 2, 38, 60, 56, 18, 42, 96, 44, 12, 12, 72, 48, 4, 2, 6, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 16, 1, 84, 144, 68, 18, 96, 144, 72, 6, 74, 114, 8, 20, 96, 168, 80, 6, 2, 126, 84, 32, 108, 132, 120, 36, 90, 36
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2019

Keywords

Links

Crossrefs

Cf. A034448, A048250, A323160, A323163.

Programs

Formula

a(n) = gcd(A034448(n), A048250(n)).

A323160 a(n) = gcd(n, A323159(n)) = gcd(n, A034448(n), A048250(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 2, 3, 2, 1, 6, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2019

Keywords

Links

Crossrefs

Cf. A034448, A048250, A323159, A323163, A323166.

Programs

  • PARI
    A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A323160(n) = gcd(n, gcd(A034448(n), A048250(n)));

Formula

a(n) = gcd(n, A323159(n)) = gcd(A048250(n), A323166(n)).
a(n) = gcd(n, A034448(n), A048250(n)).

A325316 a(n) = A048250(n) OR A162296(n), where OR is the bitwise-OR, A003986.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 31, 20, 26, 32, 36, 24, 60, 31, 42, 36, 56, 30, 72, 32, 63, 48, 54, 48, 79, 38, 60, 56, 90, 42, 96, 44, 52, 62, 72, 48, 124, 57, 91, 72, 58, 54, 108, 72, 120, 80, 90, 60, 104, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 191, 74, 114, 124, 124, 96, 168, 80
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A003986, A048250, A162296, A325317, A325318.

Programs

  • Mathematica
    Array[BitOr @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 79] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325316(n) = bitor(A048250(n),A162296(n));

Formula

a(n) = A003986(A048250(n), A162296(n)).
a(n) = A000203(n) - A325318(n) = A325317(n) + A325318(n).
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