A003958 -id:A003958 - cp's OEIS frontend

A003966 Möbius transform of A003958.

1, 0, 1, 0, 3, 0, 5, 0, 2, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 12, 0, 4, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 6, 0, 45, 0, 30, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 10, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 12, 0, 45, 0, 77, 0, 8, 0, 81, 0, 45, 0, 27, 0, 87, 0, 55, 0, 29, 0, 51, 0, 95, 0, 18

Offset: 1

Marc LeBrun

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A003958, A276833.

A003966 := proc(n) option remember; local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then p := op(1,pf) ; (op(1,p)-2)*(op(1,p)-1)^(op(2,p)-1)  ; else mul(procname(op(1,p)^op(2,p)),p=pf) ; end if; end if; end proc:
seq(A003966(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

f[p_, e_] := (p - 2) (p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2022 *)

a(n) = {my(f=factor(n)); for (i=1, #f~, p = f[i, 1]; f[i, 1] = (p-2)*(p-1)^(f[i,2]-1); f[i, 2] = 1); factorback(f);} \\ Michel Marcus, Feb 27 2015

A003958(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1]--); factorback(f);
A003966(n) = sumdiv(n,d,moebius(n/d)*A003958(d)); \\ Antti Karttunen, Oct 24 2018

for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X+X)*(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

Multiplicative with a(p^e) = (p-2)(p-1)^(e-1). - David W. Wilson, Sep 01 2001
Dirichlet inverse b(n) is multiplicative with b(p^e) = 2-p for prime p and e > 0 (A276833). - Werner Schulte, Oct 25 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/(105*zeta(3)) = 0.1563923... . - Amiram Eldar, Oct 23 2022
From Vaclav Kotesovec, Feb 11 2023: (Start)
Dirichlet g.f.: 1/zeta(s) * Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + (p^(1-s)-2) / (1 - p + p^s)), (with a product that converges for s=2). (End)

More terms from Antti Karttunen, Oct 24 2018

A064522 For an integer n with prime factorization p1p2p3* ... pn let n = (p1-1)(p2-1)(p3-1)* ... (pn-1) (A003958); sequence gives n such that n is divisible by n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 36, 40, 48, 64, 72, 80, 84, 96, 120, 128, 144, 160, 168, 192, 216, 240, 256, 272, 288, 312, 320, 336, 384, 400, 432, 440, 480, 504, 512, 544, 576, 624, 640, 672, 720, 768, 800, 864, 880, 960, 1008, 1024, 1088, 1152, 1248, 1280

Offset: 1

Vladeta Jovovic, Oct 07 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A064476, A003958.

nsm(n)= { local(f,p=1); f=factor(n); for(i=1, matsize(f)[1], p*=(f[i, 1] - 1)^f[i, 2]); return(p) } { n=0; for (m=1, 10^9, if (m%nsm(m) == 0, write("b064522.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 17 2009

A348998 a(n) = A348928(A276086(n)), where A348928(n) = gcd(n, A003958(n)), and A003958 is multiplicative with a(p^e) = (p-1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18

Offset: 0

Antti Karttunen, Nov 07 2021

After each primorial number (A002110), the apparent periodicity grows more complex.

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A002110, A003958, A276086, A348928, A348999.

A348998(n) = { my(m1=1, m2=1, p=2); while(n, m1 *= (p^(n%p)); m2 *= ((p-1)^(n%p)); n = n\p; p = nextprime(1+p)); gcd(m1,m2); };

A353795 Numbers k such that k divides A353794(k), where A353794(n) = A003958(A003973(n)) * A064989(A003973(n)).

Original entry on oeis.org

1, 4, 12, 24, 36, 44, 72, 96, 112, 132, 180, 220, 360, 384, 396, 400, 480, 560, 660, 784, 832, 864, 1044, 1056, 1188, 1200, 1344, 1920, 1980, 2088, 2352, 2376, 2496, 2800, 3168, 3600, 3920, 4320, 4736, 5220, 5280, 5376, 5824, 5940, 6800, 6912, 7056, 7200, 7488, 8400, 8800, 9504, 9900, 10000, 10440, 10800, 11484

Offset: 1

Antti Karttunen, May 12 2022

nonn

Of 2608 initial terms, only 188 are not in A353796. The first three of these are: 400, 784, 832.

Cf. A000010, A003958, A003961, A003973, A064989, A353794.

Cf. also A353796.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353794(n) = { my(s=sigma(A003961(n))); (A003958(s)*A064989(s)); };
isA353795(n) = !(A353794(n)%n);

A003967 Inverse Möbius transform of A003958.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 4, 7, 10, 11, 9, 13, 14, 15, 5, 17, 14, 19, 15, 21, 22, 23, 12, 21, 26, 15, 21, 29, 30, 31, 6, 33, 34, 35, 21, 37, 38, 39, 20, 41, 42, 43, 33, 35, 46, 47, 15, 43, 42, 51, 39, 53, 30, 55, 28, 57, 58, 59, 45, 61, 62, 49, 7, 65, 66, 67, 51, 69, 70, 71, 28, 73

Offset: 1

Marc LeBrun

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

Cf. A003958, A341635 (Dirichlet inverse).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003967(n) = sumdiv(n,d,A003958(d)); \\ Antti Karttunen, Feb 11 2022

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)/(1 - p*X + X))[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2022

Multiplicative with a(p^e) = e+1 if p = 2; ((p-1)^(e+1)-1)/(p-2) if p > 2. - David W. Wilson, Sep 01 2001
Dirichlet g.f.: zeta(s) * Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)). - Ilya Gutkovskiy, Feb 11 2022
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / (1890 * zeta(3)). - Vaclav Kotesovec, Feb 11 2022

More terms from David W. Wilson, Aug 29 2001

A324044 a(n) = A003958(n) - A033879(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, -1, 2, 0, 6, 0, 2, 2, 0, 0, 7, 0, 6, 2, 2, 0, 14, -3, 2, -6, 6, 0, 20, 0, 0, 2, 2, 2, 23, 0, 2, 2, 14, 0, 24, 0, 6, 4, 2, 0, 30, -5, 9, 2, 6, 0, 20, 2, 14, 2, 2, 0, 56, 0, 2, 2, 0, 2, 32, 0, 6, 2, 28, 0, 55, 0, 2, 6, 6, 2, 36, 0, 30, -25, 2, 0, 68, 2, 2, 2, 14, 0, 70, 2, 6, 2, 2, 2, 62, 0, 11, -2, 33, 0, 44, 0, 14, 30

Offset: 1

Antti Karttunen, Feb 13 2019

sign

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

Cf. A003958, A033879.

Cf. also A319687, A323911.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A033879(n) = (2*n-sigma(n));
A324044(n) = (A003958(n)-A033879(n));

a(n) = A003958(n) - A033879(n).

A348975 a(n) = A003415(n) + A003958(n) - n, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 5, 1, 1, 0, 6, 0, 1, 1, 17, 0, 7, 0, 8, 1, 1, 0, 22, 1, 1, 8, 10, 0, 9, 0, 49, 1, 1, 1, 28, 0, 1, 1, 32, 0, 11, 0, 14, 10, 1, 0, 66, 1, 11, 1, 16, 0, 35, 1, 42, 1, 1, 0, 40, 0, 1, 12, 129, 1, 15, 0, 20, 1, 13, 0, 88, 0, 1, 12, 22, 1, 17, 0, 100, 43, 1, 0, 52, 1, 1, 1, 62, 0, 49, 1, 26, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 09 2021

No negative terms. See comments in A322582.

This is the difference between the arithmetic derivative of n [= A003415(n)] and its guaranteed lower bound A322582(n) [= n - A003958(n)].

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A003958, A168036, A322582.

Cf. also A348970 for the corresponding difference from a guaranteed upper bound.

MapAt[# + 1 &, Array[If[# < 2, 0, # Total[#2/#1 & @@@ #2]] + Times @@ Map[(#1 - 1)^#2 & @@ # &, #2] - #1 & @@ {#, FactorInteger[#]} &, 95], 1] (* Michael De Vlieger, Mar 15 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348975(n) = (A003415(n) - A322582(n));

a(n) = A003415(n) - A322582(n).

a(n) = A003958(n) + A168036(n).

A351448 Odd numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

8181, 400869, 1507005, 3918213, 11151837, 22002273, 26669007, 47319957, 58170393, 73843245, 75825981, 83488077, 94338513, 108277641, 119656197, 126889821, 137740257, 163057941, 184758813, 191992437, 199226061, 202842873, 204768225, 220926933, 228160557, 258457473, 264328677, 277602471, 300496797

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Odd numbers k such that A351442(k) = 2*A003958(k).
Any hypothetical odd term of A005820, if such a term exists, should appear in this sequence, in A347391, and in A016754 (odd squares).
None of the first 33 terms is a square, and all of them except 75825981 and 204768225 are multiples of 81. Note that 81 is one of the terms of A008848 (and of A231484), squares whose sum of divisors is also square (with A000203(81) = 121).

Index entries for sequences related to sigma(n)

Odd terms in A351447.

Cf. A000203, A003958, A005820, A008848, A016754, A231484, A339905, A347391, A351442.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351448(n) = (n%2 && (A003958(sigma(n)) == 2*A003958(n)));

A353634 Nondeficient numbers k such that phi(k) = phi(sigma(k)) and A003958(k) = A003958(sigma(k)).

Original entry on oeis.org

234728, 280904, 461168, 463112, 604136, 742664, 909872, 996008, 1065896, 1191944, 1204424, 1224392, 1465256, 1527656, 1620008, 1757288, 1758536, 1956848, 1985672, 2081768, 2102984, 2358824, 2376296, 2405552, 2518568, 2543528, 2589704, 2670824, 2820584, 2899208, 2912936, 3014024, 3151304, 3196232, 3374696, 3432104

Offset: 1

Antti Karttunen, May 04 2022

nonn

Intersection of A023196 and A353635.

Cf. A000010, A000203, A003958, A006872, A062401, A351440, A351442, A351446.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA353634(n) = { my(s=sigma(n)); if(s<(2*n),return(0)); ((eulerphi(s)==eulerphi(n)) && (A003958(s)==A003958(n))); };

A353635 Numbers k such that phi(k) = phi(sigma(k)) and A003958(k) = A003958(sigma(k)).

Original entry on oeis.org

1, 26, 74, 122, 146, 314, 386, 554, 626, 794, 842, 914, 1082, 1226, 1322, 1346, 1466, 1514, 1754, 1994, 2186, 2306, 2402, 2426, 2474, 2642, 2762, 2906, 3242, 3314, 3506, 3746, 3866, 3986, 4034, 4274, 4682, 4946, 5114, 5186, 5594, 5714, 5834, 6122, 6434, 6506, 6626, 7034, 7466, 8042, 8114, 8354, 8522, 8546, 8714, 8882

Offset: 1

Antti Karttunen, May 04 2022

nonn

Question 1: Are there any odd terms after the initial 1?

Interestingly, most of the terms seem to belong to a set where the abundancy index (ratio sigma(n)/n) converges towards 3/2. But there are exceptions, see A353634 for example.

Intersection of A006872 and A351446. A353634 lists the nondeficient terms.

Cf. A000010, A000203, A003958, A062401, A351440, A351442.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA353635(n) = { my(s=sigma(n)); ((eulerphi(s)==eulerphi(n)) && (A003958(s)==A003958(n))); };

cp's OEIS Frontend

A003966 Möbius transform of A003958.

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A064522 For an integer n with prime factorization p1*p2*p3* ... *pn let n* = (p1-1)*(p2-1)*(p3-1)* ... *(pn-1) (A003958); sequence gives n such that n is divisible by n*.

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A348998 a(n) = A348928(A276086(n)), where A348928(n) = gcd(n, A003958(n)), and A003958 is multiplicative with a(p^e) = (p-1)^e, and A276086 gives the prime product form of primorial base expansion of n.

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A353795 Numbers k such that k divides A353794(k), where A353794(n) = A003958(A003973(n)) * A064989(A003973(n)).

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A003967 Inverse Möbius transform of A003958.

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A324044 a(n) = A003958(n) - A033879(n).

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A348975 a(n) = A003415(n) + A003958(n) - n, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).

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A351448 Odd numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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A064522 For an integer n with prime factorization p1p2p3* ... pn let n = (p1-1)(p2-1)(p3-1)* ... (pn-1) (A003958); sequence gives n such that n is divisible by n.