A003959 -id:A003959 - cp's OEIS frontend

A348973 Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

1, 1, 1, 8, 1, 11, 1, 20, 15, 17, 1, 7, 1, 23, 23, 16, 1, 13, 1, 22, 31, 35, 1, 17, 35, 41, 27, 5, 1, 61, 1, 112, 47, 53, 47, 2, 1, 59, 55, 2, 1, 83, 1, 23, 7, 71, 1, 40, 63, 95, 71, 6, 1, 45, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 70, 95, 43, 1, 19, 1, 113, 65, 13, 95, 149, 1, 128, 189, 125, 1, 13, 107

Offset: 1

Antti Karttunen, Nov 06 2021

It is known that A129283(n) <= A003959(n) for all n (see A348970 for a proof), which implies that each ratio a(n)/A348974(n) is at most 1: 1/1, 1/1, 1/1, 8/9, 1/1, 11/12, 1/1, 20/27, 15/16, 17/18, 1/1, 7/9, 1/1, 23/24, 23/24, 16/27, 1/1, 13/16, 1/1, 22/27, 31/32, 35/36, 1/1, 17/27, 35/36, 41/42, 27/32, 5/6, 1/1, 61/72, 1/1, 112/243, etc.

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

Cf. A003415, A003959, A129283, A348970, A348972, A348974 (denominators).

Cf. also A345059.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Numerator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348973(n) = { my(u=n+A003415(n)); (u/gcd(A003959(n),u)); };

a(n) = A129283(n) / A348972(n) = A129283(n) / gcd(A003959(n), A129283(n)).

A348974 Denominator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 9, 1, 12, 1, 27, 16, 18, 1, 9, 1, 24, 24, 27, 1, 16, 1, 27, 32, 36, 1, 27, 36, 42, 32, 6, 1, 72, 1, 243, 48, 54, 48, 3, 1, 60, 56, 3, 1, 96, 1, 27, 8, 72, 1, 81, 64, 108, 72, 7, 1, 64, 72, 54, 80, 90, 1, 27, 1, 96, 64, 729, 84, 144, 1, 81, 96, 48, 1, 36, 1, 114, 72, 15, 96, 168, 1, 243, 256, 126, 1, 18, 108

Offset: 1

Antti Karttunen, Nov 06 2021

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

Cf. A003415, A003959, A129283, A348970, A348972, A348973 (numerators).

Cf. also A343227.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Denominator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348974(n) = { my(s=A003959(n)); (s/gcd(s,(n+A003415(n)))); };

a(n) = A003959(n) / A348972(n) = A003959(n) / gcd(A003959(n), A129283(n)).

A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 2, 6, 6, 18, 24, 72, 12, 36, 2, 6, 8, 24, 4, 12, 18, 54, 72, 216, 36, 108, 2, 6, 8, 24, 4, 12, 8, 24, 32, 96, 16, 48, 48, 144, 192, 576, 96, 288, 16, 48, 64, 192, 32, 96, 144, 432, 576, 1728, 288, 864, 16, 48, 64, 192, 32, 96, 2, 6, 8, 24, 4, 12, 12, 36, 48, 144, 24, 72, 4, 12, 16, 48, 8, 24, 36, 108, 144

Offset: 0

Antti Karttunen, Nov 07 2021

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

Cf. A003959, A034448, A276086, A348733, A348949, A348996.

Cf. also A346471 for similar construction. (Compare the scatter plots).

A348997(n) = { my(m1=1, m2=1, p=2); while(n, if(n%p, m1 *= ((1+p)^(n%p)); m2 *= (1+(p^(n%p)))); n = n\p; p = nextprime(1+p)); gcd(m1, m2); };

a(n) = A348733(A276086(n)) = gcd(A348949(n), A348996(n)).

A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 8, 1, 2, 0, 18, 0, 2, 2, 41, 0, 22, 0, 22, 2, 2, 0, 98, 1, 2, 12, 26, 0, 40, 0, 172, 2, 2, 2, 148, 0, 2, 2, 130, 0, 48, 0, 34, 28, 2, 0, 426, 1, 34, 2, 38, 0, 158, 2, 162, 2, 2, 0, 278, 0, 2, 32, 645, 2, 64, 0, 46, 2, 56, 0, 706, 0, 2, 36, 50, 2, 72, 0, 590, 91, 2, 0, 350, 2, 2, 2, 226, 0, 348

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A322582 with A348507.

Question: Is a(n) >= A305809(n) for all n?

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

Cf. A003958, A003959, A305809, A322582, A348507, A349129.

f1[p_, e_] := (p - 1)^e; s1[1] = 0; s1[n_] := n - Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p + 1)^e; s2[1] = 0; s2[n_] := Times @@ f2 @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, s1[#]*s2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A322582(n) = (n-A003958(n));
A348507(n) = (A003959(n)-n);
A349139(n) = sumdiv(n,d,A322582(d)*A348507(n/d));

a(n) = Sum_{d|n} A322582(d) * A348507(n/d).

A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 3, 4, 7, 15, 31, 40, 63, 127, 255, 511, 639, 1023, 2047, 2175, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. Terms 1, 4 and 40 are probably the only terms that are not in A004767.

Cf. A000225 (subsequence), A000265, A003959, A004767.

For similar sequences, see A336700, A387410, A387411, A387415, A387418.

A000265(n) = (n>>valuation(n,2));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
isA387419(n) = !((1+A000265(A003959(n)))%A000265(1+n));

A348732 a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 18, 6, 0, 0, 16, 0, 0, 0, 64, 0, 18, 0, 24, 0, 0, 0, 72, 10, 0, 36, 32, 0, 0, 0, 210, 0, 0, 0, 94, 0, 0, 0, 108, 0, 0, 0, 48, 36, 0, 0, 256, 14, 30, 0, 56, 0, 108, 0, 144, 0, 0, 0, 96, 0, 0, 48, 664, 0, 0, 0, 72, 0, 0, 0, 342, 0, 0, 40, 80, 0, 0, 0, 384, 174, 0, 0, 128, 0, 0, 0, 216, 0, 108

Offset: 1

Antti Karttunen, Oct 31 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..21125

Cf. A003959, A005117 (positions of zeros), A034448, A034460, A048146, A348029, A348507.

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348732(n) = (A003959(n)-A034448(n));

a(n) = A003959(n) - A034448(n).
a(n) = A348507(n) - A034460(n).
a(n) = A048146(n) + A348029(n).

A348508 a(n) = A003959(n) - 2*n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

-1, -1, -2, 1, -4, 0, -6, 11, -2, -2, -10, 12, -12, -4, -6, 49, -16, 12, -18, 14, -10, -8, -22, 60, -14, -10, 10, 16, -28, 12, -30, 179, -18, -14, -22, 72, -36, -16, -22, 82, -40, 12, -42, 20, 6, -20, -46, 228, -34, 8, -30, 22, -52, 84, -38, 104, -34, -26, -58, 96, -60, -28, 2, 601, -46, 12, -66, 26, -42, 4, -70, 288

Offset: 1

Antti Karttunen, Oct 30 2021

sign

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

Cf. A003959, A033879, A138636, A168036, A252748, A348029, A348970.

f[p_, e_] := (p + 1)^e; a[1] = -1; a[n_] := Times @@ f @@@ FactorInteger[n] - 2*n; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348508(n) = (A003959(n) - 2*n);

a(n) = A003959(n) - 2*n.
a(n) = A348507(n) - n.
a(n) = A348029(n) - A033879(n).
From Antti Karttunen, Dec 05 2021: (Start)
a(n) = A168036(n) + A348970(n).
For all n >= 1, a(A138636(n)) = 12.
(End)
a(p) = 1 - p if p prime. - Bernard Schott, Feb 17 2022

A348512 a(n) = A003959(sigma(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 4, 9, 8, 12, 36, 27, 24, 14, 48, 36, 72, 24, 108, 108, 32, 48, 56, 54, 96, 243, 144, 108, 216, 32, 96, 162, 216, 72, 432, 243, 128, 324, 192, 324, 112, 60, 216, 216, 288, 96, 972, 108, 288, 168, 432, 324, 288, 80, 128, 432, 192, 192, 648, 432, 648, 486, 288, 216, 864, 96, 972, 378, 128, 288, 1296, 162, 384, 972, 1296

Offset: 1

Antti Karttunen, Oct 29 2021

Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A051027.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348512(n) = A003959(sigma(n));

Multiplicative with a(p^e) = A003959(1 + p + ... + p^e).

a(n) = A003959(A000203(n)).

A003969 Inverse Möbius transform of A003959.

Original entry on oeis.org

1, 4, 5, 13, 7, 20, 9, 40, 21, 28, 13, 65, 15, 36, 35, 121, 19, 84, 21, 91, 45, 52, 25, 200, 43, 60, 85, 117, 31, 140, 33, 364, 65, 76, 63, 273, 39, 84, 75, 280, 43, 180, 45, 169, 147, 100, 49, 605, 73, 172, 95, 195, 55, 340, 91, 360, 105, 124, 61, 455, 63, 132, 189

Offset: 1

Marc LeBrun

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

Cf. A003959, A005596, A072691.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, ((f[i,1] + 1)^(f[i,2] + 1) - 1)/f[i,1]); } \\ Amiram Eldar, Oct 23 2022

Multiplicative with a(p^e) = ((p+1)^(e+1)-1)/p. - David W. Wilson, Sep 01 2001

Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691/A005596 = 2.199369... . - Amiram Eldar, Oct 23 2022

More terms from David W. Wilson, Aug 29 2001

A387710 Numbers k for which A003959(k) < 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 117, 118, 119, 121, 122, 123, 125

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959, A387711.
Subsequence of A005100.
Subsequences: A000040, A001358\{4, 6}, A246281.
Positions of 0's in A387715.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387710(n) = (A003959(n)<(2*n));

cp's OEIS Frontend

A348973 Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

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A348974 Denominator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

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A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

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A349139 a(n) = Sum_{d|n} A322582(d) * A348507(n/d), where A322582(n) = n - A003958(n) and A348507(n) = A003959(n) - n.

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A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A348732 a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

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A348508 a(n) = A003959(n) - 2*n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A348512 a(n) = A003959(sigma(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

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A003969 Inverse Möbius transform of A003959.