A336839 -id:A336839 - cp's OEIS frontend

A336918 Numbers k such that A000005(k) divides A003973(k); numbers k for which A336839(k) = 1.

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 101

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Numbers k such that A003961(k) is in A003601. Numbers which become (or stay as) arithmetic numbers when all primes in their prime factorization are replaced by the next larger primes.

Numbers k for which A003973(k) is equal to A000005(k)*A336838(k).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Wikipedia, Arithmetic number
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A003601, A003961, A003973, A336838, A336930.
Positions of ones in A336839.
Cf. A336919 (complement).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA336918(n) = !(sigma(A003961(n))%numdiv(n));

A336919 Numbers k such that A000005(k) does not divide A003973(k); numbers k for which A336839(k) > 1.

Original entry on oeis.org

4, 9, 16, 18, 20, 32, 36, 44, 45, 48, 49, 64, 68, 72, 80, 81, 90, 98, 99, 100, 112, 116, 124, 144, 153, 160, 162, 164, 169, 176, 180, 192, 196, 198, 208, 220, 225, 236, 240, 244, 245, 252, 256, 261, 279, 284, 288, 292, 304, 306, 320, 324, 336, 338, 340, 352, 356, 360, 361, 369, 392, 396, 400, 404, 405, 428, 432, 441

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Numbers k such that A003961(k) is not in A003601, but in A049642.

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

Cf. A000005, A003601, A003961, A003973, A049642, A336839.

Cf. A336918 (complement).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA336919(n) = !!(sigma(A003961(n))%numdiv(n));

A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).

The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));

A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

Original entry on oeis.org

0, 2, 4, 14, 6, 36, 10, 68, 44, 52, 12, 192, 16, 84, 92, 284, 18, 326, 22, 274, 148, 100, 28, 840, 90, 132, 344, 438, 30, 648, 36, 1094, 176, 148, 212, 1622, 40, 180, 232, 1192, 42, 1032, 46, 520, 802, 228, 52, 3324, 230, 654, 260, 684, 58, 2376, 252, 1896, 316, 244, 60, 3156, 66, 292, 1278, 4010, 332, 1224, 70, 766

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

All terms are even because A003973 and A336845 match parity-wise. Also in the sum formulas, only even terms are summed (only one of which is zero).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A003961, A048673, A094471, A336837, A336839, A336840, A336845, A336856.
Cf. A336846 [= gcd(a(n), A003973(n))].
Twice the terms of A336854.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));

A336841(n) = sumdiv(n,d,(A003961(n)-A003961(d)));

A336841(n) = sumdiv(A003961(n),d,(A003961(n)-d));

a(n) = A336845(n) - A003973(n) = (A000005(n)*A003961(n)) - A000203(A003961(n)).
a(n) = A094471(A003961(n)).
a(n) = Sum_{d|n} (A003961(n)-A003961(d)) = Sum_{d|A003961(n)} (A003961(n)-d).
a(n) = 2*A336854(n) = 2*Sum_{d|n} (A048673(n)-A048673(d)).
a(n) = ((A003961(n)+1)*A000005(n)) - 2*A336840(n).
a(n) = 2 * ((A000005(n)*A048673(n)) - A336840(n)).
a(n) = A000005(n) * (A336837(n)/A336839(n)) = A336837(n) * A336856(n).

A336838 Numerator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 2, 3, 13, 4, 6, 6, 10, 31, 8, 7, 13, 9, 12, 12, 121, 10, 62, 12, 52, 18, 14, 15, 30, 19, 18, 39, 26, 16, 24, 19, 182, 21, 20, 24, 403, 21, 24, 27, 40, 22, 36, 24, 91, 124, 30, 27, 363, 133, 38, 30, 39, 30, 78, 28, 60, 36, 32, 31, 52, 34, 38, 62, 1093, 36, 42, 36, 130, 45, 48, 37, 310, 40, 42, 57, 52, 42, 54, 42

Offset: 1

Antti Karttunen, Aug 07 2020

Ratio r(n) = a(n)/A336839(n) is multiplicative. For example r(3) = 3/1, r(4) = 13/3, thus r(12) = r(3)*r(4) = 13/1.

Conjecture: For all primes p with an odd exponent e, a(p^e) is a multiple of A048673(p). Note that q+1 is a divisor of (q+1)^e - sigma(q^e) = (q+1)^e - (1 + q + q^2 + ... + q^e) when e is odd, thus also A048673(p) = (q+1)/2 is, where q = A003961(p), thus the conjecture holds, unless the denominator (A336839) has enough prime factors of A048673(p).

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A003961, A003973, A048673, A057020, A336840, A336918.

Cf. A336839 (denominators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336838(n) = numerator(sigma(A003961(n))/numdiv(n));

a(n) = A057020(A003961(n)).
a(n) = numerator(A003973(n)/A000005(n)).
a(n) = A003973(n) / A336856(n) = A003973(n) / gcd(A000005(n), A003973(n)).
a(p) = A048673(p) for all primes p.
a(p^3) = 2*A048673(p)^3 - 2*A048673(p)^2 + A048673(p). [The denominator A336839(p^3) = 1 for all p]

A336856 Prime-shifted analog of gcd(d(n), sigma(n)): a(n) = gcd(A000005(n), A003973(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 4, 1, 4, 2, 6, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 8, 2, 8, 2, 2, 2, 4, 2, 2, 1, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 1, 4, 8, 2, 2, 4, 8, 2, 4, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4, 8, 2, 4, 4, 6, 4, 4, 4, 12, 2, 2, 2, 3, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Cf. A000005, A003961, A003973, A009205, A336837, A336838, A336839, A336840, A336841.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A336856(n) = gcd(numdiv(n), A003973(n));

a(n) = A009205(A003961(n)).
a(n) = gcd(A000005(n), A003973(n)) = gcd(A000005(n), A336841(n)).
a(n) = gcd(A000005(n), 2*A336840(n)).
a(n) = A003973(n) / A336838(n) = A000005(n) / A336839(n).
For n > 1, a(n) = A336841(n) / A336837(n).
For all primes p, and n >= 0, a(p^((2^n)-1)) = 2^n.

A336837 Numerator of ratio A336841(n)/A000005(n).

Original entry on oeis.org

0, 1, 2, 14, 3, 9, 5, 17, 44, 13, 6, 32, 8, 21, 23, 284, 9, 163, 11, 137, 37, 25, 14, 105, 30, 33, 86, 73, 15, 81, 18, 547, 44, 37, 53, 1622, 20, 45, 58, 149, 21, 129, 23, 260, 401, 57, 26, 1662, 230, 109, 65, 114, 29, 297, 63, 237, 79, 61, 30, 263, 33, 73, 213, 4010, 83, 153, 35, 383, 100, 183, 36, 1715, 39, 81

Offset: 1

Antti Karttunen, Aug 07 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A003961, A003973, A336838, A336841, A336856.

Cf. A336839 (denominators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));
A336837(n) = numerator(A336841(n)/numdiv(n));

a(n) = A336841(n) / A336856(n) = A336841(n) / gcd(A000005(n), A336841(n)).

cp's OEIS Frontend

A336918 Numbers k such that A000005(k) divides A003973(k); numbers k for which A336839(k) = 1.

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A336919 Numbers k such that A000005(k) does not divide A003973(k); numbers k for which A336839(k) > 1.

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A336840 Inverse Möbius transform of A048673.

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A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

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A336838 Numerator of the arithmetic mean of the divisors of A003961(n).

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A336856 Prime-shifted analog of gcd(d(n), sigma(n)): a(n) = gcd(A000005(n), A003973(n)).

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A336837 Numerator of ratio A336841(n)/A000005(n).

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