A048050 -id:A048050 - cp's OEIS frontend

A216982 Anti-Chowla's function: sum of anti-divisors of n except the largest.

0, 0, 0, 0, 2, 0, 5, 3, 2, 7, 5, 5, 10, 7, 8, 3, 17, 16, 5, 11, 8, 21, 19, 7, 22, 7, 24, 27, 5, 16, 21, 37, 26, 7, 29, 8, 25, 45, 26, 28, 14, 38, 27, 11, 56, 27, 29, 24, 39, 47, 8, 59, 53, 16, 37, 19, 36, 57, 51, 67, 16, 37, 70, 3, 41, 42, 87, 67, 8, 55

Offset: 1

Juri-Stepan Gerasimov, Feb 19 2013

nonn

Numbers n such that Chowla's function(n) = a(n): 1, 2, 3, 10, 15, 28, 75, 88, 231, 284, 602,...
Places n where a(n) is zero: 1, 2, 3, 4, 6, 96,...
Fixed points of this sequence: 17, 53, 127, 217, 385, 2321,...
Places n where a(n) equals the largest anti-divisor: 1, 2, 7, 10, 31, 37, 39, 55, 78, 160, 482, 937, 1599, 2496,...
Numbers n such that n -/+ 1 and a(n -/+ 1) are all primes: 6, 18, 72, 102, 108, 198, 270, 432, 570, 882,...

			Anti-divisors of 7 are 2, 3, 5, so a(7) = 2 + 3 = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048050, A066272, A066417, A066481, A130799, A218767.

A216982 := proc(n)
        A066417(n)-A066481(n) ;
end proc: # R. J. Mathar, Feb 23 2013

a(n)=if(n<5,0,my(k=valuation(n, 2)); sigma(2*n+1)+sigma(2*n-1)+sigma(n>>k)<<(k+1)-2-20*n\/3) \\ Charles R Greathouse IV, Mar 05 2013

a(n) = A066417(n) - A066481(n).

A218767 Total number of divisors and anti-divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 6, 5, 7, 5, 8, 6, 7, 7, 7, 7, 10, 5, 9, 7, 9, 7, 10, 8, 7, 9, 11, 5, 11, 7, 12, 9, 7, 9, 11, 7, 11, 9, 12, 6, 13, 7, 9, 13, 9, 7, 13, 9, 12, 7, 13, 9, 11, 9, 11, 9, 11, 9, 18, 6, 9, 13, 9, 9, 13, 11, 13, 7, 13, 7, 18, 9, 9, 11, 11, 13, 13, 5, 15, 11, 11, 9, 16, 12, 9

Offset: 1

Juri-Stepan Gerasimov, Feb 05 2013

nonn

Or tau(n) + anti-tau(n), where anti-tau = A066272.
Total sum of divisors and anti-divisors of n or sigma(n) + A066417(n): 1, 3, 6, 10, 11, 16, 18, 23, 21, 32, 24, 41, 33, 40, 42, 45, 46, 67, 38, 66, 54, 72, 58, 83, 70, 66, 82, 102, 54, 108,...
Numbers n such that sigma(n) = n + anti-sigma(n): A074751.
Numbers n such that Chowla's function(n) = anti-sigma(n): 1, 2, 16, 60, 72,...
Number of divisors of n minus number of anti-divisors of n or tau(n) - anti-tau(n): 1, 2, 1, 2, 0, 3, -1, 2, 1, 1, -1, 4, -2, 1, 1, 3, -3, 2, -1, 3, 1, -1, -3, 6, -2, 1, -1, 1, -1, 5, -3, 0, -1, 1, -1, 7, -3, -3, -1, 4, -2, 3, -3, 3, -1,...
Product of number of divisors of n and number of anti-divisors of n, or tau(n)*anti-tau(n): 0, 0, 2, 3, 4, 4, 6, 8, 6, 12, 6, 12, 8, 12, 12, 10, 10, 24, 6, 18, 12, 20, 10, 16, 15, 12, 20, 30, 6, 24,...
Number of ways to write n as k*(k - m) with k divisor and m anti-divisor of n: 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0,...
Numbers which are not of the form k*(k - m), k divisor, m anti-divisor (i.e., where the number of ways is zero): 1, 2, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 26, 29,

Cf. A000005, A000203, A001065, A066272, A066417, A027750, A048050, A130799.

A218767 := proc(n)
        numtheory[tau](n)+A066272(n) ;
end proc: # R. J. Mathar, Feb 16 2013

a(n) = A000005(n) + A066272(n).

A239313 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros, except the first column which lists 0 together with the nonnegative integers, and the first element of column k is in row k*(k+1)/2.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 3, 3, 4, 0, 1, 5, 5, 0, 6, 0, 0, 7, 7, 3, 8, 0, 0, 1, 9, 9, 0, 0, 10, 0, 5, 0, 11, 11, 0, 0, 12, 0, 0, 3, 13, 13, 7, 0, 1, 14, 0, 0, 0, 0, 15, 15, 0, 0, 0, 16, 0, 9, 5, 0, 17, 17, 0, 0, 0, 18, 0, 0, 0, 3, 19, 19, 11, 0, 0, 1, 20, 0, 0, 7, 0, 0

Offset: 1

Omar E. Pol, Mar 15 2014

Alternating row sums give the Chowla's function, i.e., sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A048050(n).
Row n has length A003056(n) hence column k starts in row A000217(k).
Column 1 gives 0 together with A001477.
Column 2 is A193356.
The number of positive terms in row n is A001227(n), if n >= 3. - Omar E. Pol, Apr 18 2016

			Triangle begins (row n = 1..24):
0;
0;
1,   1;
2,   0;
3,   3;
4,   0,  1;
5,   5,  0;
6,   0,  0;
7,   7,  3;
8,   0,  0,  1;
9,   9,  0,  0;
10,  0,  5,  0;
11, 11,  0,  0;
12,  0,  0,  3;
13, 13,  7,  0,  1;
14,  0,  0,  0,  0;
15, 15,  0,  0,  0;
16,  0,  9,  5,  0;
17, 17,  0,  0,  0;
18,  0,  0,  0,  3;
19, 19, 11,  0,  0,  1;
20,  0,  0,  7,  0,  0;
21, 21,  0,  0,  0,  0;
22,  0, 13,  0,  0,  0;
...
For n = 15 the divisors of 15 are 1, 3, 5, 15 therefore the sum of divisors of 15 except 1 and 15 is 3 + 5 = 8. On the other hand the 15th row of triangle is 13, 13, 7, 0, 1, hence the alternating row sum is 13 - 13 + 7 - 0 + 1 = 8, equalling the sum of divisors of 15 except 1 and 15.
If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of divisors of n, except 1 and n. Example: the sum of divisors of 24 except 1 and 24 is 2 + 3 + 4 + 6 + 8 + 12 = 35, and the alternating sum of the 24th row of triangle is 22 - 0 + 13 - 0 + 0 - 0 = 35.

Cf. A000203, A000217, A001065, A001227, A001477, A193356, A196020, A211343, A212119, A228813, A231345, A231347, A235791, A235794, A236104, A236106, A236112.

T(n,k) = A196020(n,k), if k >= 2. - Omar E. Pol, Apr 18 2016

A261023 Least number k such that prime(n) = sigma(k) - k - 1.

Original entry on oeis.org

4, 9, 6, 10, 121, 22, 289, 34, 529, 841, 58, 1369, 30, 82, 2209, 42, 3481, 118, 4489, 5041, 70, 6241, 6889, 78, 9409, 10201, 202, 60, 214, 102, 16129, 17161, 18769, 84, 138, 298, 24649, 26569, 27889, 29929, 32041, 358, 36481, 238, 186, 394, 44521, 49729, 51529

Offset: 1

Paolo P. Lava, Aug 07 2015

For any prime k <= p^2. In fact if k = p^2 we have that sigma(p) = sigma(p^2) - p^2, that is 1 + p = 1 + p + p^2 - p^2.

			sigma(2) = 3 and 4 is the least number such that sigma(4) - 4 = 7 - 4 = 3.
sigma(13) = 14 and 22 is the least number such that sigma(22) - 22 = 36 - 22 = 14.

Robert Israel and Paolo P. Lava, Table of n, a(n) for n = 1..1229 (first 100 from Paolo P. Lava)

Cf. A008864, A048050, A070015, A158913.

with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do
if isprime(n) then for k from 1 to q do
if sigma(n)=sigma(k)-k then print(k); break; fi; od;
fi; od; end: P(10^9);

Table[k = 1; While[DivisorSigma[1, Prime@ p] != DivisorSigma[1, k] - k, k++]; k, {p, 60}] (* Michael De Vlieger, Aug 07 2015 *)

a(n) = my(k = 1, p = prime(n)); while(sigma(k)-k-1 != p, k++); k; \\ Michel Marcus, Aug 12 2015

first(m)=my(v=vector(m),k);for(i=1,m,k=1;while(!(prime(i)==sigma(k)-k-1),k++);v[i]=k;);v; \\ Anders Hellström, Aug 14 2015

a(n) = A070015(A008864(n)). - Robert Israel, Aug 14 2015

A288654 a(n) = (sigma(n)-n-1)*(3-omega(n)).

Original entry on oeis.org

-3, 0, 0, 4, 0, 5, 0, 12, 6, 7, 0, 15, 0, 9, 8, 28, 0, 20, 0, 21, 10, 13, 0, 35, 10, 15, 24, 27, 0, 0, 0, 60, 14, 19, 12, 54, 0, 21, 16, 49, 0, 0, 0, 39, 32, 25, 0, 75, 14, 42, 20, 45, 0, 65, 16, 63, 22, 31, 0, 0, 0, 33, 40, 124, 18, 0, 0, 57, 26, 0, 0, 122

Offset: 1

Wesley Ivan Hurt, Jun 12 2017

If n is prime, then a(n) = 0. If n is semiprime, then a(n) = sopfr(n).

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

Cf. A000203 (sigma), A001221 (omega), A001414 (sopfr), A048050 (Chowla's function).

Table[(DivisorSigma[1, n] - n - 1) (3 - PrimeNu[n]), {n, 100}]

A288654(n) = ((sigma(n)-n-1)*(3-omega(n))); \\ Antti Karttunen, Mar 04 2018

A326830 Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 5, 0, 9, 3, 17, 0, 46, 6, 68, 23, 153, 27, 297, 67, 534, 188, 978, 276, 1932, 620, 3250, 1313, 6033, 2246, 10854, 4361, 18776, 8639, 32831, 14835, 58230, 27635, 98052, 50980, 169522, 88243, 289720, 157179, 486232, 280206, 818006, 478014

Offset: 0

Ilya Gutkovskiy, Oct 20 2019

nonn

Euler transform of A048050.

Convolution of A326830 and A002865 is A318784. - Vaclav Kotesovec, Oct 26 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000203, A001001, A001157, A048050, A061256, A318784, A326831.

with(numtheory):
b:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

nmax = 47; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

G.f.: Product_{k>=1} 1 / (1 - x^k)^A048050(k).
G.f.: exp(Sum_{k>=1} (A001001(k) - A000203(k) - A001157(k) + 1) * x^k / k).
a(n) ~ exp(3^(2/3) * ((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/2 - Pi^2 * (3/((Pi^2 - 6)*Zeta(3)))^(1/3) * n^(1/3)/4 - Pi^4 / (32*(Pi^2 - 6)*Zeta(3)) - 1/8) * A^(3/2)* (2*Pi)^(1/24) / (3^(1/8) * ((Pi^2 - 6)*Zeta(3))^(3/8) * n^(1/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2019

A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 5, 0, 7, 3, 17, 0, 37, 6, 58, 23, 120, 21, 235, 67, 390, 161, 726, 230, 1349, 521, 2225, 1055, 3990, 1714, 7040, 3341, 11604, 6294, 20053, 10500, 34252, 19115, 56055, 34168, 94306, 56998, 157078, 99515, 254766, 171484, 419287, 283565

Offset: 0

Ilya Gutkovskiy, Oct 20 2019

nonn

Weigh transform of A048050.

Convolution of A326831 and A025147 is A319107. - Vaclav Kotesovec, Oct 26 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A048050, A192065, A319107, A326830.

with(numtheory):
g:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

nmax = 47; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, (-1)^(k/d + 1) d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

G.f.: Product_{k>=1} (1 + x^k)^A048050(k).

a(n) ~ exp(3*(2*(Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3) / (2^(7/3) * ((Pi^2 - 6)*Zeta(3))^(1/3)) - Pi^4 / (96*(Pi^2 - 6)*Zeta(3))) * 2^(19/24) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 26 2019

A331037 Numbers k such that the sum of the divisors of k (except for 1 and k) plus the sum of the digits of k is equal to k.

Original entry on oeis.org

1, 2, 3, 5, 7, 14, 52, 76, 2528, 9536, 9664, 35456, 138496, 8456192, 33665024, 33673216, 537444352, 2148958208, 137454419968

Offset: 1

Joseph E. Marrow, Jan 08 2020

Additional terms include 537444352, 2148958208, 137454419968, 35184644718592, 9007202811510784. Are there any terms > 1 not of the form 2^k*p where p is prime and k>0? - David A. Corneth, Jan 08 2020
Terms not of the form 2^k*p do exist, for example 2^15*65713*24194197 and 2^19*1739719*2639431. - Giovanni Resta, Jan 08 2020
a(20) > 10^13. - Giovanni Resta, Jan 14 2020

			The first term that is not 1 or a single-digit prime is obtained by adding the proper divisors of 14 other than 1 (2,7) to its digits (1,4): (2+7) + (1+4) = 14.
The second such term is 52: the proper divisors of 52 other than 1 (2,4,13,26) and its digits (5,2) sum to (2+4+13+26) + (5+2) = 52.

Cf. A000203, A007953, A048050.

Cf. A331093 (sum of divisors - digit sum = the number).

Select[Range[10^7], DivisorSigma[1, #] - # - If[# == 1, 0, 1] + Plus @@ IntegerDigits[#] == # &] (* Amiram Eldar, Jan 12 2020 *)

is(n) = n == sigma(n)-1-if(n>1,n,0)+sumdigits(n) \\ Rémy Sigrist, Jan 08 2020

a(14)-a(16) from Rémy Sigrist, Jan 08 2020

a(17)-a(19) from Giovanni Resta, Jan 14 2020

A331093 Numbers such that the sum of their divisors, excluding 1 and the number itself, minus the sum of their digits equals the number.

Original entry on oeis.org

12, 114256, 6988996, 8499988, 8689996, 8789788, 8877988, 8988868, 8999956, 9696988, 9759988, 9899596, 9948988, 9996868, 9998884, 9999892, 15996988, 16878988, 17799796, 17887996, 17988796, 17999884, 18579988, 18768988, 18869788, 18895996, 18958996, 18995788, 19398988, 19587988, 19698868, 19777996, 19799668

Offset: 1

Joseph E. Marrow, Jan 08 2020

After the second term, it seems that the digit sum is 55.
All terms after a(2) appear to be of the form 2^2 * 7 * p, where p is a prime. - Scott R. Shannon, Jan 09 2020
If there exists a third term not of the form 2^2*7*p, it is larger than 10^13. - Giovanni Resta, Jan 14 2020

			a(3) = 6988996 as the sum of the divisors of 6988996, excluding 1 and 6988996, equals 6989051, the sum of its digits equals 55, and 6989051 - 55 = 6988996.

Cf. A000203, A007953, A048050.

Cf. A331037 (sum of divisors + digit sum = number).

Select[Range[10^7], DivisorSigma[1, #] - Plus @@ IntegerDigits[#] == 2 # + 1 &] (* Amiram Eldar, Jan 08 2020 *)

isok(n) = sigma(n) - n - 1 - sumdigits(n) == n; \\ Michel Marcus, Jan 09 2020

Terms a(7) and beyond from Scott R. Shannon, Jan 09 2020

A335022 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1) * d.

Original entry on oeis.org

0, 0, 0, -2, 0, 1, 0, -6, 3, 3, 0, -9, 0, 5, 8, -14, 0, 4, 0, -11, 10, 9, 0, -29, 5, 11, 12, -13, 0, 5, 0, -30, 14, 15, 12, -30, 0, 17, 16, -39, 0, 9, 0, -17, 32, 21, 0, -69, 7, 18, 20, -19, 0, 13, 16, -49, 22, 27, 0, -61, 0, 29, 40, -62, 18, 17, 0, -23, 26, 21, 0, -98, 0, 35, 48, -25

Offset: 1

Ilya Gutkovskiy, May 19 2020

sign

Difference between the sum of the odd nontrivial divisors of n and the sum of the even nontrivial divisors of n.

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

Cf. A002129, A048050, A058344, A335021.

Table[DivisorSum[n, (-1)^(# + 1) # &, 1 < # < n &], {n, 1, 76}]
nmax = 76; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d>1) && (dMichel Marcus, May 20 2020

G.f.: Sum_{k>=2} (-1)^(k + 1) * k * x^(2*k) / (1 - x^k).

cp's OEIS Frontend

A216982 Anti-Chowla's function: sum of anti-divisors of n except the largest.

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A218767 Total number of divisors and anti-divisors of n.

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A239313 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros, except the first column which lists 0 together with the nonnegative integers, and the first element of column k is in row k*(k+1)/2.

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A261023 Least number k such that prime(n) = sigma(k) - k - 1.

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A288654 a(n) = (sigma(n)-n-1)*(3-omega(n)).

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A326830 Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.

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A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

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A331037 Numbers k such that the sum of the divisors of k (except for 1 and k) plus the sum of the digits of k is equal to k.

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