A067342 -id:A067342 - cp's OEIS frontend

A077650 Initial decimal digit of sigma(n), the sum of divisors of n.

1, 3, 4, 7, 6, 1, 8, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 4, 3, 3, 2, 6, 3, 4, 4, 5, 3, 7, 3, 6, 4, 5, 4, 9, 3, 6, 5, 9, 4, 9, 4, 8, 7, 7, 4, 1, 5, 9, 7, 9, 5, 1, 7, 1, 8, 9, 6, 1, 6, 9, 1, 1, 8, 1, 6, 1, 9, 1, 7, 1, 7, 1, 1, 1, 9, 1, 8, 1, 1, 1, 8, 2, 1, 1, 1, 1, 9, 2, 1, 1, 1, 1, 1, 2, 9, 1, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Labos Elemer, Nov 19 2002

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000030, A000203, A067342.

Table[First[IntegerDigits[DivisorSigma[1, w]]], {w, 1, 128}]

a(n) = digits(sigma(n))[1]; \\ Michel Marcus, Nov 18 2017

a(n) = A000030(A000203(n)).

A067343 Sum of decimal digits of n equals sum of decimal digits of sum of divisors of n.

Original entry on oeis.org

1, 15, 24, 64, 69, 78, 90, 114, 133, 147, 153, 186, 198, 258, 270, 276, 288, 306, 339, 360, 366, 393, 429, 474, 492, 495, 507, 522, 582, 588, 609, 618, 627, 639, 708, 717, 738, 762, 763, 801, 817, 834, 846, 871, 906, 933, 960, 978, 990, 1062, 1080, 1083

Offset: 1

Labos Elemer, Jan 16 2002

Cf. A067342, A007953, A000203.

sd(n) = digs = digits(n); sum(i=1, #digs, digs[i]);
isok(n) = sd(n) == sd(sigma(n)); \\ Michel Marcus, Dec 26 2013

A067342(n) = A007953(A000203(n)) = A007953(n).

A256635 a(n) = the smallest number k such that the base-10 digital sum of sigma(k) is n.

Original entry on oeis.org

1, 19, 2, 3, 13, 5, 4, 7, 10, 12, 28, 18, 192, 67, 42, 273, 52, 138, 324, 336, 196, 300, 372, 438, 2716, 997, 1590, 3468, 2512, 3260, 5817, 5692, 4112, 17472, 10852, 15840, 18496, 27252, 22860, 24300, 31572, 35172, 61488, 165652, 138438, 265252, 285652, 292860

Offset: 1

Jaroslav Krizek, Apr 06 2015

a(n) = the smallest number k such that A007953(A000203(k)) = n.

Note that A007953(A000203(k)) is also A067342(k).

			For n = 5; digital sum of sigma(13) = digital sum of 14 = 5. The number 13 is the smallest number with this property so a(5) = 13.

Max Alekseyev, Table of n, a(n) for n = 1..100 (terms for n = 1..66 from Chai Wah Wu)

Cf. A000203, A007953, A067342, A256642.

A256635:=func; [A256635(n):n in[1..50]];

N := 10^6: # return all values before the first > N
for n from 1 to N do
   v:= convert(convert(numtheory:-sigma(n),base,10),`+`);
   if not assigned(A[v]) then A[v]:= n fi;
od:
for count from 1 while assigned(A[count]) do od:
seq(A[i],i=1..count-1); # Robert Israel, Apr 09 2015

f[n_] := Block[{k = 1}, While[Plus @@ IntegerDigits[DivisorSigma[1, k]] != n, k++]; k]; Array[f, 48] (* Michael De Vlieger, Apr 07 2015 *)

a(n) = {my(k = 1); while(sumdigits(sigma(k)) != n, k++); k;} \\ Michel Marcus, Apr 09 2015

from sympy.ntheory.factor_ import divisor_sigma
def A256635(n):
    k = 1
    while sum(int(d) for d in str(divisor_sigma(k))) != n:
        k += 1
    return k # Chai Wah Wu, Apr 18 2015

A277216 Product of decimal digits of sum of divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 2, 8, 5, 3, 8, 2, 16, 4, 8, 8, 3, 8, 27, 0, 8, 6, 18, 8, 0, 3, 8, 0, 30, 0, 14, 6, 18, 32, 20, 32, 9, 24, 0, 30, 0, 8, 54, 16, 32, 56, 14, 32, 8, 35, 27, 14, 72, 20, 0, 14, 0, 0, 0, 0, 48, 12, 54, 0, 14, 32, 16, 48, 12, 54, 16, 14, 45, 28, 4, 8

Offset: 1

Jaroslav Krizek, Oct 05 2016

Conjecture: a(n) = n only for numbers 1 and 210; sigma(210) = 576; a(210) = 5*7*6 = 210.

			a(12) = 16 because sigma(12) = 28; 2*8 = 16.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A067342 (sum of decimal digits of sigma(n)).

Cf. A000203, A007954, A277217.

[&*Intseq(SumOfDivisors(n)): n in [1..100000]];

seq( convert(convert(numtheory:-sigma(n),base,10),`*`), n=1..100); # Robert Israel, Oct 06 2016

Table[Times @@ IntegerDigits@ DivisorSigma[1, n], {n, 75}] (* Michael De Vlieger, Oct 06 2016 *)

a(n) = d = digits(sigma(n)); prod(k=1, #d, d[k]); \\ Michel Marcus, Oct 05 2016

a(n) = A007954(A000203(n)).

A277217 Numbers k for which the sum of digits of sigma(k) = the product of digits of sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 86, 126, 131, 206, 207, 311, 1123, 1213, 2113, 4111, 10921, 12211, 16581, 21121, 21211, 22111, 39660, 51558, 52940, 60812, 61504, 63548, 68822, 81303, 83409, 87081, 87451, 89708, 94523, 97307, 106118, 108527, 110387, 111611, 120831, 160271

Offset: 1

Jaroslav Krizek, Oct 05 2016

Numbers k such that A067342(k) = A277216(k).
Prime terms: 2, 3, 5, 7, 131, 311, 1123, 1213, 2113, 4111, 12211, ...
Corresponding values of sigma(a(n)): 1, 3, 4, 7, 6, 8, 132, 312, 132, 312, 312, 312, 1124, 1214, 2114, ...
Only 196 terms less than 35*10^8. - Robert G. Wilson v, Oct 07 2016
Alternatively, numbers k such that sigma(k) is in A034710. - Charlie Neder, Dec 27 2018

			86 is a term because sigma(86) = 132; sum and product of digits of 132 = 6.

Robert G. Wilson v, Table of n, a(n) for n = 1..196 (first 100 terms from Jaroslav Krizek)

Cf. A000203, A007953, A007954, A034710.

Cf. A067342 (sum of decimal digits of sigma(n)), A277216 (product of decimal digits of sigma(n)).

[n: n in [1..100000] | &+Intseq(SumOfDivisors(n)) eq &*Intseq(SumOfDivisors(n))];

Select[Range@ 200000, Total@ # == Times @@ # &@ IntegerDigits@ DivisorSigma[1, #] &] (* Michael De Vlieger, Oct 06 2016 *)

isok(n) = my(d=digits(sigma(n))); vecprod(d) == vecsum(d); \\ Michel Marcus, Mar 02 2019

A067344 Sum of decimal digits of square of divisors of n equals sum of square of digits of n.

Original entry on oeis.org

1, 21, 41, 120, 242, 312, 323, 401, 501, 1040, 1114, 1141, 1204, 1214, 1233, 1241, 1304, 1503, 2033, 2115, 2133, 2140, 2403, 3010, 3014, 3124, 3211, 3304, 3322, 4001, 4012, 4121, 4301, 4310, 5130, 10044, 10214, 10242, 10320, 10324, 11042, 11115

Offset: 1

Labos Elemer, Jan 16 2002

			n=51223, SquareSumDigit=25+1+4+4+9=43, Sigma[2,51223]=2623908580 with digit sum=43.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A067342, A067343, A007953, A001157.

Do[s=Apply[Plus, IntegerDigits[DivisorSigma[2, n]]]- Apply[Plus, IntegerDigits[n]^2]; If[Equal[s, 0], Print[n]], {n, 1, 10000}]
Select[Range[12000],Total[Flatten[IntegerDigits[#]^2]]== Total[ IntegerDigits[ DivisorSigma[2,#]]]&] (* Harvey P. Dale, Sep 12 2012 *)

A007953[A001157(n)]=A007953[n]^2

A071422 a(n) = a(n-1) + sum of decimal digits of sigma(n), the sum of divisors of n.

Original entry on oeis.org

1, 4, 8, 15, 21, 24, 32, 38, 42, 51, 54, 64, 69, 75, 81, 85, 94, 106, 108, 114, 119, 128, 134, 140, 144, 150, 154, 165, 168, 177, 182, 191, 203, 212, 224, 234, 245, 251, 262, 271, 277, 292, 300, 312, 327, 336, 348, 355, 367, 379, 388, 405, 414, 417, 426, 429

Offset: 1

Labos Elemer, May 27 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A037123, A000788, A051351.

Partial sums of A067342.

s=0; Do[s=s+Apply[Plus, IntegerDigits[DivisorSigma[1, n]]]; Print[s], {n, 1, 128}]
nxt[{n_,a_}]:={n+1,a+Total[IntegerDigits[DivisorSigma[1,n+1]]]}; Transpose[ NestList[nxt,{1,1},60]][[2]] (* Harvey P. Dale, Jan 25 2013 *)

cp's OEIS Frontend

A077650 Initial decimal digit of sigma(n), the sum of divisors of n.

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A067343 Sum of decimal digits of n equals sum of decimal digits of sum of divisors of n.

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A256635 a(n) = the smallest number k such that the base-10 digital sum of sigma(k) is n.

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A277216 Product of decimal digits of sum of divisors of n.

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A277217 Numbers k for which the sum of digits of sigma(k) = the product of digits of sigma(k).

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A067344 Sum of decimal digits of square of divisors of n equals sum of square of digits of n.

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A071422 a(n) = a(n-1) + sum of decimal digits of sigma(n), the sum of divisors of n.

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