A118503 -id:A118503 - cp's OEIS frontend

A134599 Sum of digital sums (base 3) of the prime factors of n.

0, 2, 1, 4, 3, 3, 3, 6, 2, 5, 3, 5, 3, 5, 4, 8, 5, 4, 3, 7, 4, 5, 5, 7, 6, 5, 3, 7, 3, 6, 3, 10, 4, 7, 6, 6, 3, 5, 4, 9, 5, 6, 5, 7, 5, 7, 5, 9, 6, 8, 6, 7, 7, 5, 6, 9, 4, 5, 5, 8, 5, 5, 5, 12, 6, 6, 5, 9, 6, 8, 7, 8, 5, 5, 7, 7, 6, 6, 7, 11, 4, 7, 3, 8, 8, 7, 4, 9, 5, 7, 6, 9, 4, 7, 6, 11, 5, 8, 5, 10, 5, 8, 5, 9, 7

Offset: 1

Hieronymus Fischer, Nov 11 2007

			a(6) = 3, since 6 = 2*3 and so a(6) = ds_3(2) + ds_3(3) = 2 + 1.

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

Cf. A053735, A080773 (base 2), A118503.

f[p_, e_] := e * DigitSum[p, 3]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 30 2025 *)

a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * sumdigits(f[i, 1], 3));} \\ Amiram Eldar, Jul 30 2025

If p_1*p_2*p_3* ... *p_m = n is the unique prime factorization of n, then a(n) = Sum_{k=1..m} ds_3(p_k), where ds_3 is the digital sum base 3.

Totally additive with a(p) = A053735(p). - Amiram Eldar, Jul 30 2025

a(1) = 0 prepended by Amiram Eldar, Jul 30 2025

A289142 Numbers whose sum of prime factors (taken with multiplicity) is divisible by 3.

Original entry on oeis.org

1, 3, 8, 9, 14, 20, 24, 26, 27, 35, 38, 42, 44, 50, 60, 62, 64, 65, 68, 72, 74, 77, 78, 81, 86, 92, 95, 105, 110, 112, 114, 116, 119, 122, 125, 126, 132, 134, 143, 146, 150, 155, 158, 160, 161, 164, 170, 180, 185, 186, 188, 192, 194, 195, 196, 203, 204

Offset: 1

David James Sycamore, Jun 26 2017

nonn

U{S(n); 3|n}, where S(n)= {x; sopfr(x)=n}; numbers placed in ascending order.
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Robert Israel, Jul 03 2017
From Antti Karttunen, Jun 11 2024, with minor edits Jun 30 2024: (Start)
Numbers such that the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3.
For n that is not a multiple of 3, sopfr(n) [= A001414(n)] is a multiple of 3 if and only if the arithmetic derivative of n [= A003415(n)] is a multiple of 3. See A373475 for a proof.
This sequence (as a multiplicative semigroup) is generated by the union of A369659 with {3}.
(End)

			sopfr(42) = 2 + 3 + 7 = 12 = 4*3, sopfr(95) = 5 + 19 = 24 = 8 * 3, sopfr(180) = 2 + 2 + 3 + 3 + 5 = 15 = 5 * 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002476, A003627, A036349, A036350, A046363, A373371 (characteristic function).
Positions of multiples of 3 in A001414 (sopfr) and in A118503.
Subsequences that are formed by intersecting this sequence with other multiplicative semigroups: A102217, A369659, A373373, A373473, A373475, A373478, A373597.
Cf. also A373385, A373602, A374052.

select(n -> add(t[1]*t[2],t=ifactors(n)[2]) mod 3 = 0, [$1..1000]); # Robert Israel, Jul 03 2017

Join[{1},Select[Range[250],Mod[Total[Times@@@FactorInteger[#]],3]==0&]] (* Harvey P. Dale, Mar 16 2020 *)

s(n)=my(f=factor(n),p=f[,1],e=f[,2]);sum(k=1,#p,e[k]*p[k]);
for(n=1,200,if(s(n)%3==0,print1(n,","))); \\ Joerg Arndt, Jun 26 2017

isA289142 = A373371; \\ Antti Karttunen, Jun 08 2024

For n >= 2, a(n) = A102217(n-1)/3. - Antti Karttunen, Jun 08 2024

Corrected by Robert Israel, Jul 03 2017

A102216 2-Suzanne numbers.

Original entry on oeis.org

4, 8, 15, 22, 26, 35, 42, 44, 60, 62, 64, 68, 84, 88, 99, 118, 121, 123, 129, 136, 138, 141, 143, 145, 152, 158, 161, 165, 169, 174, 176, 183, 187, 189, 194, 196, 198, 200, 202, 206, 208, 215, 231, 235, 240, 242, 246, 248, 255, 273, 275, 279, 280, 282, 284

Offset: 1

Eric W. Weisstein, Dec 30 2004

From Amiram Eldar, Apr 23 2021: (Start)
Composite numbers k such that the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) are both even.
The Monica and Suzanne sets were named by Smith (1996) after his two cousins, Monica and Suzanne Hammer. (End)

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Suzanne Set.

Subsequence of A102218.

Cf. A007953, A018252, A102217, A118503.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; suz2Q[n_] := CompositeQ[n] && And @@ EvenQ[{s[n], sp[n]}]; Select[Range[300], suz2Q] (* Amiram Eldar, Apr 23 2021 *)

A102217 3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

Original entry on oeis.org

9, 24, 27, 42, 60, 72, 78, 81, 105, 114, 126, 132, 150, 180, 186, 192, 195, 204, 216, 222, 231, 234, 243, 258, 276, 285, 315, 330, 336, 342, 348, 357, 366, 375, 378, 396, 402, 429, 438, 450, 465, 474, 480, 483, 492, 510, 540, 555, 558, 564, 576, 582, 585

Offset: 1

Eric W. Weisstein, Dec 30 2004

nonn

Composite numbers k such that the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) are both divisible by 3. - Amiram Eldar, Apr 23 2021

The new secondary definition is equal to the original because taking the decimal digit sum preserves congruence modulo 3. This is a multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jun 08 2024

			From _Antti Karttunen_, Jun 08 2024: (Start)
42 = 2*3*7 is a term as it is a multiple of 3, and also 2+3+7 = 12 is a multiple of 3.
60 = 2*2*3*5 is a term is it is a multiple of 3, and also 2+2+3+5 = 12 is a multiple of 3.
(End)

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Suzanne Set.

Subsequence of A177927.
Intersection of A008585 and A289142 without the initial 3.
Positions of multiples of 3 in A082299, after A082299(3).
Cf. A001414, A007953, A018252, A102216, A118503, A373371.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; suz3Q[n_] := CompositeQ[n] && And @@ Divisible[{s[n], sp[n]}, 3]; Select[Range[600], suz3Q] (* Amiram Eldar, Apr 23 2021 *)

isA102217(n) = if(n<=3 || (n%3), 0, my(f=factor(n)); 0==(sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1]))%3)); \\ Antti Karttunen, Jun 08 2024

isA102217(n) = (n>3 && !(n%3) && A373371(n)); \\ Antti Karttunen, Jun 08 2024

a(n) = 3*A289142(1+n). - Antti Karttunen, Jun 08 2024

Alternative definition added and keyword:base removed by Antti Karttunen, Jun 08 2024

A177927 3-Monica numbers.

Original entry on oeis.org

4, 9, 10, 22, 24, 25, 27, 34, 42, 46, 55, 58, 60, 72, 78, 81, 82, 85, 94, 105, 106, 114, 115, 118, 121, 126, 128, 132, 142, 145, 150, 166, 178, 180, 186, 187, 192, 195, 202, 204, 205, 214, 216, 222, 224, 226, 231, 234, 235, 243, 253, 256, 258, 262, 265, 274, 276, 285, 289, 295

Offset: 1

Chris Fry, Dec 26 2010

3-Monica numbers are composite positive integers k for which 3 divides S(k)-Sp(k), where S(k) denotes the sum of the digits of k and Sp(k) denotes the sum of the digits in an extended prime factorization of k.

			S(10)=1+0=1, 10=2*5, Sp(10)=2+5=7, S(10)-Sp(10)=-6 which is divisible by 3.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 93.
E. W. Weisstein, The CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pages 1192-1193.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Monica Set.

Cf. A006753 (Smith numbers are a subset of every n-Monica sequence).
Cf. A102217 (n-Suzanne numbers are a subset of n-Monica numbers).
Cf. A102219 (This list of '3-Monica' numbers is incorrect. It does not contain all the Smith numbers and appears to be based on S(n)+Sp(n) ==0 (mod 3), instead of S(n)-Sp(n) == 0 (mod 3)).
Cf. A007953, A118503.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; mon3Q[n_] := CompositeQ[n] && Divisible[s[n] - sp[n], 3]; Select[Range[300], mon3Q] (* Amiram Eldar, Apr 23 2021 *)

A102218 2-Monica numbers.

Original entry on oeis.org

4, 8, 10, 12, 14, 15, 22, 26, 27, 35, 42, 44, 45, 54, 56, 58, 60, 62, 63, 64, 65, 68, 78, 84, 85, 88, 90, 92, 94, 96, 99, 102, 108, 111, 118, 119, 121, 122, 123, 126, 129, 133, 136, 138, 141, 143, 145, 152, 155, 158, 159, 160, 161, 164, 165, 166, 169, 174, 175

Offset: 1

Eric W. Weisstein, Dec 30 2004

Composite numbers k such that the difference between the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) is even. - Amiram Eldar, Apr 23 2021

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Monica Set.

Cf. A007953, A018252, A102216, A102219, A118503, A177927.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; mon2Q[n_] := CompositeQ[n] && EvenQ[s[n] - sp[n]]; Select[Range[200], mon2Q] (* Amiram Eldar, Apr 23 2021 *)

A118541 Product of digits of prime factors of n, with multiplicity.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 3, 14, 15, 16, 7, 18, 9, 20, 21, 2, 6, 24, 25, 6, 27, 28, 18, 30, 3, 32, 3, 14, 35, 36, 21, 18, 9, 40, 4, 42, 12, 4, 45, 12, 28, 48, 49, 50, 21, 12, 15, 54, 5, 56, 27, 36, 45, 60, 6, 6, 63, 64, 15, 6, 42, 28, 18, 70, 7, 72, 21, 42, 75, 36, 7, 18

Offset: 0

Jonathan Vos Post, May 06 2006

See also: A007954 Product of digits of n. See also: A118503 Sum of digits of prime factors of n, with multiplicity.

			a(22) = 2 because 22 = 2 * 11 and the digital product of 2 * the digital product of 11 = 2 * ! * 1 = 2.
a(121) = 1 because 121 = 11^2 = 11 * 11, multiplying the digits of the prime factors with multiplicity gives A007954(11) +A007954(11) = 1 * 1 = 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001221, A001222, A007953, A007954, A095402, A118503.

Table[Times @@ Flatten@ Map[IntegerDigits, Table[#1, {#2}] & @@@ FactorInteger@ n], {n, 0, 78}] (* Michael De Vlieger, Jun 16 2016 *)

\\ here b(n) is A007954.
b(n)={my(v=digits(n)); prod(i=1, #v, v[i])}
a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); b(p)^e)} \\ Andrew Howroyd, Jul 23 2018

Completely multiplicative with a(p) = A007954(p) for prime p.

a(36) corrected by Giovanni Resta, Jun 16 2016

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A195191 Smallest n-Smith number.

Original entry on oeis.org

32, 402, 2401, 2030, 10112, 10, 200, 10200, 10010, 100200, 1000110, 1000200, 100, 20000, 10200000, 1001000, 100200000, 1000110000, 1000200000, 1000, 2000000, 10200000000, 100100000, 100200000000, 1000110000000, 1000000000100, 10000, 200000000

Offset: 2

Kausthub Gudipati, Sep 11 2011

The smallest number for which the sum of the digits of its prime factors equals n multiplied by the sum of its digits.

			The first term of A104390, the first term of A104391, the first term of A103125 etc.

Shyam Sunder Gupta, Smith numbers

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc:
A118503 := proc(n) a := 0 ; for p in ifactors(n)[2] do a := a+ op(2,p)*A007953(op(1,p)) ; end do: a ; end proc:
A195191 := proc(n) for k from 1 do if A118503(k) = n*A007953(k) then return k; end if; end do: end proc: # R. J. Mathar, Sep 14 2011

a(12)-a(29) from Donovan Johnson, Sep 15 2011

A118585 Sum of squares of digits of prime factors of n, with multiplicity.

Original entry on oeis.org

0, 4, 9, 8, 25, 13, 49, 12, 18, 29, 2, 17, 10, 53, 34, 16, 50, 22, 82, 33, 58, 6, 13, 21, 50, 14, 27, 57, 85, 38, 10, 20, 11, 54, 74, 26, 58, 86, 19, 37, 17, 62, 25, 10, 43, 17, 65, 25, 98, 54, 59, 18, 34, 31, 27, 61, 91, 89, 106, 42

Offset: 1

Jonathan Vos Post, May 07 2006

Differs from A067666 if any prime factor exceeds 1 digit. Fixed points include 16, 27. See also: A067666 Sum of squares of prime factors of n (counted with multiplicity). See also: A003132 Sum of squares of digits of n. See also: A118503 Sum of digits of prime factors of n, with multiplicity.

			a(22) = 6 because 22 = 2 * 11 and the sum of squares of digits of prime factors is 2^2 + 1^2 + 1^2.
a(121) = 4 because 121 = 11^2 = 11 * 11, so 1^2 + 1^2 + 1^2 + 1^2 = 4.

Cf. A001221, A001222, A003132, A005063, A007953, A007954, A067666, A095402, A118503.

Join[{0},Table[Total[Flatten[IntegerDigits/@(Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]])]^2],{n,2,60}]] (* Harvey P. Dale, Nov 17 2022 *)

a(n) = SUM[i=1..k] (e_i)*A003132(p_i) where prime decomposition of n = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k).

a(0) removed by Andrey Zabolotskiy, Jun 08 2024

A385932 Composite numbers m such that the sum of digits of m divides the sum of digits of prime factors of m (counted with multiplicity).

Original entry on oeis.org

4, 10, 22, 27, 32, 42, 58, 60, 70, 85, 94, 100, 104, 121, 152, 166, 200, 202, 231, 265, 274, 315, 316, 319, 322, 330, 342, 346, 355, 361, 378, 382, 391, 402, 406, 430, 438, 450, 454, 483, 510, 517, 526, 535, 540, 562, 576, 588, 602, 610, 612, 627, 632, 634, 636, 645, 648

Offset: 1

Stefano Spezia, Jul 12 2025

Equivalently, numbers m such that A007953(m) | A118503(m).

Union of the k-Smith numbers for all the positive integers k.

			10 = 2*5 is a term since it is a 7-Smith number: 1 + 0 = 1 | 7 = 2 + 5;
60 = 2^2*3*5 is term since it is a 2-Smith number: 6 + 0 = 6 | 12 = 2 + 2 + 3 + 5;
382 = 2*191 is a term since it is a Smith number (k=1): 3 + 8 + 2 = 13 | 13 = 2 + 1 + 9 + 1;
635 = 5*127 is not a term since 6 + 3 + 5 = 14 does not divide 15 = 5 + 1 + 2 + 7.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.1.14 and 3.1.16 on pages 84-85.

Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).

Supersequence of A006753, A103125, A103126, A104390, A104391.

Cf. A002808, A007953, A118503.

fQ[n_]:=!PrimeQ[n] && n>1 && Divisible[Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]] & /@ FactorInteger[n]]], Total[IntegerDigits[n]]]; Select[ Range@ 650, fQ]

cp's OEIS Frontend

A134599 Sum of digital sums (base 3) of the prime factors of n.

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A289142 Numbers whose sum of prime factors (taken with multiplicity) is divisible by 3.

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A102216 2-Suzanne numbers.

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A102217 3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

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A177927 3-Monica numbers.

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A102218 2-Monica numbers.

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A118541 Product of digits of prime factors of n, with multiplicity.

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