A102217 -id:A102217 - cp's OEIS frontend

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

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 *)

A118503 Sum of digits of prime factors of n, with multiplicity.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 2, 7, 4, 9, 8, 8, 8, 8, 10, 9, 10, 4, 5, 9, 10, 6, 9, 11, 11, 10, 4, 10, 5, 10, 12, 10, 10, 12, 7, 11, 5, 12, 7, 6, 11, 7, 11, 11, 14, 12, 11, 8, 8, 11, 7, 13, 13, 13, 14, 12, 7, 6, 13, 12, 9, 7, 13, 12, 8, 14, 8, 12, 10, 12, 13, 14, 9, 9, 16, 13

Offset: 1

Jonathan Vos Post, May 06 2006

This is to A095402 (Sum of digits of all distinct prime factors of n) as bigomega = A001222 is to omega = A001221. See also: A007953 Digital sum (i.e., sum of digits) of n.

			a(22) = 4 because 22 = 2 * 11 and the digital sum of 2 + the digital sum of 11 = 2 + 2 = 4.
a(121) = 4 because 121 = 11^2 = 11 * 11, summing the digits of the prime factors with multiplicity gives A007953(11) + A007953(11) = 2 + 2 = 4.
a(1000) = 21 because = 2^3 * 5^3 = 2 * 2 * 2 * 5 * 5 * 5 and 2 + 2 + 2 + 5 + 5 + 5 = 21, as opposed to A095402(1000) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)

Cf. A001221, A001222, A007953, A095402, A102217, A289142 (positions of multiples of 3's).

A118503 := proc(n) local a; a := 0 ; for p in ifactors(n)[2] do a := a+ op(2, p)*A007953(op(1, p)) ; end do: a ; end proc: # R. J. Mathar, Sep 14 2011

sdpf[n_]:=Total[Flatten[IntegerDigits/@Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[n]]]]; Join[{0},Array[sdpf,100,2]] (* Harvey P. Dale, Sep 19 2013 *)

A118503(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1])); }; \\ Antti Karttunen, Jun 08 2024

a(n) = Sum_{i=1..k} (e_i)*A007953(p_i) where prime decomposition of n = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k).

a(0) removed by Joerg Arndt at the suggestion of 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 *)

A373479 Numbers k such that A001414(k) and A003415(k) are both multiples of 3, but A083345(k) is not, where A001414 is the sum of prime factors with repetition, A003415 is the arithmetic derivative, and A083345(n) = A003415(n)/gcd(n,A003415(n)).

Original entry on oeis.org

9, 27, 72, 81, 126, 180, 216, 234, 243, 315, 342, 378, 396, 450, 540, 558, 576, 585, 612, 648, 666, 693, 702, 729, 774, 828, 855, 945, 990, 1008, 1026, 1044, 1071, 1098, 1125, 1134, 1188, 1206, 1287, 1314, 1350, 1395, 1422, 1440, 1449, 1476, 1530, 1620, 1665, 1674, 1692, 1728, 1746, 1755, 1764, 1827, 1836, 1854

Offset: 1

Antti Karttunen, Jun 07 2024

nonn

All terms are multiples of 9.

Not equal to 9*A289142, nor (after the initial term 9), equal to 3*A102217, although most of the terms are.

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

Setwise difference A373478 \ A373475.
Subsequence of A008591.
Cf. A001414, A003415, A083345, A102217, A289142.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
isA373479(n) = (!(A001414(n)%3) && !(A003415(n)%3) && (A083345(n)%3));

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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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A118503 Sum of digits of prime factors of n, with multiplicity.

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A177927 3-Monica numbers.

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A373479 Numbers k such that A001414(k) and A003415(k) are both multiples of 3, but A083345(k) is not, where A001414 is the sum of prime factors with repetition, A003415 is the arithmetic derivative, and A083345(n) = A003415(n)/gcd(n,A003415(n)).

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