A095402 -id:A095402 - cp's OEIS frontend

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

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

A095418 Excess of sum of all decimal digits of distinct prime factors for n-th repunit over corresponding digit-sum for repunit itself (which is n).

Original entry on oeis.org

-1, 0, 10, 0, 10, 20, 30, 17, 32, 26, 34, 35, 49, 53, 42, 51, 43, 74, 0, 56, 95, 77, 0, 81, 38, 94, 97, 106, 104, 80, 109, 123, 108, 96, 97, 132, 100, 65, 145, 136, 141, 184, 145, 173, 123, 99, 139, 172, 196, 120, 170, 176, 179, 213, 161, 169, 122, 201, 217, 184, 211, 216

Offset: 1

Labos Elemer, Jun 22 2004

			n=60: concatenated distinct-prime factor-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its digit sum is 244, so a(60) = 244 - 60 = 184.
The value of this excess-sum is zero if n=2,4,19,23.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A095402, A002275, A095370, A095413, A095414, A095415, A095416, A095417.

a[1] = -1; a[n_] := Total@ Flatten[IntegerDigits /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Array[a, 62] (* Giovanni Resta, Jul 19 2018 *)

a(n) = A095402(A002275(n)) - n = A095417(n) - n.

Data corrected by Giovanni Resta, Jul 19 2018

A095417 Sum of all decimal digits of distinct prime factors for n-th repunit.

Original entry on oeis.org

0, 2, 13, 4, 15, 26, 37, 25, 41, 36, 45, 47, 62, 67, 57, 67, 60, 92, 19, 76, 116, 99, 23, 105, 63, 120, 124, 134, 133, 110, 140, 155, 141, 130, 132, 168, 137, 103, 184, 176, 182, 226, 188, 217, 168, 145, 186, 220, 245, 170, 221, 228, 232, 267, 216, 225, 179, 259

Offset: 1

Labos Elemer, Jun 22 2004

			n=60: concatenated distinct-prime factor-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its digit sum=a(60)=244.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A095402, A002275, A095370, A095413-A095418.

a[1]=0; a[n_] := Total[ Flatten[ IntegerDigits /@ First /@ FactorInteger[(10^n - 1)/9]]]; Array[a, 60] (* Giovanni Resta, Jul 19 2018 *)

a(n) = A095402(A002275(n)).

Data corrected by Giovanni Resta, Jul 19 2018

A095403 Sum of digits of n minus the sum of digits of all distinct prime factors of n.

Original entry on oeis.org

1, 0, 0, 2, 0, 1, 0, 6, 6, -6, 0, -2, 0, -4, -2, 5, 0, 4, 0, -5, -7, 0, 0, 1, 2, 2, 6, 1, 0, -7, 0, 3, 1, -3, -4, 4, 0, -1, 5, -3, 0, -6, 0, 4, 1, 3, 0, 7, 6, -2, -5, 1, 0, 4, 3, 2, -1, 0, 0, -4, 0, 2, -1, 8, 2, 5, 0, 4, 7, -7, 0, 4, 0, -1, 4, 1, 5, 6, 0, 1, 6, 3, 0, 0, 0, 5, 1, 12, 0, -1, -1, 4, 5, 0, -1, 10, 0, 8, 13, -6, 0, -10, 0, -1, -9, -3, 0, 4, 0, -7, -10

Offset: 1

Labos Elemer, Jun 21 2004

			n=1000: A007953[1000]=1, prime set={2,5}, A095402[1000]=7, a[1000]=1-7=-6

Cf. A007953, A051351.

ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] sd[x_] :=Apply[Plus, IntegerDigits[x]] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] sdp[x_] :=Apply[Plus, tdp[x]] a=Table[sd[w], {w, 1, 150}];b=Table[sdp[w], {w, 1, 150}];b-a

a[n]=A007953[n]-A095402[n]

A095405 Numbers n such that Sum-of-digits-of-n = Sum-of-digits-of-all-distinct-prime-factors-of-n.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 29, 31, 37, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 79, 83, 84, 85, 89, 94, 97, 101, 103, 107, 109, 113, 127, 131, 136, 137, 139, 149, 151, 157, 160, 163, 166, 167, 173, 179, 181, 191, 193, 197, 199, 202, 211, 223, 227, 229, 233, 234

Offset: 1

Labos Elemer, Jun 21 2004

			n=85: digit sum=13, prime factor-digit sum=5+1+7=13, so 85 is here.

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

Cf. A007953, A051351, A095402, A095403, A095404, A095406.

ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] sd[x_] :=Apply[Plus, IntegerDigits[x]] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] sdp[x_] :=Apply[Plus, tdp[x]] a=Table[sd[w], {w, 1, 256}];b=Table[sdp[w], {w, 1, 150}];b-a; Flatten[Position[Sign[b-a], 0]]
Select[Range[2,300],Total[Flatten[IntegerDigits/@FactorInteger[#][[All, 1]]]] == Total[IntegerDigits[#]]&] (* Harvey P. Dale, Sep 29 2019 *)

Solutions to A007953[x]=A095402[x].

A141346 Sum of the distinct digits of the prime factors of n.

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jun 26 2008

Motivated by seeking an explanation for A080592. For n >= 2, 1 <= a(n) <= 45. For n >= 1, a(n) <= A095402(n).

			a(44) = 3 as 44 = 2^2 * 11 and the sum of the distinct digits of the prime factors is 1 + 2 (whereas A095402(44) = 4 = 1 + 1 + 2).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A095402, A080592, A008472.

Join[{0},Rest[Total[Union[Flatten[IntegerDigits/@Transpose[ FactorInteger[ #]][[1]]]]]&/@Range[90]]] (* Harvey P. Dale, Nov 30 2011 *)

from sympy import factorint
def a(n):
    s = set()
    for p in factorint(n): s |= set(str(p))
    return sum(map(int, s))
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Dec 12 2023

A095404 Numbers n such that Sum-of-digits-of-n > Sum-of-digits-of-all-distinct-prime-factors-of-n.

Original entry on oeis.org

1, 4, 6, 8, 9, 16, 18, 24, 25, 26, 27, 28, 32, 33, 36, 39, 44, 45, 46, 48, 49, 52, 54, 55, 56, 62, 64, 65, 66, 68, 69, 72, 75, 76, 77, 78, 80, 81, 82, 86, 87, 88, 92, 93, 96, 98, 99, 108, 117, 121, 124, 125, 128, 129, 135, 138, 143, 144, 147, 148, 155, 156, 159, 162, 164

Offset: 1

Labos Elemer, Jun 21 2004

			n=24: digit sum=6, prime factor-digit sum=2+3=5, so 24 is here;
n=153: digit sum=9, prime factor-digit sum=3+5+3=11>9, so 153 is here.

Cf. A007953, A051351, A095402, A095403, A095405, A095406.

ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] sd[x_] :=Apply[Plus, IntegerDigits[x]] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] sdp[x_] :=Apply[Plus, tdp[x]] a=Table[sd[w], {w, 1, 256}];b=Table[sdp[w], {w, 1, 150}];b-a; Flatten[Position[Sign[b-a], 1]]

Solutions to A007953[x]>A095402[x].

A095406 Numbers n such that Sum-of-digits-of-n < Sum-of-digits-of-all-distinct-prime-factors-of-n.

Original entry on oeis.org

10, 12, 14, 15, 20, 21, 30, 34, 35, 38, 40, 42, 50, 51, 57, 60, 63, 70, 74, 90, 91, 95, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 118, 119, 120, 122, 123, 126, 130, 132, 133, 134, 140, 141, 142, 145, 146, 150, 152, 153, 154, 158, 161, 170, 171, 174

Offset: 1

Labos Elemer, Jun 21 2004

			n=38: digit sum=11, prime factor-digit sum=2+1+9=12>11, so 38 is here;
n=10^j:digit sum=1, prime factor-digit sum=2+5=7?1. so 10^j is here for all j [this implies that the sequence is infinite].

Cf. A007953, A051351, A095402, A095403, A095404, A095405.

ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] sd[x_] :=Apply[Plus, IntegerDigits[x]] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] sdp[x_] :=Apply[Plus, tdp[x]] a=Table[sd[w], {w, 1, 256}];b=Table[sdp[w], {w, 1, 150}];b-a; Flatten[Position[Sign[b-a], -1]]
Select[Range[200],Total[IntegerDigits[#]]Harvey P. Dale, May 06 2012 *)

Solutions to A007953[x] < A095402[x].

A176707 Sum of digits of all distinct prime factors of n-th semiprime.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Apr 24 2010

			a(1)=2 because 1st semiprime=2*2 and 2=2; a(2)=5 because 2nd semiprime=2*3 and 2<3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A095402.

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc:
A176707 := proc(n) s := A001358(n) ; add( A007953(p), p = numtheory[factorset](s) ) ; end proc: seq(A176707(n),n=1..120) ; # R. J. Mathar, Apr 25 2010

from sympy import factorint
def aupton(terms):
  alst, m = [], 4
  while len(alst) < terms:
    f = factorint(m)
    if sum(f.values()) == 2: # semiprime
      alst.append(sum(sum(map(int, str(p))) for p in f.keys()))
    m += 1
  return alst
print(aupton(81)) # Michael S. Branicky, Feb 05 2021

a(13), a(34) etc. corrected by - R. J. Mathar, Apr 25 2010

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

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

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A095418 Excess of sum of all decimal digits of distinct prime factors for n-th repunit over corresponding digit-sum for repunit itself (which is n).

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A095417 Sum of all decimal digits of distinct prime factors for n-th repunit.

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A095403 Sum of digits of n minus the sum of digits of all distinct prime factors of n.

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A095405 Numbers n such that Sum-of-digits-of-n = Sum-of-digits-of-all-distinct-prime-factors-of-n.

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A141346 Sum of the distinct digits of the prime factors of n.

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A095404 Numbers n such that Sum-of-digits-of-n > Sum-of-digits-of-all-distinct-prime-factors-of-n.

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A095406 Numbers n such that Sum-of-digits-of-n < Sum-of-digits-of-all-distinct-prime-factors-of-n.

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