A000026 -id:A000026 - cp's OEIS frontend

A230542 Numbers that cannot be divided by their multiplicative projection (A000026).

8, 9, 18, 24, 25, 32, 36, 40, 45, 49, 50, 56, 63, 64, 75, 81, 88, 90, 96, 98, 99, 100, 104, 117, 120, 121, 125, 126, 128, 136, 147, 150, 152, 153, 160, 162, 168, 169, 171, 175, 180, 184, 192, 196, 198, 200, 207, 224, 225, 232, 234, 242, 243, 245, 248, 250, 252

Offset: 1

Paolo P. Lava, Oct 24 2013

nonn

Subset of A159836. For n < 1000 only 6 terms (144, 216, 256, 400, 576, 648, 720 and 768) are missing with respect to A159836.

			Prime factors of 88 are 2^3 and 11; its multiplicative projection is 2*3*11 = 66 and 66 does not divide 88. Therefore 88 is in the sequence.
Prime factors of 108 are 2^2 and 3^3; its multiplicative projection is 2*2*3*3 = 36 and 108 / 36 = 3. therefore 108 is not in the sequence.

Paolo P. Lava, Table of n, a(n) for n = 1..5000

Cf. A000026, A159836.

with(numtheory); P:=proc(q) local a,k,n;
for n from 1 to q do a:=ifactors(n)[2];  b:=mul(a[k][2]*a[k][1],k=1..nops(a));
if not type(n/b,integer) then print(n); fi; od; end: P(10^6);

selQ[n_] := !Divisible[n, Times @@ Flatten[FactorInteger[n]]];
Select[Range[1000], selQ] (* Jean-François Alcover, Apr 08 2020 *)

ok(n)={my(f=factor(n)); n % prod(k=1, matsize(f)[1], f[k, 1]*f[k, 2]) <> 0} \\ Andrew Howroyd, Feb 26 2018

Missing a(85) in b-file inserted by Andrew Howroyd, Feb 26 2018

A109444 Cumulative sum of mosaic numbers (A000026).

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 34, 40, 50, 61, 73, 86, 100, 115, 123, 140, 152, 171, 191, 212, 234, 257, 275, 285, 311, 320, 348, 377, 407, 438, 448, 481, 515, 550, 574, 611, 649, 688, 718, 759, 801, 844, 888, 918, 964, 1011, 1035, 1049, 1069, 1120, 1172, 1225

Offset: 1

Jonathan Vos Post, Aug 26 2005

Integers in this sequence which are themselves primes include a(2) = 3, a(11) = 61, a(12) = 73, a(20) = 191, a(23) = 257, a(26) = 311, a(62) = 1697, a(67) = 1949, a(68) = 2017. Integers in this sequence which are perfect powers greater than 1 of composites include a(14) = 100, a(53) = 1225.

Cf. A000026.

Accumulate[Array[Times@@Flatten[FactorInteger[#]]&,60]] (* Harvey P. Dale, Sep 16 2018 *)

A000026(Product (p_j^k_j)) = Product (p_j * k_j).

a(n) = Sum_{i=1..n} A000026(i).

A008474 If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j).

Original entry on oeis.org

0, 3, 4, 4, 6, 7, 8, 5, 5, 9, 12, 8, 14, 11, 10, 6, 18, 8, 20, 10, 12, 15, 24, 9, 7, 17, 6, 12, 30, 13, 32, 7, 16, 21, 14, 9, 38, 23, 18, 11, 42, 15, 44, 16, 11, 27, 48, 10, 9, 10, 22, 18, 54, 9, 18, 13, 24, 33, 60, 14, 62, 35, 13, 8, 20, 19, 68, 22, 28, 17, 72, 10, 74, 41, 11, 24, 20, 21

Offset: 1

Olivier Gérard

nonn

a(1) = 0 by convention, but could equally be taken to be 1 or 2.
Since only the primes p_j with nonzero exponents k_j in the factorization of n are considered in Sum (p_j + k_j), to the empty product (1) should correspond the empty sum (0). a(1) = 0 is thus the most natural choice. - Daniel Forgues, Apr 06 2010
Conjecture: for m > 4, by iterating the map m -> A008474(m) one always reaches 5 [tested up to m = 320000]. - Ivan N. Ianakiev, Nov 10 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.
Eric Weisstein's World of Mathematics, Prime Factorization

Cf. A107737, A107738, A000026, A027748, A124010, A250030.

a008474 n = sum $ zipWith (+) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Feb 11 2012, Aug 27 2011

A008474 := proc(n) local e,j; e := ifactors(n)[2]:
add(e[j][1]+e[j][2],j=1..nops(e)) end:
seq (A008474(n), n=1..60);
# Peter Luschny, Jan 17 2011

A008474[n_]:=Plus@@Flatten[FactorInteger[n]]; Table[A008474[n], {n, 200}] (* Zak Seidov, May 23 2005 *)

{for(k=1, 79,
M=factor(k); smt =0;
for(i=1, matsize(M)[1], for(j=1, matsize(M)[2], smt=smt+M[i,j]));
print1(smt, ", "))} \\\ Douglas Latimer, Apr 27 2012

a(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]+f[i,2]) \\ Charles R Greathouse IV, Jun 03 2015

from sympy import factorint
def a(n): return 0 if n == 1 else sum(p+k for p, k in factorint(n).items())
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Mar 28 2022

Additive with a(p^e) = p + e.

More terms from Zak Seidov, May 23 2005

A279513 Multiplicative with a(p^k) = p*a(k) for any prime p and k>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 6, 6, 10, 11, 12, 13, 14, 15, 8, 17, 12, 19, 20, 21, 22, 23, 18, 10, 26, 9, 28, 29, 30, 31, 10, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 44, 30, 46, 47, 24, 14, 20, 51, 52, 53, 18, 55, 42, 57, 58, 59, 60, 61, 62, 42, 12, 65, 66, 67, 68

Offset: 1

Rémy Sigrist, Dec 13 2016

To compute a(n): multiply (with multiplicity) each prime number appearing in the prime tower factorization of n (see A182318 for definition).
a(n) = n if n is squarefree.
a(n) <= A000026(n) for any n>0.
First differs from A000026 at n=256: a(256)=12 and A000026(256)=16.
If n = p_1 * p_2 * ... * p_k with p_1, p_2, ..., p_k primes, then a(p_1 ^ p_2 ^ ... ^ p_k) = n.

			a(6!) = a(2^(2^2)*3^2*5) = 2*2*2*3*2*5 = 240.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.33 Hall-Montgomery Constant, p. 207.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Richard Alan Gilman and Rainer Tschiersch, Problem 6660, The American Mathematical Monthly, Vol. 100, No. 3 (1993), pp. 296-298.

Cf. A000026, A182318, A279510.

a:= proc(n) option remember; `if`(n=1, 1,
      mul(i[1]*a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2020

a[n_] := a[n] = If[n==1, 1, Times @@ (#[[1]] a[#[[2]]]& /@ FactorInteger[n] )]; Array[a, 256] (* Jean-François Alcover, Mar 31 2017 *)

a(n) =  my (f=factor(n)); return (prod(i=1, #f~, f[i,1]*a(f[i,2])))

Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/p^2 + (p-1)*Sum_{k>=2} a(k)/p^(2*k)) = 0.8351076361... (Gilman and Tschiersch, 1993; Finch, 2003; the constant was calculated by Kevin Ford). - Amiram Eldar, Nov 04 2022

A078779 Union of S, 2S and 4S, where S = odd squarefree numbers (A056911).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 101

Offset: 1

Benoit Cloitre, Jan 11 2003

nonn

Numbers n such that the cyclic group Z_n is a DCI-group.
Numbers n such that A008475(n) = A001414(n).
A193551(a(n)) = A000026(a(n)) = a(n). - Reinhard Zumkeller, Aug 27 2011
Union of squarefree numbers and twice the squarefree numbers (A005117). - Reinhard Zumkeller, Feb 11 2012
The complement is A046790. - Omar E. Pol, Jun 11 2016

T. D. Noe, Table of n, a(n) for n = 1..7098
B. Alspach and M. Mishna, Enumeration of Cayley graphs and digraphs, Discr. Math., 256 (2002), 527-539.
M. Mishna, Home Page
M. Muzychuk, On Adam's conjecture for circulant graphs, Discr. Math. 167 (1997), 497-510.

Cf. A121176, A121684, A008475, A001414, A046790.

a078779 n = a078779_list !! (n-1)
a078779_list = m a005117_list $ map (* 2) a005117_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y     = x : m xs ys'
                           | x == y    = x : m xs ys
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Feb 11 2012, Aug 27 2011

is(n)=issquarefree(n/gcd(n,2)) \\ Charles R Greathouse IV, Nov 05 2017

a(n) = (Pi^2/7)*n + O(sqrt(n)). - Vladimir Shevelev, Jun 08 2016

Edited by N. J. A. Sloane, Sep 13 2006

A381201 a(n) is the product of the elements of the set of bases and exponents in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 6, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 24, 14, 10, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 12, 65, 66, 67, 34, 69, 70

Offset: 1

Paolo Xausa, Feb 16 2025

The prime factorization of 1 is the empty set, so a(1) = 1 by convention (empty product).

			a(12) = 6 because 12 = 2^2*3^1, the set of these bases and exponents is {1, 2, 3} and 1*2*3 = 6.
a(31500) = 210 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 1*2*3*5*7 = 210.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000026, A336965, A381202, A381203, A381204, A381205.

A381201[n_] := Times @@ Union[Flatten[FactorInteger[n]]];
Array[A381201, 100]

a(n) = my(f=factor(n)); vecprod(Vec(setunion(Set(f[,1]), Set(f[,2])))); \\ Michel Marcus, Feb 18 2025

A091967 a(n) is the n-th term of sequence A_n, ignoring the offset, or -1 if A_n has fewer than n terms.

Original entry on oeis.org

0, 2, 1, 0, 2, 3, 0, 6, 6, 4, 44, 1, 180, 42, 16, 1096, 7652, 13781, 8, 24000, 119779, 458561, 152116956851941670912, 1054535, -53, 26, 27, 59, 4806078, 2, 35792568, 3010349, 2387010102192469724605148123694256128, 2, 0, -53, 43, 0, -4097, 173, 37338, 111111111111111111111111111111111111111111, 30402457, 413927966

Offset: 1

Proposed by several people, including Jeff Burch and Michael Joseph Halm

sign

This version ignores the offset of A_n and just counts from the beginning of the terms shown in the OEIS entry.
Thus a(8) = 6 because A_8 begins 1,1,2,2,3,4,5,6,... [even though A_8(8) is really 7].
The value a(n) = -1 could arise in two different ways, but it will be easy to decide which. - N. J. A. Sloane, Nov 27 2016
From M. F. Hasler, Sep 22 2013: (Start)
The value of a(91967) can be chosen at will.
Note that this sequence may change if the initial terms in A_n are altered, which does happen from time to time, usually because of the addition of an initial term.
After a(47), currently unknown, the sequence continues with a(48) = A48(47) = 1497207322929, a(49) = A49(48) = unknown, a(50) = A50(49) = unknown, a(51) = A51(50) = 1125899906842625, a(52)=97, a(53) = -1 (since A000053 has only 29 terms). (End)
a(58) = A000058(57) = 138752...985443 (29334988649136302 digits) is too large to include in the b-file. - Pontus von Brömssen, May 21 2022

			a(1) = 0 since A000001 has offset 0, and begins with A000001(0) = 0.
a(26) = 26 because the 26th term of A000026 = 26.

Pontus von Brömssen, Table of n, a(n) for n = 1..57 (terms n = 1..46 from M. F. Hasler)
OEIS, List of finite sequences, sorted by A-number.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences whose definition involves A_n (or An).

See A051070, A107357, A102288 for other versions.

Cf. A000001, A000002, A000003, A000004, A000005, A000006, A000007, A000008, A000009, A000010, A000011, A000012, A000013, A000014, A000015, etc.

Corrected and extended by Jud McCranie; further extended by N. J. A. Sloane and E. M. Rains, Dec 08 1998
Corrected and extended by N. J. A. Sloane, May 25 2005
a(26), a(36) and a(42) corrected by M. F. Hasler, Jan 30 2009
a(43) and a(44) added by Daniel Sterman, Nov 27 2016
a(1) corrected by N. J. A. Sloane, Nov 27 2016 at the suggestion of Daniel Sterman
Definition and comments changed by N. J. A. Sloane, Nov 27 2016

A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 12, 4, 6, 16, 20, 12, 24, 24, 32, 5, 32, 12, 36, 24, 48, 40, 44, 16, 12, 48, 8, 36, 56, 64, 60, 6, 80, 64, 96, 18, 72, 72, 96, 32, 80, 96, 84, 60, 48, 88, 92, 20, 18, 24, 128, 72, 104, 16, 160, 48, 144, 112, 116, 96, 120, 120, 72, 7, 192, 160, 132, 96, 176, 192

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(20) = a(2^2*5) = (2 - 1)*(2 + 1) * (5 - 1)*(1 + 1) = 24.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000010, A000026, A000040, A000302 (numbers n such that a(n) is odd), A001221, A006093, A007947, A023900, A034444, A059975, A062355, A173557, A304407, A304409, A304411, A304412.

a:= n-> mul((i[1]-1)*(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021

a[n_] := Times @@ ((#[[1]] - 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 70}]
Table[DivisorSigma[0, n] EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]], {n, 70}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*(e+1))} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*abs(A023900(n)) = A000005(n)*A173557(n) = A000005(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*phi(n) if n is a squarefree (A005117), where omega() = A001221 and phi() = A000010.

A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

Original entry on oeis.org

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]

a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A007947(n).
a(p^k) = p*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 2/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, Jun 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(p^s - 1))^2.
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...
and gamma is the Euler-Mascheroni constant A001620. (End)

A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 9, 8, 18, 12, 24, 14, 24, 24, 12, 18, 24, 20, 36, 32, 36, 24, 36, 12, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000 .
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.

Cf. A072691, A330596.

a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
a(p^k) = (p + 1)*k where p is a prime and k > 0.
a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - Amiram Eldar, Nov 30 2022

cp's OEIS Frontend

A230542 Numbers that cannot be divided by their multiplicative projection (A000026).

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A109444 Cumulative sum of mosaic numbers (A000026).

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A008474 If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j).

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A279513 Multiplicative with a(p^k) = p*a(k) for any prime p and k>0.

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A078779 Union of S, 2S and 4S, where S = odd squarefree numbers (A056911).

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A381201 a(n) is the product of the elements of the set of bases and exponents in the prime factorization of n.

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A091967 a(n) is the n-th term of sequence A_n, ignoring the offset, or -1 if A_n has fewer than n terms.

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