A008578 -id:A008578 - cp's OEIS frontend

A359587 Fully multiplicative with a(p) = A008578(1+A329697(p)).

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 3, 3, 4, 1, 2, 4, 5, 2, 6, 3, 5, 2, 4, 3, 8, 3, 5, 4, 5, 1, 6, 2, 6, 4, 5, 5, 6, 2, 3, 6, 7, 3, 8, 5, 7, 2, 9, 4, 4, 3, 5, 8, 6, 3, 10, 5, 7, 4, 5, 5, 12, 1, 6, 6, 7, 2, 10, 6, 7, 4, 5, 5, 8, 5, 9, 6, 7, 2, 16, 3, 5, 6, 4, 7, 10, 3, 5, 8, 9, 5, 10, 7, 10, 2, 3, 9, 12, 4, 5, 4, 5, 3, 12

Offset: 1

Antti Karttunen, Jan 08 2023

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A001222, A008578, A056239, A087436, A318995, A329697, A336396, A336466.

A008578(n) = if(1==n,1,prime(n-1));
A329697(n) = if(!bitand(n, n-1), 0, 1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A359587(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = A008578(1+A329697(f[i, 1]))); factorback(f); };

For n >= 1: (Start)
a(A000265(n)) = a(2*n) = a(n).
A001222(a(n)) = A087436(n),
A056239(a(n)) = A329697(n),
A318995(a(n)) = A336396(n) = A329697(A336466(n)).
(End)

A362463 Array of numbers read by upward antidiagonals: leading row lists the primes as they were in the 19th century (A008578); the following rows give absolute values of differences of previous row.

Original entry on oeis.org

1, 1, 2, 0, 1, 3, 1, 1, 2, 5, 0, 1, 0, 2, 7, 1, 1, 2, 2, 4, 11, 0, 1, 2, 0, 2, 2, 13, 1, 1, 2, 0, 0, 2, 4, 17, 0, 1, 2, 0, 0, 0, 2, 2, 19, 1, 1, 2, 0, 0, 0, 0, 2, 4, 23, 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29, 1, 1, 0, 2, 2, 2, 2, 2, 2, 4, 2, 31, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 4, 6, 37, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2, 4, 41

Offset: 1

N. J. A. Sloane, May 08 2023

The Gilbreath transform (cf. A362451) of A008578.

Analogous to A036262. The Gilbreath conjecture is that the initial terms of the rows are 1,(1,0)* = A135528.

			The array begins:
  1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67
  1 1 2 2 4  2  4  2  4  6  2  6  4  2  4  6  6  2  6  4
  0 1 0 2 2  2  2  2  2  4  4  2  2  2  2  0  4  4  2  2
  1 1 2 0 0  0  0  0  2  0  2  0  0  0  2  4  0  2  0  2
  0 1 2 0 0  0  0  2  2  2  2  0  0  2  2  4  2  2  2  0
  1 1 2 0 0  0  2  0  0  0  2  0  2  0  2  2  0  0  2  2
  0 1 2 0 0  2  2  0  0  2  2  2  2  2  0  2  0  2  0  0
  1 1 2 0 2  0  2  0  2  0  0  0  0  2  2  2  2  2  0  0
  0 1 2 2 2  2  2  2  2  0  0  0  2  0  0  0  0  2  0  0
  1 1 0 0 0  0  0  0  2  0  0  2  2  0  0  0  2  2  0  0
The first few antidiagonals are:
  1,
  1, 2,
  0, 1, 3,
  1, 1, 2, 5,
  0, 1, 0, 2, 7,
  1, 1, 2, 2, 4, 11,
  0, 1, 2, 0, 2, 2, 13,
  1, 1, 2, 0, 0, 2, 4, 17,
  0, 1, 2, 0, 0, 0, 2, 2, 19,
  1, 1, 2, 0, 0, 0, 0, 2, 4, 23,
  0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29,

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Cf. A008578, A036262, A135528, A362451.

A362463[dmax_]:=With[{d=Reverse[NestList[Abs[Differences[#]]&,Join[{1},Prime[Range[dmax-1]]],dmax-1]]},Array[Diagonal[d,#]&,dmax,1-dmax]];A362463[20] (* Generates 20 antidiagonals *) (* Paolo Xausa, May 08 2023 *)

A363627 a(n) = greatest product < n of some subset of the divisors of n, or if n is in A008578 then a(n) = n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 8, 13, 7, 5, 8, 17, 12, 19, 10, 7, 11, 23, 18, 5, 13, 9, 14, 29, 20, 31, 16, 11, 17, 7, 27, 37, 19, 13, 32, 41, 36, 43, 22, 27, 23, 47, 36, 7, 25, 17, 26, 53, 36, 11, 32

Offset: 1

Denis Ivanov, Jun 12 2023

nonn

a(n) = n <=> n in A008578.

For composite n, a(n) < n < A363501(n) and where both bounds are products of divisors of n and as tight as possible.

			n = 4; divisors: [1,2,4]; subsets: [[], [1], [2], [4], [1, 2], [1, 4], [2, 4], [1, 2, 4]]; products: [1, 1, 2, 4, 2, 4, 8, 8]; the maximal product that is lesser than 4 is 2, so a(4) = 2.

Cf. A008578, A363501.

If[PrimeQ@n || n == 1, n,
  Last@Select[Union[Times @@@ Subsets[Divisors@n]], # < n &]];

a(n) = my(d=divisors(n), nb = #d, m=1); forsubset(nb, s, my(p=vecprod(vector(#s, k, d[s[k]]))); if (p1, m, n); \\ Michel Marcus, Jun 17 2023

A365708 a(n) for n >= 2 is the smallest term of A008578 m dividing n such that n/m is not a square, a(1) = 2.

Original entry on oeis.org

2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

David James Sycamore, Sep 16 2023

nonn

In other words, a(n) is the least prime divisor of n for n = m^2, (m > 1), otherwise a(n) = 1. (Every term is 1 or a prime.) Compare with A007913.

			a(1) = 2, since 2 is the smallest m such that 1/m is not a square.
a(2) = 1, since 2/1 = 2 is not a square (likewise for all primes p, a(p) = 1).
a(4) = 2, since 4/1 = 4, a square, but 4/2 = 2, not a square, and so on.
If n is a square with lpf(n) = p, then a(n) = p, the least m such that n/m is not a square.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007913, A008578, A020639.

{2}~Join~Table[SelectFirst[Divisors[n], ! IntegerQ@ Sqrt[n/#] &], {n, 2, 120}] (* Michael De Vlieger, Sep 17 2023 *)

a(n) = fordiv(n, d, if (!issquare(n/d), return(d))); n+1; \\ Michel Marcus, Sep 17 2023

A029896 Convolution of Thue-Morse sequence A001285 with A008578 = {1, primes}.

Original entry on oeis.org

1, 4, 9, 16, 27, 43, 63, 89, 118, 153, 193, 243, 297, 358, 427, 492, 571, 661, 755, 853, 961, 1074, 1181, 1312, 1448, 1586, 1747, 1900, 2057, 2221, 2389, 2575, 2780, 2993, 3189, 3417, 3641, 3870, 4127, 4384

Offset: 0

N. J. A. Sloane

nonn

A094742 Duplicate of A008578.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83

Offset: 1

dead

A131690 a(n) = Product_{k=1..n} A008578(k)^A008578(n-k+1).

Original entry on oeis.org

1, 2, 12, 360, 151200, 2095632000, 7551819475200000, 7286477990937425280000000, 16326289449604557795871699200000000000, 48235535472088469901966394717904245153920000000000000, 1927704301314417844667587261525561805756528196513768633600000000000000000

Offset: 1

Darse Billings, Sep 14 2007

nonn

Exponents of the prime factorization are the primes in reverse order. Similar to A087315, but where the largest prime factor has an exponent of one instead of two (and 1^n is understood to be the first term).

			a(5) = 1^7 * 2^5 * 3^3 * 5^2 * 7^1 = 151200.

Cf. A008578, A087315, A076265, A076265.

s={};Do[p1=Join[{1},Prime[Range[n-1]]];AppendTo[s,Product[p1[[k]]^p1[[n-k+1]],{k,n}]],{n,11}];s (* James C. McMahon, Feb 27 2025 *)

a(n) = Product_{k=1..n} A008578(k)^A008578(n-k+1) where A008578 is the sequence of primes prepended with 1.

a(11) from James C. McMahon, Feb 27 2025

A159900 Concatenation of the first n elements of A008578.

Original entry on oeis.org

1, 12, 123, 1235, 12357, 1235711, 123571113, 12357111317, 1235711131719, 123571113171923, 12357111317192329, 1235711131719232931, 123571113171923293137, 12357111317192329313741

Offset: 1

Jaroslav Krizek, Apr 25 2009

Cf. A158611.

Prepend 1 to the terms of A019518(n) - R. J. Mathar, Apr 28 2009.

Slightly edited by R. J. Mathar, Apr 28 2009

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A162988 n appears A008578(n) times.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Jaroslav Krizek, Jul 19 2009

nonn

a(n) ~ 2 sqrt(n / log n)

Formula and more terms from Charles R Greathouse IV, Nov 02 2009

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A175248 Noncomposites (A008578) with noncomposite (A008578) subscripts.

Original entry on oeis.org

1, 2, 3, 7, 13, 29, 37, 53, 61, 79, 107, 113, 151, 173, 181, 199, 239, 271, 281, 317, 349, 359, 397, 421, 457, 503, 541, 557, 577, 593, 613, 701, 733, 769, 787, 857, 863, 911, 953, 983, 1021, 1061, 1069, 1151, 1163, 1193, 1213, 1291, 1399, 1429, 1439, 1459

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

Noncomposite numbers of order 2.

a(n) = noncomposite(noncomposite(n)) = A008578(A008578(n)). a(1) = 1, a(n) = A055003(n) for n >=2. a(n) U A175249(n+1) = A008578 for n >= 1.

			a(5) = 13 because a(5) = q(q(5)) = q(7) = 13, q = noncomposite.

Cf. A006450, A102615.

a(n)=A008578(A008578(n)).

More terms from Juri-Stepan Gerasimov, Apr 15 2010

Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A359587 Fully multiplicative with a(p) = A008578(1+A329697(p)).

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A362463 Array of numbers read by upward antidiagonals: leading row lists the primes as they were in the 19th century (A008578); the following rows give absolute values of differences of previous row.

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A363627 a(n) = greatest product < n of some subset of the divisors of n, or if n is in A008578 then a(n) = n.

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A365708 a(n) for n >= 2 is the smallest term of A008578 m dividing n such that n/m is not a square, a(1) = 2.

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A029896 Convolution of Thue-Morse sequence A001285 with A008578 = {1, primes}.

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A094742 Duplicate of A008578.

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A131690 a(n) = Product_{k=1..n} A008578(k)^A008578(n-k+1).

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A159900 Concatenation of the first n elements of A008578.

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A162988 n appears A008578(n) times.

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A175248 Noncomposites (A008578) with noncomposite (A008578) subscripts.

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