A257993 -id:A257993 - cp's OEIS frontend

A328762 Numbers n for which 2 < A257993(A276086(A276086(n))) < A257993(n), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

210, 1470, 5250, 6510, 7140, 8400, 9450, 10710, 14490, 15750, 16380, 17640, 18690, 19950, 23730, 24990, 25620, 26880, 27930, 29190, 30030, 31290, 32340, 33600, 37380, 38640, 39270, 40530, 41580, 42840, 46620, 47880, 48510, 49770, 50820, 52080, 55860, 57120, 57750, 59010, 60270, 61530, 63420, 65730, 69510, 70770, 72660, 74970

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

All terms are multiples of 5 (and thus of 30), because when applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, implying that the primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), A276086 will yield a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401" (with the least significant zero either in position 2 or 3), thus A328578(n) = A257993(A276086(A276086(n))) cannot simultaneously be larger than 2 and smaller than A257993(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Setwise difference A328587 \ A328632.

Cf. A257993, A276086, A328578, A328589.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328762(n) = { my(u=A328578(n)); ((u > 2) && (u < A257993(n))); };

A276094 a(n) = n modulo A002110(A257993(n)), a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 2, 1, 4, 1, 18, 1, 2, 1, 4, 1, 24, 1, 2, 1, 4, 1, 30, 1, 2, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 2, 1, 4, 1, 18, 1, 2, 1, 4, 1, 24, 1, 2, 1, 4, 1, 60, 1, 2, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 2, 1, 4, 1, 18, 1, 2, 1, 4, 1, 24, 1, 2, 1, 4, 1, 90, 1, 2, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 2, 1, 4, 1, 18, 1

Offset: 0

Antti Karttunen, Aug 22 2016

Antti Karttunen, Table of n, a(n) for n = 0..2310

Cf. A000040, A002110, A257993, A276084, A276088, A276093.

{0}~Join~Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; Mod[n, Product[Prime@ i, {i, k}]], {n, 79}] (* Michael De Vlieger, Jun 22 2017 *)

from sympy import nextprime, primepi, primorial
def a053669(n):
    p = 2
    while True:
        if n%p: return p
        else: p=nextprime(p)
def a257993(n): return primepi(a053669(n))
def a002110(n): return 1 if n<1 else primorial(n)
def a(n): return 0 if n==0 else n%a002110(a257993(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

a(0) = 0, and for n >= 1, a(n) = n modulo A002110(A257993(n)).
or a(n) = A276088(n) * A002110(A276084(n)).
Other identities. For all n >= 0:
a(n) = n - A276093(n).

A328634 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 4.

Original entry on oeis.org

4, 8, 16, 32, 64, 152, 184, 210, 242, 274, 362, 394, 440, 448, 452, 484, 572, 604, 634, 638, 646, 662, 694, 782, 814, 872, 904, 992, 1024, 1070, 1078, 1082, 1114, 1202, 1234, 1264, 1268, 1276, 1292, 1324, 1412, 1444, 1470, 1502, 1534, 1622, 1654, 1700, 1708, 1712, 1744, 1832, 1864, 1894, 1898, 1906, 1922, 1954, 2042

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 30, but not of 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Row 4 of A328631.

Cf. A257993, A276086, A328578.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328634(n) = (4==A328578(n));

A328635 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 5.

Original entry on oeis.org

10, 14, 38, 58, 62, 112, 166, 176, 214, 218, 226, 240, 360, 650, 658, 660, 780, 844, 848, 856, 1080, 1200, 1280, 1288, 1474, 1478, 1486, 1500, 1620, 1910, 1918, 1920, 2040, 2104, 2108, 2116, 2312, 2314, 2318, 2386, 2396, 2440, 2450, 2504, 2520, 2546, 2580, 2700, 2732, 2744, 2752, 2950, 3000, 3120, 3176, 3362, 3374, 3382, 3420

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 210, but not of 11.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Row 5 of A328631.

Cf. A257993, A276086, A328578.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328635(n) = (5==A328578(n));

A328636 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 6.

Original entry on oeis.org

20, 22, 26, 40, 44, 46, 68, 70, 86, 92, 220, 224, 238, 248, 270, 272, 286, 356, 370, 424, 428, 500, 538, 544, 584, 622, 630, 682, 728, 766, 836, 896, 910, 934, 980, 1018, 1124, 1162, 1208, 1230, 1232, 1246, 1306, 1376, 1390, 1460, 1520, 1558, 1604, 1642, 1706, 1748, 1786, 1856, 1870, 1930, 2000, 2038, 2084, 2144, 2182, 2228, 2266

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 2310, but not of 13.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Row 6 of A328631.

Cf. A257993, A276086, A328578.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328636(n) = (6==A328578(n));

A328761 The least integer k >= 0 for which A328578(k) = A257993(A276086(A276086(k))) = n.

Original entry on oeis.org

1, 0, 2, 4, 10, 20, 28, 82, 116, 148, 208, 418, 838, 1048, 1466, 1858, 2276, 4612, 9028, 11546, 16162, 18478, 23098, 27688, 30028, 90086, 147838, 180178, 240232, 297988, 330328, 390358, 450238, 480478, 1021016, 2039728, 3033028, 3573566, 4594558, 5105098, 6606598, 7147136, 8168152, 8678668, 9699688, 29099062, 48498238, 58198138, 77597516, 96996896, 106696588, 126095968

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

The sequence seems to be monotonic from the second term onward. If this holds, then this sequence (after the initial 1) gives also the positions of records in A328578. See also the scatterplot of A328578.

Index entries for sequences related to primorial base

Column 1 of A328631.

Cf. A257993, A276086, A328578.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
A328761(n) = for(k=0,oo,if(A328578(k)==n,return(k)));

A328761list(up_to) = { my(v=vector(up_to), k); for(n=0,oo,k=A328578(n); if(k>#v, return(v)); if(!v[k],v[k] = n; print("Found ",k, " at n=",n)); if(k>=#v,return(v))); };
v328761 = A328761list(52);
A328761(n) = if(2==n,0,v328761[n]);

A286382 Compound filter: a(n) = P(A257993(n), A278226(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 16, 12, 67, 9, 16, 23, 436, 80, 1771, 18, 67, 80, 1771, 668, 16111, 48, 277, 302, 7141, 2630, 64621, 156, 1129, 1178, 28681, 10442, 258841, 14, 16, 23, 436, 80, 1771, 31, 436, 467, 21946, 1832, 87991, 94, 1771, 1832, 87991, 16292, 793171, 328, 7141, 7262, 352381, 64982, 3173941, 1228, 28681, 28922, 1410361, 259562, 12698281, 25, 67, 80, 1771, 668, 16111

Offset: 1

Antti Karttunen, May 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11550
Eric Weisstein's World of Mathematics, Pairing Function
Index entries for sequences related to primorial base

Cf. A000027, A257993, A278226, A286142.

Differs from A286381 for the first time at n=24, where a(24) = 156 while A286381(24) = 14.

(define (A286382 n) (* (/ 1 2) (+ (expt (+ (A257993 n) (A278226 n)) 2) (- (A257993 n)) (- (* 3 (A278226 n))) 2)))

a(n) = (1/2)*(2 + ((A257993(n)+A278226(n))^2) - A257993(n) - 3*A278226(n)).

A328591 Even bisection of A328590: a(n) = A328578(2n) - A257993(2n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to primorial base

Cf. A257993, A276086, A328578, A328590.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
A328590(n) = (A328578(n) - A257993(n));
A328591(n) = A328590(n+n);

a(n) = A328590(2n) = A328578(2n) - A257993(2n).

A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 625, 1250, 1875, 3750, 5625, 11250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 4375, 8750, 13125, 26250, 39375, 78750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450

Offset: 0

Antti Karttunen, Aug 21 2016

Prime product form of primorial base expansion of n.
Sequence is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019
From Antti Karttunen, Feb 18 2022: (Start)
The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.
Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.
This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)
From Bill McEachen, Oct 15 2022: (Start)
From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.
Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)
The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

			For n = 24, which has primorial base representation (see A049345) "400" as 24 = 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*6 + 0*2 + 0*1, thus a(24) = prime(3)^4 * prime(2)^0 * prime(1)^0 = 5^4 = 625.
For n = 35 = "1021" as 35 = 1*A002110(3) + 0*A002110(2) + 2*A002110(1) + 1*A002110(0) = 1*30 + 0*6 + 2*2 + 1*1, thus a(35) = prime(4)^1 * prime(2)^2 * prime(1) = 7 * 3*3 * 2 = 126.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Program in LODA-assembly
Antti Karttunen, Program in LODA-assembly [Cached copy]
Index entries for sequences related to primorial base

Cf. A276085 (a left inverse) and also A276087, A328403.
Cf. A000040, A001221, A001222, A002110, A020639, A049345, A053669, A055396, A057588, A071178, A143293, A257993, A267263, A276084, A276088, A276092, A276093, A276147, A276150, A276151, A276153, A276156, A283477, A324198 (= gcd(n, a(n))), A328584 (= lcm(n, a(n))), A324646, A324289, A328386, A328403, A328475, A328571, A328572, A328578, A328612, A328613, A328620, A328624, A328627, A328763, A328766, A328828, A328835, A328841, A328842, A328843, A328844, A329041, A324580 [= n*a(n)], A324895 (largest proper divisor of a(n)), A351252, A353486 (reduced modulo 4), A358840 (modulo 6), A353489, A353516.
Cf. A048103 (terms sorted into ascending order), A100716 (natural numbers not present in this sequence).
Cf. A278226 (associated filter-sequence), A286626 (and its rgs-version), A328477.
Cf. A328316 (iterates started from zero).
Cf. A327858, A327859, A327860, A327963, A328097, A328098, A328099, A328110, A328112, A328382 for various combinations with arithmetic derivative (A003415).
Cf. also A327167, A329037.
Cf. A019565 and A054842 for base-2 and base-10 analogs and A276076 for the analogous "factorial base exp-function", from which this differs for the first time at n=24, where a(24)=625 while A276076(24)=7.
Cf. A327969, A351088, A351458 for sequences with conjectures involving this sequence.

b = MixedRadix[Reverse@ Prime@ Range@ 12]; Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 51}] (* Michael De Vlieger, Aug 23 2016, Version 10.2 *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ f@ n], {n, 0, 73}] (* Michael De Vlieger, Aug 30 2016, Pre-Version 10 *)
a[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen's Sage code *)

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; }; \\ Antti Karttunen, May 12 2017

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; \\ (Better than above one, avoids unnecessary construction of primorials). - Antti Karttunen, Oct 14 2019

from sympy import prime
def a(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m # Indranil Ghosh, May 12 2017, after Antti Karttunen's PARI code

from sympy import nextprime
def a(n):
    m, p = 1, 2
    while n > 0:
        n, r = divmod(n, p)
        m *= p**r
        p = nextprime(p)
    return m
print([a(n) for n in range(74)])  # Peter Luschny, Apr 20 2024

def A276086(n):
    m=1
    i=1
    while n>0:
        p = sloane.A000040(i)
        m *= (p**(n%p))
        n = floor(n/p)
        i += 1
    return (m)
# Antti Karttunen, Oct 14 2019, after Indranil Ghosh's Python code above, and my own leaner PARI code from Oct 14 2019. This avoids unnecessary construction of primorials.

(define (A276086 n) (let loop ((n n) (t 1) (i 1)) (if (zero? n) t (let* ((p (A000040 i)) (d (modulo n p))) (loop (/ (- n d) p) (* t (expt p d)) (+ 1 i))))))

(definec (A276086 n) (if (zero? n) 1 (* (expt (A053669 n) (A276088 n)) (A276086 (A276093 n))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

(definec (A276086 n) (if (zero? n) 1 (* (A053669 n) (A276086 (- n (A002110 (A276084 n))))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

a(0) = 1; for n >= 1, a(n) = A053669(n) * a(A276151(n)) = A053669(n) * a(n-A002110(A276084(n))).
a(0) = 1; for n >= 1, a(n) = A053669(n)^A276088(n) * a(A276093(n)).
a(n) = A328841(a(n)) + A328842(a(n)) = A328843(n) + A328844(n).
a(n) = a(A328841(n)) * a(A328842(n)) = A328571(n) * A328572(n).
a(n) = A328475(n) * A328580(n) = A328476(n) + A328580(n).
a(A002110(n)) = A000040(n+1). [Maps primorials to primes]
a(A143293(n)) = A002110(n+1). [Maps partial sums of primorials to primorials]
a(A057588(n)) = A276092(n).
a(A276156(n)) = A019565(n).
a(A283477(n)) = A324289(n).
a(A003415(n)) = A327859(n).
Here the text in brackets shows how the right hand side sequence is a function of the primorial base expansion of n:
A001221(a(n)) = A267263(n). [Number of nonzero digits]
A001222(a(n)) = A276150(n). [Sum of digits]
A067029(a(n)) = A276088(n). [The least significant nonzero digit]
A071178(a(n)) = A276153(n). [The most significant digit]
A061395(a(n)) = A235224(n). [Number of significant digits]
A051903(a(n)) = A328114(n). [Largest digit]
A055396(a(n)) = A257993(n). [Number of trailing zeros + 1]
A257993(a(n)) = A328570(n). [Index of the least significant zero digit]
A079067(a(n)) = A328620(n). [Number of nonleading zeros]
A056169(a(n)) = A328614(n). [Number of 1-digits]
A056170(a(n)) = A328615(n). [Number of digits larger than 1]
A277885(a(n)) = A328828(n). [Index of the least significant digit > 1]
A134193(a(n)) = A329028(n). [The least missing nonzero digit]
A005361(a(n)) = A328581(n). [Product of nonzero digits]
A072411(a(n)) = A328582(n). [LCM of nonzero digits]
A001055(a(n)) = A317836(n). [Number of carry-free partitions of n in primorial base]
Various number theoretical functions applied:
A000005(a(n)) = A324655(n). [Number of divisors of a(n)]
A000203(a(n)) = A324653(n). [Sum of divisors of a(n)]
A000010(a(n)) = A324650(n). [Euler phi applied to a(n)]
A023900(a(n)) = A328583(n). [Dirichlet inverse of Euler phi applied to a(n)]
A069359(a(n)) = A329029(n). [Sum a(n)/p over primes p dividing a(n)]
A003415(a(n)) = A327860(n). [Arithmetic derivative of a(n)]
Other identities:
A276085(a(n)) = n.          [A276085 is a left inverse]
A020639(a(n)) = A053669(n). [The smallest prime not dividing n -> the smallest prime dividing n]
A046523(a(n)) = A278226(n). [Least number with the same prime signature as a(n)]
A246277(a(n)) = A329038(n).
A181819(a(n)) = A328835(n).
A053669(a(n)) = A326810(n), A326810(a(n)) = A328579(n).
A257993(a(n)) = A328570(n), A328570(a(n)) = A328578(n).
A328613(a(n)) = A328763(n), A328620(a(n)) = A328766(n).
A328828(a(n)) = A328829(n).
A053589(a(n)) = A328580(n). [Greatest primorial number which divides a(n)]
A276151(a(n)) = A328476(n). [... and that primorial subtracted from a(n)]
A111701(a(n)) = A328475(n).
A328114(a(n)) = A328389(n). [Greatest digit of primorial base expansion of a(n)]
A328389(a(n)) = A328394(n), A328394(a(n)) = A328398(n).
A235224(a(n)) = A328404(n), A328405(a(n)) = A328406(n).
a(A328625(n)) = A328624(n), a(A328626(n)) = A328627(n). ["Twisted" variants]
a(A108951(n)) = A324886(n).
a(n) mod n = A328386(n).
a(a(n)) = A276087(n), a(a(a(n))) = A328403(n). [2- and 3-fold applications]
a(2n+1) = 2 * a(2n). - Antti Karttunen, Feb 17 2022

Name edited and new link-formulas added by Antti Karttunen, Oct 29 2019

Name changed again by Antti Karttunen, Feb 05 2022

A055396 Smallest prime dividing n is a(n)-th prime (a(1)=0).

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 1, 14, 1, 2, 1, 15, 1, 4, 1, 2, 1, 16, 1, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 1, 19, 1, 2, 1, 20, 1, 21, 1, 2, 1, 4, 1, 22, 1, 2, 1, 23, 1, 3, 1, 2, 1, 24, 1, 4, 1, 2, 1, 3, 1

Offset: 1

Henry Bottomley, May 15 2000

nonn

Grundy numbers of the game in which you decrease n by a number prime to n, and the game ends when 1 is reached. - Eric M. Schmidt, Jul 21 2013
a(n) = the smallest part of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(21) = 2; indeed, the partition having Heinz number 21 = 3*7 is [2,4]. - Emeric Deutsch, Jun 04 2015
a(n) is the number of numbers whose largest proper divisor is n, i.e., for n>1, number of occurrences of n in A032742. - Stanislav Sykora, Nov 04 2016
For n > 1, a(n) gives the number of row where n occurs in arrays A083221 and A246278. - Antti Karttunen, Mar 07 2017

			a(15) = 2 because 15=3*5, 3<5 and 3 is the 2nd prime.

John H. Conway, On Numbers and Games, 2nd Edition, p. 129.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021.
Wikipedia, Nimber (explains the term Grundy number).
Index entries for sequences computed from indices in prime factorization
Index entries for sequences generated by sieves

Cf. A004280, A020639, A032742, A038179, A049084, A055399, A061395, A215366, A243055, A257993, A276086.

Cf. also A078898, A246277, A250469 and arrays A083221 and A246278.

a055396 = a049084 . a020639  -- Reinhard Zumkeller, Apr 05 2012

with(numtheory):
a:= n-> `if`(n=1, 0, pi(min(factorset(n)[]))):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 03 2013

a[1] = 0; a[n_] := PrimePi[ FactorInteger[n][[1, 1]] ]; Table[a[n], {n, 1, 96}](* Jean-François Alcover, Jun 11 2012 *)

a(n)=if(n==1,0,primepi(factor(n)[1,1])) \\ Charles R Greathouse IV, Apr 23 2015

from sympy import primepi, isprime, primefactors
def a049084(n): return primepi(n)*(1*isprime(n))
def a(n): return 0 if n==1 else a049084(min(primefactors(n))) # Indranil Ghosh, May 05 2017

From Reinhard Zumkeller, May 22 2003: (Start)
a(n) = A049084(A020639(n)).
A000040(a(n)) = A020639(n); a(n) <= A061395(n).
(End)
From Antti Karttunen, Mar 07 2017: (Start)
A243055(n) = A061395(n) - a(n).
a(A276086(n)) = A257993(n).
(End)

cp's OEIS Frontend

A328762 Numbers n for which 2 < A257993(A276086(A276086(n))) < A257993(n), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A276094 a(n) = n modulo A002110(A257993(n)), a(0) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A328634 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A328635 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A328636 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A328761 The least integer k >= 0 for which A328578(k) = A257993(A276086(A276086(k))) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A286382 Compound filter: a(n) = P(A257993(n), A278226(n)), where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A328591 Even bisection of A328590: a(n) = A328578(2n) - A257993(2n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

Views

Author

Keywords

Comments