A257993 -id:A257993 - cp's OEIS frontend

A328578 Index of the least prime not dividing A276086(A276086(n)): a(n) = A257993(A276087(n)).

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

Offset: 0

Antti Karttunen, Oct 20 2019

Index of the least significant zero digit in the primorial base expansion of A276086(n), when the rightmost digit is in the position 1.

The scatter plot shows both regular looking as well as more chaotic regions. This can be more clearly seen in related A328579. See also A328839.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Rémy Sigrist, Colored logarithmic scatterplot of the ordinal transform of the sequence for n = 0..510510
Rémy Sigrist, Density plot of the sequence for n = 0..510510
Index entries for sequences related to primorial base

Cf. A000720, A055396, A276086, A276087, A328403, A328570, A328579 (the corresponding prime), A328590, A328763, A328766, A328839.
Cf. A328585 (where equal with A257993), A328587 (less than), A328588 (greater than).
Cf. A328761 (the first occurrence of each n).
Cf. also array A328631 and its rows A005408, A328632, A328633, A328634, A328635, A328636 (positions of terms 1 .. 6 in this sequence).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };
A328578(n) = A328570(A276086(n));

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A328578(n) = A257993(A276086(A276086(n)));

a(n) = A328570(A276086(n)) = A257993(A276087(n)) = A055396(A328403(n)).
a(n) = A000720(A328579(n)).
a(n) = A257993(n) + A328590(n).
a(n) = A055396(A328763(n)).
For all n >= 0, a(A328761(n)) = n.

A286142 Compound filter: a(n) = T(A257993(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 2, 12, 2, 31, 2, 38, 7, 23, 2, 94, 2, 23, 16, 138, 2, 94, 2, 80, 16, 23, 2, 328, 7, 23, 29, 80, 2, 532, 2, 530, 16, 23, 16, 706, 2, 23, 16, 302, 2, 499, 2, 80, 67, 23, 2, 1228, 7, 80, 16, 80, 2, 328, 16, 302, 16, 23, 2, 1957, 2, 23, 67, 2082, 16, 499, 2, 80, 16, 467, 2, 2704, 2, 23, 67, 80, 16, 499, 2, 1178, 121, 23, 2, 1894, 16, 23, 16, 302, 2, 1957, 16

Offset: 1

Antti Karttunen, May 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000027, A046523, A257993, A286144, A286160, A286161, A286162, A286163, A286164.

Differs from A286143 for the first time at n=24, where a(24) = 328, while A286143(24) = 355.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 - Boole[n == 1] & @@ {Module[{i = 1}, While[! CoprimeQ[Prime@ i, n], i++]; i], Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]}, {n, 92}] (* Michael De Vlieger, May 04 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A257993(n) = { for(i=1,n,if(n%prime(i),return(i))); }
A286142(n) = (1/2)*(2 + ((A257993(n)+A046523(n))^2) - A257993(n) - 3*A046523(n));
for(n=1, 10000, write("b286142.txt", n, " ", A286142(n)));

from sympy import factorint, prime, primepi, gcd
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a053669(n):
    x=1
    while True:
        if gcd(prime(x), n) == 1: return prime(x)
        else: x+=1
def a257993(n): return primepi(a053669(n))
def a(n): return T(a257993(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286142 n) (* (/ 1 2) (+ (expt (+ (A257993 n) (A046523 n)) 2) (- (A257993 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A257993(n)+A046523(n))^2) - A257993(n) - 3*A046523(n)).

A328585 Numbers n for which A257993(n) is equal to A257993(A276086(A276086(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.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 103, 105, 107, 108, 109, 111, 113, 115, 117, 119, 121, 123, 125, 126, 127, 129, 131, 133, 135, 137, 138, 139

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Numbers n for which A257993(n) is equal to A328578(n).

All odd numbers are included, as A257993(2n+1) = A328578(2n+1) = 1 for all n >= 0.

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

Union of A005408 (odd numbers) and A328586 (even terms).
Positions of zeros in A328590.
Cf. A257993, A276086, A328578, A328587, A328588.
Cf. also A328316, A328317.

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)));
isA328585(n) = (A328578(n) == A257993(n));

A328587 Numbers n for which A257993(A276086(A276086(n))) is less than 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.

Original entry on oeis.org

12, 24, 30, 42, 54, 60, 72, 84, 90, 102, 114, 120, 132, 144, 150, 162, 174, 180, 192, 204, 210, 216, 228, 246, 258, 276, 288, 306, 318, 336, 348, 366, 378, 396, 408, 420, 432, 444, 450, 462, 474, 480, 492, 504, 510, 522, 534, 540, 552, 564, 570, 582, 594, 600, 612, 624, 636, 648, 666, 678, 696, 708, 726, 738, 756, 768, 786, 798, 816

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Numbers n for which A328578(n) is less than A257993(n).
All terms are multiples of 6. The final digit {0, 2, 4, 6, 8} of the decimal representation seems to be quite evenly distributed.
Other multiples of six are in A328586 and A328589.
210 is the first term not present in A328632.

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

Union of A328632 \ {0} and A328762.
Positions of negative terms in A328590.
Cf. A257993, A276086, A328578, A328585, A328586, A328588.

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)));
isA328587(n) = (A328578(n) < A257993(n));

A286144 Compound filter: a(n) = T(A000010(n), A257993(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 3, 5, 10, 8, 21, 14, 21, 14, 55, 19, 78, 27, 36, 44, 136, 34, 171, 44, 78, 65, 253, 53, 210, 90, 171, 90, 406, 63, 465, 152, 210, 152, 300, 103, 666, 189, 300, 152, 820, 103, 903, 230, 300, 275, 1081, 169, 903, 230, 528, 324, 1378, 208, 820, 324, 666, 434, 1711, 187, 1830, 495, 666, 560, 1176, 251, 2211, 560, 990, 324, 2485, 349, 2628, 702, 820, 702

Offset: 1

Antti Karttunen, May 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000010, A000027, A257993, A286142, A286143, A286152, A286160, A286161, A286162, A286163, A286164.

Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {EulerPhi@ n, Module[{i = 1}, While[! CoprimeQ[Prime@ i, n], i++]; i]}, {n, 74}] (* Michael De Vlieger, May 04 2017 *)

A000010(n) = eulerphi(n);
A257993(n) = { for(i=1,n,if(n%prime(i),return(i))); }
A286144(n) = (2 + ((A000010(n)+A257993(n))^2) - A000010(n) - 3*A257993(n))/2;
for(n=1, 10000, write("b286144.txt", n, " ", A286144(n)));

from sympy import prime, primepi, gcd, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a053669(n):
    x=1
    while True:
        if gcd(prime(x), n) == 1: return prime(x)
        else: x+=1
def a257993(n): return primepi(a053669(n))
def a(n): return T(totient(n), a257993(n)) # Indranil Ghosh, May 05 2017

(define (A286144 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A257993 n)) 2) (- (A000010 n)) (- (* 3 (A257993 n))) 2)))

a(n) = (1/2)*(2 + ((A000010(n)+A257993(n))^2) - A000010(n) - 3*A257993(n)).

A328586 Even numbers n for which A257993(n) is equal to A257993(A276086(A276086(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.

Original entry on oeis.org

6, 18, 36, 48, 66, 78, 96, 108, 126, 138, 156, 168, 186, 198, 222, 234, 252, 264, 282, 294, 312, 324, 342, 354, 372, 384, 402, 414, 426, 438, 456, 468, 486, 498, 516, 528, 546, 558, 576, 588, 606, 618, 642, 654, 672, 684, 702, 714, 732, 744, 762, 774, 792, 804, 822, 834, 846, 858, 876, 888, 906, 918, 936, 948, 966, 978, 996, 1008

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

All terms are multiples of 6, but very few multiples of 5 (and thus of 10) are present: the first ones are at a(169) = 2520 and a(254) = 3780. Among the first 10000 terms, there are only 28 ending with decimal digit 0, while those that end with either 2 or 4 are 2450 both, and with either 6 or 8, both have 2536 each.

Other multiples of six are in A328587 and A328589.

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

Cf. A257993, A276086, A328578, A328316, A328317, A328633.

Even terms in A328585. Cf. also A328587, A328588, 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)));
isA328586(n) = ((!(n%2))&&(A328578(n) == A257993(n)));

A328588 Numbers n for which A257993(A276086(A276086(n))) is larger than 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.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212, 214, 218, 220, 224, 226, 230, 232, 236, 238, 240, 242

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Numbers n for which A328578(n) > A257993(n).

A047235 (numbers that are congruent to {2, 4} mod 6, thus even numbers that are not multiples of 3, with A257993(n) = 1) is a subsequence, because in primorial base (A049345) such numbers end with digits "10" or "20". A276086 will convert such a number to a number of the form p_k^e_k * ... * 7^b * 5^a * 3^{1,2} * 2^0 (an odd multiple of three, thus of the form 6k+3) which in primorial base will end with digits "11", thus on the second iteration A276086 will convert that to a number of the form p_k^e_k * ... * 7^b * 5^a * 3^1 * 2^1, with the least missing prime having an index (A257993) at least 3, which is larger than the original 1. Thus all terms of A047235 are included in this sequence.

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

Cf. A049345, A047235, A257993, A276086, A328578, A328585, A328586, A328587.
Union of A047235 (terms of the form 6k+2 and 6k+4) and A328589 (gives the terms that are multiples of 6).
Positions of positive terms in A328590.
Differs from A047235 for the first time at n=81, with a(81) = 240, a term not present in A047235.

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)));
isA328588(n) = (A328578(n) > A257993(n));

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

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 0, 2, 0, 3, 0, -1, 0, 3, 0, 2, 0, 0, 0, 4, 0, 4, 0, -1, 0, 4, 0, 5, 0, -2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 4, 0, -1, 0, 4, 0, 4, 0, 0, 0, 5, 0, 5, 0, -1, 0, 5, 0, 3, 0, -2, 0, 3, 0, 2, 0, 0, 0, 4, 0, 4, 0, -1, 0, 5, 0, 5, 0, 0, 0, 5, 0, 6, 0, -1, 0, 4, 0, 6, 0, -2, 0, 4, 0, 5, 0, 0, 0, 5, 0, 5, 0, -1, 0, 5, 0

Offset: 1

Antti Karttunen, Oct 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A257993, A276086, A328578, A328585 (positions of zeros), A328587 (of negative terms), A328588 (of positive terms).

Cf. A328591 (even bisection).

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));

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

A328589 Numbers n that are multiples of 6 and for which A257993(A276086(A276086(n))) is larger than 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.

Original entry on oeis.org

240, 270, 300, 330, 360, 390, 630, 660, 690, 720, 750, 780, 810, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1500, 1530, 1560, 1590, 1620, 1650, 1890, 1920, 1950, 1980, 2010, 2040, 2070, 2550, 2580, 2610, 2640, 2670, 2700, 2940, 2970, 3000, 3030, 3060, 3090, 3120, 3360, 3390, 3420, 3450, 3480, 3510, 3540, 3810, 3840, 3870, 3900, 3930, 3960, 4200

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Multiples of six such that the least nondivisor prime of the original n is less than the least nondivisor prime of the number obtained after two iterations of A276086 is.

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, so 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 then A328578(n) = A257993(A276086(A276086(n))) is definitely not larger than 3, while A257993(6k) >= 3 for all k >= 1.

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

Subsequence of A328588.
Other multiples of 6 are either in A328586 or in A328587.
Cf. A049345, A084967, A257993, A276086, A328578, A328585, A328588, A328590, A328762.

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)));
isA328589(n) = ((!(n%6))&&(A328578(n) > A257993(n)));

A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, 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.

Original entry on oeis.org

2, 6, 18, 34, 36, 48, 66, 78, 96, 108, 122, 126, 138, 154, 156, 168, 186, 198, 212, 222, 234, 244, 252, 264, 282, 294, 312, 324, 332, 342, 354, 364, 372, 384, 402, 414, 422, 426, 438, 454, 456, 468, 486, 498, 516, 528, 542, 546, 558, 574, 576, 588, 606, 618, 632, 642, 654, 664, 672, 684, 702, 714, 732, 744, 752, 762, 774, 784, 792, 804

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 6, but not of 5.
Question: Is the even bisection of A328316, starting from A328316(4) as: 6, 18, 43218, ..., a subsequence of this sequence? See also A328317.
Subsequence such that both k and A276087(k) are in this sequence starts as: 2, 6, 18, 34, 36, 48, 66, 154, 156, 186, 234, 244, 294, 312, 324, 354, 364, 384, 426, 438, 454, 456, 542, 546, 558, 588, 606, ...
When A276086 is applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, implying that the number's primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), the original number will be converted to 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 in position A328578(n), which is seen to be always either 3 or 2.

			294 = 7^2 * 3 * 2 has primorial base expansion (A049345) "12400", which, when converted to a prime product form (A276086) yields 11^1 * 7^2 * 5^4 * 3^0 * 2^0 = 336875. This in turn has primorial base representation [11,2,9,1,0,2,1], which when converted to prime product form gives 17^11 * 13^2 * 11^9 * 7^1 * 5^0 * 3^2 * 2^1 = 1720796647657111567992931482, which has the required property of being a multiple of 6 but not of 5, thus 294 is included in this sequence.

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

Row 3 of A328631.
Cf. A049345, A257993, A276086, A276087, A328578.
Cf. also A328316, A328317, A328589, A328762.

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)));
isA328633(n) = (3==A328578(n));

isA328633(n) = { my(u=A276086(A276086(n))); ((u%5)&&!(u%6)); };

cp's OEIS Frontend

A328578 Index of the least prime not dividing A276086(A276086(n)): a(n) = A257993(A276087(n)).

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A286142 Compound filter: a(n) = T(A257993(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A328585 Numbers n for which A257993(n) is equal to A257993(A276086(A276086(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.

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A328587 Numbers n for which A257993(A276086(A276086(n))) is less than 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.

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A286144 Compound filter: a(n) = T(A000010(n), A257993(n)), where T(n,k) is sequence A000027 used as a pairing function.

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A328586 Even numbers n for which A257993(n) is equal to A257993(A276086(A276086(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.

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A328588 Numbers n for which A257993(A276086(A276086(n))) is larger than 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.

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

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