A276087 -id:A276087 - cp's OEIS frontend

A328405 The length of primorial base expansion (number of significant digits) of A276086(A276086(n)), where A276086(n) converts primorial base expansion of n into its prime product form.

2, 2, 3, 2, 4, 4, 3, 4, 4, 3, 5, 5, 5, 6, 6, 6, 5, 5, 7, 6, 9, 8, 10, 14, 11, 12, 14, 12, 12, 15, 3, 4, 5, 4, 5, 6, 4, 5, 7, 3, 8, 5, 9, 9, 8, 7, 12, 7, 8, 12, 8, 7, 12, 14, 16, 15, 15, 15, 11, 12, 5, 6, 8, 7, 7, 8, 5, 7, 9, 9, 14, 12, 12, 9, 12, 7, 15, 15, 12, 12, 18, 13, 20, 17, 11, 13, 15, 14, 17, 13, 8, 9, 11, 14, 11, 13, 11, 10, 10, 10

Offset: 0

Antti Karttunen, Oct 16 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A061395, A143293, A235224, A276086, A276087, A328394, A328403, A328404, A328406.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[IntegerLength[Nest[f, #, 2], b] &, 100, 0]] (* Michael De Vlieger, Oct 17 2019 *)

A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
A328405(n) = A235224(A276087(n));

a(n) = A235224(A276087(n)) = A061395(A328403(n)).

For all n, A000040(a(n)) > A328394(n).

A328579 a(n) = A053669(A276086(A276086(n))).

Original entry on oeis.org

3, 2, 5, 2, 7, 2, 5, 2, 7, 2, 11, 2, 3, 2, 11, 2, 7, 2, 5, 2, 13, 2, 13, 2, 3, 2, 13, 2, 17, 2, 3, 2, 7, 2, 5, 2, 5, 2, 11, 2, 13, 2, 3, 2, 13, 2, 13, 2, 5, 2, 17, 2, 17, 2, 3, 2, 17, 2, 11, 2, 3, 2, 11, 2, 7, 2, 5, 2, 13, 2, 13, 2, 3, 2, 17, 2, 17, 2, 5, 2, 17, 2, 19, 2, 3, 2, 13, 2, 19, 2, 3, 2, 13, 2, 17, 2, 5, 2, 17, 2, 17, 2, 3, 2, 17, 2

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A000040, A053669, A276086, A276087, A326810, A328569, A328578.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A326810(n) = { my(i=1, p=2); while(n && (n%p), n = n\p; p = nextprime(1+p)); (p); };
A328579(n) = A326810(A276086(n));

A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
A328579(n) = A053669(A276086(A276086(n)));

a(n) = A053669(A276087(n)).
a(n) = A326810(A276086(n)).
a(n) = A000040(A328578(n)).
For all even n, a(n) > A328569(n).

A328836 Numbers k such that A276086(k) is a sum of distinct primorial numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 30, 39, 212, 249, 421, 2312, 2559, 30045, 32589, 510511, 512820, 543099, 1021050, 9729723, 10242789, 233335659, 446185742

Offset: 1

Antti Karttunen, Oct 29 2019

Numbers k such that A276086(k) is in A276156, i.e., numbers k for which A328828(A276086(k)) is zero, i.e., numbers k such that in the primorial base expansion of A276086(k) there are no digits larger than 1.
Numbers k for which A276087(k) is squarefree.
No more terms below 2^31.

Index entries for sequences related to primorial base

Sequence A328833 sorted into ascending order.
Cf. A008966, A276086, A276087, A276156, A328828, A328832, A328833.
Positions of zeros in A328829 and in A328844, positions of ones in A328389.
Cf. A143293 (a subsequence).
All the terms of A328313 are included in this sequence, like also in A328837.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328836(n) = !A328828(A276086(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)); };

A328569 Exponent of least prime factor in A276086(A276086(n)), where A276086 converts the primorial base expansion of n into its prime product form.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Equally, the least significant nonzero digit in primorial base expansion of A276086(n).

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003557, A051903, A067029, A276086, A276087, A276088, A328389, A328400, A328476, A328570, A328579.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); };
A328569(n) = A276088(A276086(n));

a(n) = A276088(A276086(n)) = A067029(A276087(n)).
max(a(n),1+A051903(A328400(A003557(A276086(A328476(n)))))) = A328389(n). [A328400 is optional in the formula]
For all even n, a(n) < A328579(n).

A328766 Number of nonleading zeros in primorial base expansion of A276086(n).

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 3, 1, 1, 0, 3, 1, 3, 1, 1, 0, 3, 0, 3, 1, 1, 0, 3, 0, 3, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 4, 1, 1, 0, 4, 0, 4, 1, 1, 0, 4, 0, 4, 1, 1, 0, 4, 1, 4, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 4, 1, 2, 0, 4, 0, 4, 1, 1, 0, 4, 0, 4, 2, 2, 1, 4, 0, 4, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 5, 1, 1, 0, 4

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A079067, A143293, A276086, A276087, A328620, A328763.

Cf. also A328578.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328620(n) = { my(s=0, p=2); while(n, s += (0==(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328766(n) = A328620(A276086(n));

a(n) = A328620(A276086(n)) = A079067(A276087(n)).
a(n) = A001221(A328763(n)) - 1.
For all n >= 1, a(A143293(n-1)) = n. [Note however that these are not the first occurrences of each n, that is, A143293 does not give the indices of records]

A328829 Index of the least significant digit > 1 in the primorial base expansion of A276086(n), 0 if no such digit exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 2, 2, 3, 0, 3, 4, 3, 3, 3, 4, 3, 5, 2, 2, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 0, 3, 3, 3, 4, 4, 2, 2, 3, 0, 3, 5, 3, 3, 3, 5, 3, 5, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5, 3, 6, 3, 4, 3, 3, 3, 3, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5, 3, 5, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5, 3, 5, 3, 3, 4, 3, 3, 3, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5

Offset: 0

Antti Karttunen, Oct 29 2019

nonn

a(n) = index of the least non-unitary prime divisor of A276087(n) or 0 if no such prime-divisor exists.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A276086, A276087, A277885, A328828, A328836 (positions of zeros).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
A328829(n) = A328828(A276086(n));

A277885(n) = if(1==n,0,my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1,return(primepi(f[i,1])))); (0));
A328829(n) = A277885(A276086(A276086(n)));

a(n) = A328828(A276086(n)) = A277885(A276087(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));

cp's OEIS Frontend

A328405 The length of primorial base expansion (number of significant digits) of A276086(A276086(n)), where A276086(n) converts primorial base expansion of n into its prime product form.

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A328579 a(n) = A053669(A276086(A276086(n))).

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A328836 Numbers k such that A276086(k) is a sum of distinct primorial numbers.

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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.

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A328569 Exponent of least prime factor in A276086(A276086(n)), where A276086 converts the primorial base expansion of n into its prime product form.

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A328766 Number of nonleading zeros in primorial base expansion of A276086(n).

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A328829 Index of the least significant digit > 1 in the primorial base expansion of A276086(n), 0 if no such digit exists.

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

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