cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 54 results. Next

A328461 a(n) = A276156(n) / A002110(A007814(n)).

Original entry on oeis.org

1, 1, 3, 1, 7, 4, 9, 1, 31, 16, 33, 6, 37, 19, 39, 1, 211, 106, 213, 36, 217, 109, 219, 8, 241, 121, 243, 41, 247, 124, 249, 1, 2311, 1156, 2313, 386, 2317, 1159, 2319, 78, 2341, 1171, 2343, 391, 2347, 1174, 2349, 12, 2521, 1261, 2523, 421, 2527, 1264, 2529, 85, 2551, 1276, 2553, 426, 2557, 1279, 2559, 1, 30031, 15016, 30033, 5006
Offset: 1

Views

Author

Antti Karttunen, Oct 16 2019

Keywords

Comments

A276156(n) converts the binary expansion of n to a number whose primorial base representation has the same digits of 0's and 1's, thus each one of its terms is a unique sum of distinct primorial numbers. In this sequence that sum is then divided by the largest primorial that divides it, which only depends on the position of the least significant 1-bit in the binary expansion of the original n, that is, the 2-adic valuation of n.

Links

Crossrefs

Cf. A000265, A002110, A007814, A111701, A276154, A276156.
Cf. A328462 (bisection, also row 1 of array A328464 which shows the same information in tabular form).
Cf. A328471, A328472, A328473, A328474.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328461(n) = (A276156(n)/A002110(valuation(n,2)));

Formula

a(n) = A276156(n) / A002110(A007814(n)).
a(n) = A111701(A276156(n)).

A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 9, 16, 6, 1, 31, 19, 36, 8, 1, 33, 106, 41, 78, 12, 1, 37, 109, 386, 85, 144, 14, 1, 39, 121, 391, 1002, 155, 222, 18, 1, 211, 124, 421, 1009, 2432, 235, 324, 20, 1, 213, 1156, 426, 1079, 2443, 4200, 341, 438, 24, 1, 217, 1159, 5006, 1086, 2575, 4213, 7430, 457, 668, 30, 1, 219, 1171, 5011, 17018, 2586, 4421, 7447, 12674, 691, 900, 32, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 16 2019

Keywords

Comments

Array is read by falling antidiagonals with n (row) and k (column) ranging as: (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
Row n contains all such sums of distinct primorials whose least significant summand is A002110(n-1), with each sum divided by that least significant primorial, which is also the largest primorial which divides that sum.

Examples

			Top left 9 X 11 corner of the array:
1: | 1,  3,   7,   9,    31,    33,    37,    39,    211,    213,    217
2: | 1,  4,  16,  19,   106,   109,   121,   124,   1156,   1159,   1171
3: | 1,  6,  36,  41,   386,   391,   421,   426,   5006,   5011,   5041
4: | 1,  8,  78,  85,  1002,  1009,  1079,  1086,  17018,  17025,  17095
5: | 1, 12, 144, 155,  2432,  2443,  2575,  2586,  46190,  46201,  46333
6: | 1, 14, 222, 235,  4200,  4213,  4421,  4434,  96578,  96591,  96799
7: | 1, 18, 324, 341,  7430,  7447,  7753,  7770, 215442, 215459, 215765
8: | 1, 20, 438, 457, 12674, 12693, 13111, 13130, 392864, 392883, 393301
9: | 1, 24, 668, 691, 20678, 20701, 21345, 21368, 765050, 765073, 765717

Links

Crossrefs

Cf. A328463 (transpose).
Cf. A000265, A002110, A007814, A135764, A276154, A276156,
Rows 1 - 5: A328462, A328465, A328466, A328467, A328468.
Column 2: A008864.
Column 3: A023523 (after its initial term).
Column 4: A286624.
Cf. also arrays A276945, A286625.

Programs

  • PARI
    up_to = 105;
A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328464sq(n,k) = (A276156((2^(n-1)) * (k+k-1)) / A002110(n-1));
A328464list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A328464sq(col,(a-(col-1))))); (v); };
v328464 = A328464list(up_to);
A328464(n) = v328464[n];

Formula

A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1).
a(n) = A328461(A135764(n)). [When all sequences are considered as one-dimensional]

A351073 Maximal exponent in the prime factorization of A276156(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 5, 1, 1, 3, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 4, 3, 1, 1, 2, 1, 2, 5, 2, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 5, 1, 1, 1, 2, 1, 3, 2, 1, 1, 6, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Comments

See also comments in A143293.

Examples

			For n = 1040 = 2^10 + 2^4, A276156(n) = A002110(10) + A002110(4) = 6469693440 = 2^12 * 3 * 5 * 7^3 * 307. The largest exponent is 12, therefore a(1040) = 12.

Links

Crossrefs

Cf. A002110, A051903, A276156, A143293.

Programs

  • Mathematica
    {0}~Join~Array[Max[FactorInteger[#][[All, -1]]] &@ Total[Times @@@ Transpose@{Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ IntegerDigits[#, 2] &, 104, 2] (* Michael De Vlieger, Feb 04 2022 *)
  • PARI
    A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276156(n) = { my(s=0, p=1, r=1); while(n, if(n%2, s += r); n>>=1; p = nextprime(1+p); r *= p); (s); };
A351073(n) = A051903(A276156(n));

Formula

a(n) = A051903(A276156(n)).
For n >= 1, a(2^n) = 1.

A328474 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A276156(i)) = A046523(A276156(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 2, 7, 3, 8, 2, 3, 3, 9, 2, 10, 3, 11, 3, 3, 3, 12, 2, 10, 7, 6, 3, 13, 3, 14, 2, 15, 10, 16, 3, 6, 3, 17, 2, 3, 6, 9, 2, 10, 18, 19, 2, 6, 10, 6, 10, 20, 10, 21, 2, 22, 6, 23, 2, 3, 3, 24, 3, 18, 16, 16, 10, 6, 9, 17, 3, 3, 6, 6, 3, 10, 16, 25, 2, 3, 6, 16, 6, 22, 10, 9, 2, 26, 3, 27, 6, 3, 3, 28, 2, 6, 3, 22, 3, 10, 6, 14, 2
Offset: 1

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Links

Crossrefs

Cf. A002110, A276156, A328461, A328471, A328473.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v328474 = rgs_transform(vector(up_to, n, A046523(A276156(n))));
A328474(n) = v328474[n];

A328473 a(n) = A276156(n) - A002110(A007814(n)).

Original entry on oeis.org

0, 0, 2, 0, 6, 6, 8, 0, 30, 30, 32, 30, 36, 36, 38, 0, 210, 210, 212, 210, 216, 216, 218, 210, 240, 240, 242, 240, 246, 246, 248, 0, 2310, 2310, 2312, 2310, 2316, 2316, 2318, 2310, 2340, 2340, 2342, 2340, 2346, 2346, 2348, 2310, 2520, 2520, 2522, 2520, 2526, 2526, 2528, 2520, 2550, 2550, 2552, 2550, 2556, 2556, 2558, 0, 30030
Offset: 1

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Comments

A276156(n) converts the binary expansion of n to a number whose primorial base representation has the same digits of 0's and 1's, thus each one of its terms is a unique sum of distinct primorial numbers. This sequence is otherwise similar, but the primorial number corresponding to the least significant 1-bit of n is dropped from the sum, so the sum is not unique anymore.

Links

Crossrefs

Cf. A002110, A053589, A129760, A276151, A276156, A328461, A328474.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328473(n) = (A276156(n)-A002110(valuation(n,2)));

Formula

a(n) = A276156(A129760(n)).
a(n) = A276151(A276156(n)) = A276156(n) - A002110(A007814(n)).

A328466 Row 3 of A328464: a(n) = A276156(8n - 4) / 6.

Original entry on oeis.org

1, 6, 36, 41, 386, 391, 421, 426, 5006, 5011, 5041, 5046, 5391, 5396, 5426, 5431, 85086, 85091, 85121, 85126, 85471, 85476, 85506, 85511, 90091, 90096, 90126, 90131, 90476, 90481, 90511, 90516, 1616616, 1616621, 1616651, 1616656, 1617001, 1617006, 1617036, 1617041, 1621621, 1621626, 1621656, 1621661, 1622006, 1622011
Offset: 1

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Links

Crossrefs

Row 3 of A328464.
Cf. A002110, A276156, A328461.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328466(n) = (A276156((8*n)-4) / 6);

Formula

a(n) = (1/6) * A276156(8*n - 4).

A328467 Row 4 of A328464: a(n) = A276156(16n - 8) / 30.

Original entry on oeis.org

1, 8, 78, 85, 1002, 1009, 1079, 1086, 17018, 17025, 17095, 17102, 18019, 18026, 18096, 18103, 323324, 323331, 323401, 323408, 324325, 324332, 324402, 324409, 340341, 340348, 340418, 340425, 341342, 341349, 341419, 341426, 7436430, 7436437, 7436507, 7436514, 7437431, 7437438, 7437508, 7437515, 7453447, 7453454, 7453524
Offset: 1

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Links

Crossrefs

Row 4 of A328464.
Cf. A002110, A276156, A328461.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328467(n) = (A276156((16*n)-8) / 30);

Formula

a(n) = (1/30) * A276156(16*n - 8).

A328468 Row 5 of A328464: a(n) = A276156(32n - 16) / 210.

Original entry on oeis.org

1, 12, 144, 155, 2432, 2443, 2575, 2586, 46190, 46201, 46333, 46344, 48621, 48632, 48764, 48775, 1062348, 1062359, 1062491, 1062502, 1064779, 1064790, 1064922, 1064933, 1108537, 1108548, 1108680, 1108691, 1110968, 1110979, 1111111, 1111122, 30808064, 30808075, 30808207, 30808218, 30810495, 30810506, 30810638, 30810649
Offset: 1

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Links

Crossrefs

Row 5 of A328464.
Cf. A002110, A276156, A328461.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328468(n) = (A276156((32*n)-16) / 210);

Formula

a(n) = (1/210) * A276156(32*n - 16).

A328831 Number of distinct prime factors p such that p^p is a divisor of n-th number > 0 that is a sum of distinct primorial numbers, A276156(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2019

Keywords

Links

Crossrefs

Cf. A129251, A276156.
Cf. A328832 (gives A276156(k) for those k for which a(k) = 0).

Programs

  • PARI
    A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328831(n) = A129251(A276156(n));

Formula

a(n) = A129251(A276156(n)).

A328833 A276085 applied to the intersection of A048103 (p^p-free numbers) and A276156 (sums of distinct primorials).

Original entry on oeis.org

0, 1, 2, 3, 30, 4, 9, 6469693230, 212, 200560490130, 510511, 2312, 39, 7799922041683461553249199106329813876687996789903550945093032474868511536164700810, 7858321551080267055879092, 6469693260, 2566376117594999414479597815340071648394471, 557940830126698960967415392, 1062411448280052319722448549835623701226301211611796930357321893850294264731624591303255041960530, 421, 7420738134813, 512820, 3217644767340672907899084554132, 249
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2019

Keywords

Links

Crossrefs

Cf. A002110, A129251, A048103, A276085, A276156, A328828, A328832.
Cf. A328836 (same terms, sorted into ascending order).
Cf. A328313 (a subsequence).

Programs

  • PARI
    A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328832(n) = ((0==A129251(n)) && (0==A328828(n)));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
for(n=1,32768,if(isA328832(n),print1(A276085(n),", ")));

Formula

a(n) = A276085(A328832(n)).
Showing 1-10 of 54 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.